lectures presented at an international summer course

Lectures presented at an International Summer Course Trieste, 21 May- 8 August 1975

organized by the International Centre for Theoretical PhysicsTrieste

INTERNATIONAL A T O M IC ENERGY AGENCY, VIENNA, 1 976

COMPLEX ANALYSIS AND ITS APPLICATIONS

Vol. II

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS, TRIESTE

COMPLEX ANALYSIS AND ITS APPLICATIONS

LECTURES PRESENTED AT AN INTERNATIONAL SEMINAR COURSE

AT TRIESTE FROM 21 M AY TO 8 AUGUST 1975 ORGANIZED BY THE

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS, TRIESTE

In three volumes

VOL. II

INTERNATIONAL ATOMIC ENERGY AGENCY VIENNA, 1976

THE INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS (ICTP) in Trieste was established by the International Atomic Energy Agency (IAE A ) in 1964 under an agreement with the Italian Government, and with the assistance o f the City and University o f Trieste.

The IAEA and the United Nations Educational, Scientific and Cultural Organization (UNESCO) subsequently agreed to operate the Centre jointly from 1 January 1970.

Member States o f both organizations participate in the work o f the Centre, the main purpose o f which is to foster, through training and research, the advancement o f theoretical physics, with special regard to the needs o f developing countries.

COMPLEX ANALYSIS AND ITS APPLICATIONS IAEA, VIENNA, 1976

STI/PUB/428ISBN 92-0-130476-5

Prin ted by the I A E A in Austria

O ctober 1976

FOREWORD

The International Centre for Theoretical Physics has maintained an interdisciplinary character in its research and training programmes in different branches o f theoretical physics and related applied mathematics. In pursuance o f this objective, the Centre has since 1964 organized extended research courses in various disciplines; most o f the Proceedings o f these courses have been published by the International Atomic Energy Agency.

In 1972 the ICTP held the first o f a series o f extended summer courses in mathematics and its applications. To date, the following courses have taken place: Global Analysis and its Applications (1972), Mathematical and Numerical Methods in Fluid Dynamics (1973), Control Theory and Topics in Functional Analysis (1974), and Complex Analysis and its Applications (1975). The present volumes consist o f a collection o f the long, basic courses (Volume I) and the individual lectures (Volumes II and III) given in Trieste at the 1975 Summer Course. The contributions are partly expository and partly research-oriented — in the spirit o f the very wide range o f interest o f the participants. The programme o f lectures was organized by Professors A. Andreotti (Pisa, Italy, and Oregon, United States o f America), J. Eells (Warwick, United Kingdom) and F. Gherardelli (Florence, Italy).

Abdus Salam

CONTENTS OF VOLUME II

Several complex variables and Banach algebras (IAEA-SMR-18/47)........................ 1G.R. Allan

Analytic convexity: some comments on an example o fde Giorgi and Kccinini (IAEA-SMR-18/10 1 )....................................................... 25A. Andreotti, M. Nacinovich

Complex tori and Jacobians (IAEA-SMR-18/24)..................................................... 39M. Comalba

Introduction to complex tori (IAEA-SMR-18/13) ................................................... 101P. de la Натре

Some problems in the theory o f single-valued analytical functions andharmonic analysis in the complex domain (IAEA-SMR-18/20)........................... 145M.M. Djrbashian

A generalization o f Beurling’s estimate o f harmonic measure (IAEA-SMR-18/37)... 203 M. Essén

Pseudoconvexity and the principle o f maximum modulus (IAEA-SMR-18/64)....... 209A.A. Fadlalla

Quasiconformal mappings (IAEA-SMR-18/53)........................................................ 213F.W. Gehring

Geometric theory o f differential equations in the complex domain(IAEA-SMR-18/102) ........................................................................................... 269R. Gérard

Secretariat o f the Course 309

IA E A -S M R -1 8/47

SEVERAL COMPLEX VARIABLES AND BANACH ALGEBRAS

G.R. ALLAN School o f Mathematics,University of Leeds,Leeds, United Kingdom

Abstract

S E V E R A L C O M P L E X V A R IA B L E S A N D B A N A C H A L G E B R A S .

Th is paper aims t o present certain applications o f the th eory o f h o lom orph ic functions o f several com p lex

variables to the study o f com m utative Banach algebras. T h e m aterial falls in to the fo llo w in g sections:

( A ) In trodu ction to Banach algebras (th is w ill n o t presuppose any know ledge o f the sub ject); (B ) G roups o f

d ifferen tia l fo rm s (m a in ly concerned w ith setting up a useful language); (C ) P o lyn om ia lly convex dom ains in (En;

(D ) H o lom orph ic functiona l calculus fo r Banach algebras; (E ) Som e applications o f the functional calculus.

A. INTRODUCTION TO BANACH ALGEBRAS

Let Œ be the field o f complex numbers; an algebra A over <C will always mean a commutative and associative linear algebra with an identity element 1 ^ 0 (i.e. A is. (i) a commutative ring with 1, and (ii) a complex vector space, such that X(xy) = (Xx)y = x(Xy) for all X S (Г and x,y e A).

Example 1. Œ[X,, ..., Xn], the algebra o f all complex polynomials in n indeterminates X j , X n.

Example 2. d [ [X i , ..., Xn]], the algebra o f all formal power series in X b ..., Xn.

Example 3. Let U be an open subset o f <Tn. Then each o f the sets o f functions C(U), C°°(U), 0 (U ) , consisting respectively o f all continuous, all smooth, all holomorphic (Г-valued functions on U, is an example o f an algebra when equipped with the obvious “ pointwise” algebraic operations.

Example 4. Let К be a compact subset o f (Tn. Then C(K), the algebra o f all continuous Œ-valued functions on К is another example (pointwise operations again).

Example S. Let К be as in Example 4 and let P (K ) be the subalgebra o f C (K) consisting o f all those functions uniformly approximable on К by complex polynomials (in z ¡ , ..., zn).

Now let A be a complex algebra. An algebra-norm on A is a function x » Il x II o f A into 1R (real numbers) such that, for all x,y e A and X € (E:

(i) II x II > 0 and II x II = 0 if and only if x = 0;

(ii) II Xxll = |X| Il x II;

(iii) II x + yll < II xll + Il у II;

(iv) Il xy II < Il xll II у II;

(v) 11111=1

I f A is equipped with a given algebra-norm II .11, the pair (A ; II .11) is called a normed algebra;if, also, A is complete in the given norm then (A ; II. II) is a Banach algebra. (O f course, when noconfusion seems possible, we shall write just A rather than (A ; II .11),)

1

2 ALLAN

Of the examples o f algebras given above:

Example 1. <C[X1 ; Xn] carries many different algebra-norms - but (provided n > 1 ) it is not a Banach algebra for any o f them.

Example 2. <D[[X1;..., Xn]]: the same statement applies as in Example 1, but it is much less obvious (see Refs [1 ],[2]).

Example 3. C(U), C°°(U) cannot be given algebra-norms at all (we shall show this after Theorem 5 for C(U)); the algebra 0 (U ) can be given many different algebra-norms for certain U - but never a Banach-algebra norm (e.g., if U is connected and К is a compact subset o f U with int К i- 0 then IlfllK = SUP lf(z)l ( f £ 0 (U ) ) defines an algebra-norm on 0 (U ). On the other hand if, for example,

U is an open subset o f the plane consisting of a union o f infinitely many open discs with pairwise disjoint closures, then 0 (U ) carries no algebra-norm).

Examples 4, S. The algebras C(K), P(K ) may be given Banach-âlgebra norms, by putting

||f||K = sup |f(z)| (fe C (K )) z G K

We now give two more examples o f Banach algebras:

Example 6. Let W be the algebra o f all functions f : ]R -*■ (D representable by a Fourier series

where each fn£Œ and 2 Ifni < 00■ The norm is ||f|| = 2 Ifni an£l the algebraic operations are pointwise on IR (called the Wiener algebra).

Example 7. Let К be a compact subset o f Œ. Let R (K ) be the uniform closure in C(K) o f Œ-valued rational functions without poles in K.

Throughout this section, A is some commutative Banach algebra.The elementary theory o f Banach algebras makes frequent use o f the following trivial but

vital technical lemma (we write A -1 for the group o f invertible elements o f A ):

Lemma 1. Let A be a Banach algebra, let xGA with || 1 - x|| < 1 ; then xGA-1 and

z € K

OO

oo

(1 - X ) "

n = 0

Proof. Elementary calculation, after noting that

oo oo

I l K l - x m

0

< 111 - x||n < oo

0

BANACH ALGEBRAS 3

since ||1 - x|| < 1 and thus ^ ( 1 - x)n is norm-convergent in A since (A ; ||.||) is complete.

0

Corollary 1. A-1 is an open subset o f A.

Proof. Let x G A-1 ; then, for y S A with ||y - x|| < [|x_I II"1, we have

y = x + (y - x) = x [1 + x-1 (y - x)]

Since Цх-1 (y - x)|| < Hx'Ml-lly - x|| < 1 and x E A '1, we see from Lemma 1 that y G A-1.

Corollary 2. The mapping x » x"1 is a homeomorphism o f A '1 onto itself.

Proof. Let x G A-1 ; by the calculation o f Corollary 1, i f ||y - x|| < ||x-1|[_1 then y G A-1 and

OO

y- 1 = X' 1 [1 + x~' (y - x)]-1 = x_(n+1) (x - y )n

Thus

Ily - '-x -M l =

<

Thus y "1 -*■ x-1 as y -*■ x; which shows that x ** x-1 is a continuous mapping; but this mapping is its own inverse and is thus a homeomorphism.

Definition. Let A be a Banach algebra and let x G A; the spectrum o f x in A is

SpA (x ) = {X 6<r : X 1 - x ? A '1}

Theorem 1. SpA (x ) is a non-empty compact subset o f (Г.

Proof. SpA (x ) is closed in Œ, by Lemma 1, Corollary 1, and the continuity o f X «■ X1 - x as a mapping o f (C into A.

Also, if XGŒ and |X| > ||x||, then X 1 - x = X(1 - X”1 x) and, since ||X_1 x|| = |X|-1 ||x|| < 1, we see from Lemma 1 that X 1 - x G A-1, so that X ^ SpA (x). ThusSpA(x ) C{XG<E : |X| < j|x||}and, in particular, SpA (x ) is bounded. Thus SpA (x ) is compact; it remains to prove that SpA (x ) f 0.

Observe first that defining f(X) = {X 1 - x)-1 defines a continuous function f : CT\SpA(x ) -* A -1 (for f is the composition o f X ** X 1 - x and x >+ x_1 and we may use Lemma 1, Corollary 2). But it is trivial that, for any X,/nG<E\SpA (x):

n = 0

I-(n+ 0 ( x - y ) n

£ Цх-ЧГ + 1||у-х||П =П = 1

llx-1 II2 ||y - X»

1 — llx-1 II . Ily — x||

f(X) - f(M) = - (X - M) f(X) m

4 ALLAN

so that

(Х -М Г 1 (f (X )- fG O ) = - f (X ) f (M )^ - f(M )2

as X -*■ M, by the continuity o f f at ц. Thus f is an А -valued holomorphic function on (C\SpA (x).But also, for X ^ 0, f(X) = (X 1 - x )"1 = X"1 (1 - X-1 x )"1, so that, by Lemma 1, Corollary 2,||f(X)|| -*• 0 as |X| -*■ oo. Thus, i f SpA (x ) = 0, then f would be an entire А -valued function with ||f(X)|| -*• 0 as |X| ->oo; by Liouville’s theorem (for vector-valued functions) we could then deduce f(X) = 0 on (C — obviously a contradiction since f(X) G A-1. Hence SpA (x) i 0.

Definition. The spectral radius rA (x ) o f x in A is

rA (x ) = sup{ |XI : X eSpA (x )}

The above theorem contained a proof o f :

Lemma 2. rA (x ) < ||x|| (xG A ).

(There is a more precise result:

rA (x )= lim \\xn\\1/nn-K*>

but we shall not need it in the sequel).

Theorem 2 (Gelfand-Mazur). I f A is a Banach algebra and is a field, then A = Œ.

Proof. Let xG A ; by Theorem 1, there is a complex number XG SpA (x), i.e. such that X 1 - x £ A '1. But i f A is a field then X 1 - x = 0. Thus, the mapping X ►* X. 1 is an isometric isomorphism o f <C onto A.

Lemma 3. The closure lo f a proper ideal I o f A is again a proper ideal.

P roof Since I f A w e have I Л A-1 = 0; but A "1 is open and so I П A "1 = 0; since 1G A -1 we haveI f A. The fact that I is an ideal follows easily from the continuity o f the algebraic operations in A.

Corollary. Each maximal ideal o f A is closed in A.

Proofs Let M be a maximal ideal; by Lemma 3, M is a proper ideal, M 3 M ; by maximality o f M,M = M.

Definition. A character ф on A is an algebra homomorphism o f A onto <E; we write ФА for the set o f all characters on A.

Lemma 4. Any character ф on A is continuous and 11011 = 1.

Proof. Let x G A with ||x|[ < 1 and suppose that |ф(х)| > 1 for some character ф. Set у = ф(х)-1 x; then ||y II = |0(х)|-1 l|x|| < 1, while ф( у ) = 1. By Lemma 1, 1 - y G A*1 so that, for some zGA, z (l - y ) = 1. But then l = 0 ( l ) = 0 ( z ) ( l - ф(у)) = 0 ,a contradiction.

Thus for any x G A with ||x|| < 1 and any ф 6 ФА we have |ф(х)| < 1. This completes the proof.

I f ф is any character on A then

ker ф = {x 6 A : ф(х) = 0}

is easily seen to be a maximal ideal o f A. It is an important fact that the converse is also true:

BANACH ALGEBRAS 5

Theorem 3. Let Ж be a maximal ideal o f k ‘, then M = ker ф fo r some ф £ Фд.

Proof I f I is any closed ideal o f A it is not hard to prove that the quotient algebra A/I becomes a Banach algebra under the quotient norm

|[x]| = inf llx + ill i e i

where we have written [x] for the coset x + I. (The fact that A/I is complete is standard elementary Banach space theory — true for any closed subspace I).

I f M is maximal then M is closed (by Corollary to Lemma 3); thus A/M is a Banach algebra in the quotient norm — but is also a field since M is maximal (an elementary bit o f pure algebra). Thus, by Theorem 2, A/M = (Г and the quotient mapping x ►» x + M is thus a character ф with ker ф = M.

Corollary. Let x S A ; then

(i) x G A ”1 i f and only i f ф(х)ф 0 for every ф G ФА ;

(ii) SpA(x )= {0 (x ) : 0€ЕФа };

(iii) i f p is any complex polynomial then SpA (p (x )) = {p (X ) : X6 SpA (x)}.

Proof, (i) I f x G A "1 then 1 = ф( 1) = ф(х) 0 (x-1), so that ф(х) ф 0 for each 0 € Ф А.Suppose that x ^ A-1 ; then Ax = {ax : a G A } is a proper ideal o f A (for 1 Í Ax ) and so, by a standard Zorn’s Lemma argument, there is a maximal ideal M Э Ax. By Theorem 3, M = ker ф for some ф G ФА so that ф (x ) = 0 for such а ф. Thus, i f ф (x ) # 0 for every ф G ФА we must have x G A "1.(ii) SpA (x ) ={XG(E : X I - x ^ A -1}

= {X GŒ : 0(X 1 - x) = 0 for some ф] by part (i); thus SpA (x ) = {ф (х) : ф€ Фд}.(iii) I f p is a complex polynomial then, since a character ф is a homomorphism, ф(р(х)) = p (0 (x)). Hence

SpA (p (x )) = {0 (p (x )) :0 G ФА}

= {p (0 (x )) : 0 6 ФА}

= {p (X ):\ G S p A (x )}

We shall now determine the character sets in some particular cases.

(a) The Wiener algebra W (see Example 6 )

Let uGW be defined by u(x) = eK ; then u GW-1 with u_1 (x ) = e_lx and clearly ||u|| = ||u_1|| = 1. Thus, i f ф is any character on W, we have |<Hu)| < llu|| = 1 (by Lemma 4) and also ^ (u )!-1 = |0(u_1)| < ||u_1 II = 1 .

Thus |0(u)| = 1 — say ф(и) = eia for some aGIR. But then for a general f GW, say

= Y j fneinX where J \ n| < °°f(x )

n = -we have

r= Y f n « n

n = - O O

6 ALLAN

with norm-convergence in W. Thus

OO oo

Thus ф is simply “ evaluation at a ” ; clearly two real numbers a ,a 'give rise to the same character i f and only if a - a ’ e 2irZ, so that we may identify (for the moment just as a set) w with IR/2ttZ, i.e. with the unit circle.

As a consequence o f this determination o f w , together with the Corollary to Theorem 3, we deduce the following classical theorem o f Wiener:

I f f e w and f(x ) i 0 (all x G IR) then 1/fGW.

(b) The algebra C(K) (see Example 4)

Let К be a compact subset o f dn — or, more generally, any compact Hausdorff topological space. For any к £ K the mapping ek : C (K) -*■ (Г defined by ek(f ) = f(k ) is clearly a character on C(K) (we call it “ evaluation at k” ). We shall see that these are the only characters on C(K).

First, let I be any proper ideal o f C(K). Suppose that, for each k G К there is a function fk GI with fk(k ) f 0; then, by continuity o f fk at k, there is an open neighbourhood Uk o f k such that fjc(k') Ф 0 (all k' G Uk). By compactness o f K, there is a finite subset {k [ ,..., km} o f К such that К is covered by {Uk j,..., Ukm}. But then the function

m m

f(x ) ф 0 for all x £ K , and so f GC(K)-1 which contradicts I being a proper ideal o f C(K). Thus for some point kG K we have I Ç ker ek.

In particular, i f M is a maximal ideal then M = ker ek for some k G K. Thus, finally, i f ф is any character,then, by Theorem 3, ker ф = ker ek for some kG K. But then, for any f GC(K), f - f(k ). 1 Gker ek = ker ф, so that ф(f ) = f(k ) ф{ 1 ) = f(k ) = ek(f); thus ф = ek.

(c) The algebra P (K ) (see Example 5)

Let К be a compact subset o f <Cn. We define the polynomial hull К o f К to be

К = {z G(Tn : |p(z)| < Hp IIk , for every complex polynomial p in n variables}

(Note: here we have written z = (z b ..., zn) and

IIp IIk = s u p Ip ( z ) Iz G K

We shall use similar notation freely without further comment ).We say that К is polynomially convex if and only if К = K. For any compact К, К is a compact

polynomially convex subset o f (Cn which contains K. (Verify!)Let p be a polynomial; then ||p||£ = IIp IIk , so that, for the algebra o f polynomial functions on

K, the operation o f restriction to К is an isometric isomorphism; thus restriction from К to К also gives an isometric isomorphism o f P (K ) onto P(K). It follows that Фр(к) and Фр(к) т а У be identified and we thus consider just Фр(к>-

BANACH ALGEBRAS 7

Clearly ek, evaluation at к, is a character for any к 6K . Now let 0 be any character on P(K).By a common abuse o f notation we shall write simply zr for the co-ordinate projection z zr (z6 K ). Then zrGP(K) and we define аг= 0(zr), r = 1, n. Then a = ( a i , a n) 6 d n; but also, by Lemma 4, for any polynomial p,

IIp IIk > l * ( p ) l = l p ( a ) l

so that a£K , since К is polynomially convex. Thus, since 0(p ) = p(a) = ea(p) for each polynomial p, we also must have 0 (f) = 6a(0 for each f EP(K) by continuity o f characters.

Thus Фр(к) — Фр(к) has been identified with the set K.

Note: for n = 1 and К a compact subset o f Œ, К is the union o f К with the bounded components o f (INK (essentially the classical theorem o f Runge). No similar simple description is available for subsets o f <Tn (n > 1 ).

We shall briefly describe a standard way o f topologizing ФА for any commutative Banach algebra A. I f we consider ФА С A ', the Banach dual space o f A, then we may topologize ФА as a subspace o f A ' in its “ weak topology” , a (A ',A ) (in Bourbaki notation); this relative topology is called the Gelfand topology on ФА. The Gelfand topology is the weakest topology on ФА for which each o f the mappings ф «■ 0 (x) (x G A), o f ФА into (Г, is continuous.

I f 0o 6 ФА then a neighbourhood base o f 0O for the Gelfand topology is provided by sets o f the form:

N(0O; x , , ..., хп)= {0 е Ф А : I0(xk) - 0o(xk)| < 1 (k = 1,..., n)}

where n is a positive integer and x 1;..., x„ are elements o f A.

Theorem 4. For the Gelfand topology, ФА is a non-empty compact Hausdorff space.

Proof (outline). For each x 6 A , let Dx = {XG(T: |X| < ||x||}. Then Dx is a compact disc in <t and so, by Tychonoffs theorem, the topological product

x £ A

is a compact Hausdorff space.

We define в : Фд -*■ P by в (ф) = {0 (x ) }xEA.

Then it is an easy exercise to see that 0 is a homeomorphism o f ФА onto a closed subset o f P.The result follows.

We note that, with Фда, Фс(К). Фр(к) given their respective Gelfand topologies, the identifications o f these spaces respectively with the unit circle in <Г, К, К given above are topological and not merely set-theoretic (the proofs are easy exercises).

It is sometimes useful to change the viewpoint slightly; i f A is as before and x € A then the Gelfand transform x o f x is the function х£С (Ф А) defined by x(0) = 0(x) (0 G ФА). (The fact that x is continuous on ФА follows immediately from the definition o f the Gelfand topology.)

Theorem 5. (i) The mapping x *+xis a continuous homomorphism o f A into С(ФА);(ii) fo r any xG A :

SpA(x )= {x (0 ) : 0 G ФА}

8 ALLAN

Proof, (i) This is more or less trivial; the continuity follows because, for any x 6 A:

llxll = sup{|x(0)| : фбфд}

= sup{|0(x)| : ф G Ф д }< l|x||

by Lemma 4.

(ii) Part (ii) o f the Corollary to Theorem 3.

Example. A non-rtormable function algebra. Let U be open, U Ç (Tn.We now fulfil the promise in Example 3 to show that C(U) cannot be given an algebra-norm.

First we find the characters.Let U* = U U {°°} be the 1-point compactification o f U; then C (U*) may be identified with a

subalgebra o f C(U). Let 0 be any character on C(U); then 0|C(U*) is a character on C(U *) (not the zero functional since 1 E C (U *)) and so, by (b) above, since U* is compact, there is a point w G U* such that 0 (f) = f(w ) for each f G C(U*). We shall now show that w € U, i.e. that w is not the “ point at infinity” . We may, in many ways, find a function gGC(U ) such that g(z) f 0 (all z € U ) and |g(z)| -> °° whenever either |z| in U or z approaches 3U from within U; then g_I GC(U*) with g "1 (oo) = 0, whereas ф(g-1) = g~ '(w ) f 0 since g_1 is invertible in C(U); thus w ^ oo, so that w 6 U. We now show that 0 (f) = f(w ) for every f G C(U).

Let в be an element o f C(U) such that:(i) supp 0 is a compact subset o f U, and (ii) 0(w) = 1. Then, for any f GC(U), в f has compact support in U, so that, in particular, 0 and 0f are both in C(U*). Thus 1 = 0(w) = 0(0), while f(w ) = (00 (w ) = 0(00 = 0 (6 )0 (f) = 0(0- Thus the characters on C(U) are precisely the evaluation maps e7 (z G U). (By a character we mean a non-zero homomorphism from C(U) -»■ <E; we are not, o f course, asserting that C(U) is a Banach algebra).

Now suppose that C(U) can be given an algebra-norm ||.||; then C(U) may be completed in this norm to give a Banach algebra B, say. Then Фв may be identified with the set o f those characters on C(U) which are ||.||-continuous on C(U); since Фв is compact for the Gelfand topology а(Фв; В), it is also compact for the weaker topology о(Фв; C(U)) (which is in fact the same topology); but this is just the relativization to Фв o f the topology a(U,C(U)) on U, and this latter is the usual topology o f U as a subset o f <Tn.

Thus Фв has been identified with a compact subset К С U, namely K = {z 6 U : ег is ||.||-continuous on C(U)}.

But then, by Urysohn’s lemma, we may easily find functions u, vGC(U ) such that:

(i) u(z) = v (z )= O (a llzG K )(ii) uv = v(iii) v^O

But, by (i), 0(u) = 0 for each 0 e Фв and so 0( 1 - u) = 1 for each 0 G Фв- But then, by Corollary to Theorem 3, 1 - u£B _1. But, by (ii), v (1 -u ) = 0 and so v = v ( l - u ) - ( l ~ и Г 1= 0, contrary to (iii). This contradiction proves the result.

We now meet, in a simple case, the notion o f holomorphic functional calculus.

Theorem 6. Let x G A and let U be an open subset o f iГ with SpA (x ) С U. Then there is a unique continuous unital homomorphism 0 X : €?(U) -*■ A such that 0 x(z ) = x. Moreover, fo r any f € 0 ( U ) and any ф G ФА we have 0 (0x(f ) ) = f(0 (x)).

Proof. Let R (U ) denote the subalgebra o f 0(\J) consisting o f all rational functions with no pole in U; then, for any f G R(U), we must define 0 x(f ) = f(x ) i f a definition o f ©x is at all possible. Moreover it is clear that f ** f(x ) ( f G R (U )) defines a unital homomorphism from R(U ) into A such that z x and also that 0 (f(x )) = f(0 (x )), for any f G R (U) and any ф G ФА. By the classical

BANACH ALGEBRAS 9

approximation theorem o f Runge, R (U) is dense in 0 (U ) , for the topology r o f uniform convergence on compact subsets o f U, and the proof is therefore completed by showing that f » f(x ) is a continuous mapping from R (U ) (with topology r ) into A. To show this we note that it is elementary to find Г, a finite union o f piecewise-smooth closed curves, such that (i) Г С U, (ii) every point o f (E\U has index 0 with respect to Г, (iii) every point o f SpA (x ) has index 1 with respect to Г. Then it is again elementary that, for any f GR(U):

and that,therefore,

||f(x)||<C-||f||r (fG R (U ))

where С is independent o f f. The result is thus proved; we note that the formula (* ) must continue to hold for any f G 0 (U ) , where f(x ) is now taken to mean 0 x(f).

Corollary. I f x G A and SpA (x) is disconnected then A contains a non-trivial idempotent e.

Proof. Let К be a non-empty, proper open-and-closed subset o f SpA (x ) and let L = SpA (x)\K.Then K, L are disjoint non-empty compact subsets o f (Г.

Let V, W be disjoint open neighbourhoods o f K, L respectively and set U = V U W. Then the function E, defined to be 0 on W and 1 on V belongs to 0(\J>) and satisfies E2 = E. Thus, if we put e = ©X(E) we have e2 = e and SpA(e) = {0,1}, so that e =/ 0,1.

The rest o f this paper is mainly devoted to a several-variable extension o f Theorem 6, together with some applications. The original motive for this extension was precisely to give a several- variable version o f the above Corollary. We shall see that the extension o f Theorem 6 provides a profound tool in the theory o f Banach algebras. First we need some definitions:

Let A be as before and let x 1 (..., xnGA, where n > 1 ; write x = (x b ..., xn)G A n. Then the joint spectrum, SpA (x ) = SpA (x 1;..., xn) is defined to be

SPa (5) = {(0(Х1), ..., 0(xn)) : 0G ФА}

We note that, by an argument analogous to the proof o f Theorem 3, Corollary, we have that, also,

SpA (x ) = {X = (X ! , ..., Xn)G<Cn : idA (Xk l - xk : к = 1,..., n) f A }

where we have written idA (Xk l — xk : к = 1,..., n) to mean the ideal o f A generated by {Xk l - xk : к = 1,..., n}.

Lemma 4. (i) SpA (x ) is a non-empty compact subset o/(Tn;

(ii) I f 1 < m < n and if ir : (Tn -*■ (Em is projection onto the first m co-ordinates, then

7i-(sPA (x )) = SpA (X ] , ..., xm)

Г

n

(iii) SpA(x ) С П SpA (xk) c {XG<Dn:|Xk| < rA (xk), k = 1, ...,n}k= 1

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(iv) SpA(p (x , , xn)) = {p (X 1, ...,Xn) : GSpa(x)} fo r any complex polynomial p in n-variables.

Proof, (i) Follows from Theorem 4, since ф ** ( ф ( х , ) , 0(xn)) is a continuous mapping o f ФА onto SpA (x).

(ii), (iii), (iv) all follow more or less trivially from the definition o f joint spectrum.

We shall ultimately see that an analogue o f Theorem 6 holds in the case o f several variables, with the joint spectrum SpA (x ) playing the role o f the spectrum. For the moment we content ourselves with a much simpler result.

Theorem 7. Let x ¡ , xn G A and let D ={X,G(En: |Xk I < r k (k = 1,..., n )} be an open poly disc in (En with SpA (x ) С D. Then there is a unique continuous unital homomorphism 0 X : 0 (D ) -*■ A such that 0 x(zk) = xk (k = 1,..., n). Moreover, fo r any ф G Фд and any f G 0 (D ) we have

0 (© x (f)) = fW (x1),...,i> (xn))

(As before, zk is the kth co-ordinate projection zk(X) = Xk (k = 1,..., n).).

Proof. Choose 0 < r k' < r k (k = 1,..., n) such that SpA (x ) C{X^G(Tn: |Xk| < r k' (k = 1,..., n)} Then, for any f G 0 (D ) define

e x(f ) = — X— ... / f « K î i 1 - X !) '1 (?„1 - х „ Г d {t ... d ^~ (2m )n J J ~

Kkl = tk'(k = 1..... n )

The proof that ©x, so defined, has the required properties is very similar to that o f Theorem 6 but technically a little simpler, since we have the set o f polynomials (rather than just rational functions) dense in 0 (D ), by Taylor’s formula. We shall omit the details.

Our basic problem is to substitute for D a more general open neighbourhood which may approximate SpA(x ) more closely.

The importance o f polynomial convexity (introduced in the discussion o f P (K ) above) lies in the following result, with which we conclude the present section.

Lemma 5. I f A is generated, as a Banach algebra with identity, by X j,..., Xn, then SpA (x ) is polynomially convex.(Note: the condition o f this lemma means that polynomials in x 1;..., xn are dense in A .)

Proof. Let Л_= (X , , ..., Xn) GŒn\SpA(x). Then there are elements y ls ..., y n in A such that

n

Y Ук (xk 1 - xk) = 1

к = 1

Since polynomials in x b ..., xn are dense in A, we may approximate y j , ..., yn as closely as we wish by suitable choice o f such polynomials. In particular we may choose polynomials P i , ..., pn such that

BANACH ALGEBRAS 11

Now define the complex polynomial q by

n

Then |q(X)| = | - 1| = 1, but for any/jSSpA(x ) we have qQí) eSpA (q (x !, ...,xn)), by Lemma 4 (iv), and so Iq(i0l < llq (x i,...,xn)|| < 1. Thus /V Sp^Tx) and therefore SpA (x ) is polynomially convex.

It is useful to introduce a little abstract algebra; only the most elementary facts are presented. By a differential group we mean a pair (X ;d ), where X is an Abelian group (written additively)

and d : X -* X is a group endomorphism such that d2 = 0.The endomorphism d is the differential o f X. We define subgroups:

Z(X ) = ker d = {x 6 X : d(x) = 0 } (elements o f Z (X ) are called closed elements)

B(X) = im d = {d (x ) : x E X } (elements o f B(X) are called exact elements).

Since d2 = 0 we have B(X) Ç Z (X ) and may define

H(X) = Z(X)/B(X)

the homology group o f X. I f x S Z(X)we write [x] for its canonical image in H (X) — called the homology class o f x.

A morphism o f differential groups X, X ' is a group homomorphism f : X -*■ X ' such that

commutes. (O f course d,d' are the differentials o f X, X ' respectively.) I f f : X -*• X ' is a morphism o f differential groups then clearly fZ (X ) Ç Z (X '), fB (X) Ç B (X ') and there is thus p naturally induced group homomorphism H (f) : H (X ) -* H (X '). More precisely, H is a covariant functor from the category o f differential groups and their morphisms to the category o f Abelian groups and their homomorphisms.

The Abelian group X will be called graded (by the integers) i f for each integer n there is a subgroup Xn with

B. GROUPS OF DIFFERENTIAL FORMS

OO

® X,41

In this context, the elements o f Xn are called homogeneous o f degree n. A differential map D : X -»■ X will be said to respect the grading provided dXn Ç X n + 1 for each n. The pair (X ;d ) is

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then a graded differential group. I f X, X ' are graded differential groups then a morphism f : X -*■ X' is defined to be a morphism o f the underlying differential groups such that f(X n) Ç XJ, for each n.

I f (X , d) is graded then the homology group H (X) inherits a natural grading:

Hn(X ) = (Z (X ) n Xn)/B(X) n Xn

for each n. The group Hn(X ) is the nth homology group o f XI f f : X -»■ X ' is a morphism o f graded differential groups, then the map H (f) : H (X ) -*■ H (X ')

is also graded, i.e. H (f) maps Hn(X ) into Hn(X ') for each n.

Example: We now turn to our prime example o f a differential group. Let U be an open subset of <Tn; write C°°(U) for the algebra o f all smooth (i.e. infinitely differentiable) (С-valued functions on U. For p = 0,1,2,... we shall write¿Fp(U ) for the group o f all complex-conjugate smooth differential forms on U o f type (0, p). Thus a typical element o f <^P(U ) looks like

where each ... ¡p G C“ (U ) and the summation is over all p-tuples ( i t , ip) o f positive integers such that 1 < ii < i2 < ... < ip < n.Thus

J # ° (U ) = < r ” (ü )

[ # p(U ) = 0 ( p > n + 1)

The Э-operator m a p s ^ (U ) -»■ # p + *(U) for each p and extends uniquely, by additivity to a group endomorphism of

(where we p u t# p(U ) = 0 for p < 0).By well-known properties o f 9 (see e.g. Part I o f the paper by M.J. Field in Volume I o f these

Proceedings), we have Э2 = 0; thus (S (U ), 9) is a graded differential group.The homology groups Hp(<# (U )) are called the Dolbeault cohomology groups o f U.By the Cauchy-Riemann criteria, we have H° ( S (U )) = 0 (U ) for any open U С <Cn.

We now give an important example o f a morphism o f graded differential groups.Let Ube open in <Cn, V open in <Em and let F i,...,F m G 0 (U ) . For each z £ U define

F (z) G <Cm by F (z ) = (F j (z ) , ..., Fm(z )) and suppose that F (U ) ÇV; such an F is called a holomorphic

mapping o f U into V. If, now, со G P(V ), say

where

n OO

BANACH ALGEBRAS 13

U = Y j “ i l - 'P d í i> Л " Л Ü\

where each ip GC“ (V ) and we write { = ( £ , , { m) for the current variable in (Em, then we define

V "F * (c o )= ; (coi l ...ipo F )3 F il «.. .A a F jp

where, o f course, summation in each case is over all p-tuples (ij, ...,ip) o f positive integers such that 1 < i] < . . . < i p < n. Then F*(co) £ «?P(U ) and F* extends uniquely by additivity to a group homomorphism from <#(V) into <a(U). A simple calculation (essentially the “ chain rule” ) shows' that 3(F*co) = F*(3co), so that F* is thus a morphism o f the graded differential groups <§(V ), # (U ).

We conclude this section with a special case o f the so-called first Cousin problem.

Lemma 6 . Let U, V be open subsets o f (Dn and let Hp + 1 (S (U и V )) = 0 fo r some p > 0. Then, given со S í fP (U n V ) with 3co = 0, there exist a 6 <^p(U), p£<^P(V)twith За = 0, dp = O.such thatoo = а - /3 on U n V.

It seems worth stating the case p = 0 separately:

Corollary. Let U, V be open subsets o f (En and let H1 (S (U и V )) = 0. Then, given f 6 0 (U f î V), there exist g £ 0 (U), h в © (V ) such that f = g - h o n U n V .

Proof o f Lemma 6. We first observe that we can find в 6 C“ (U u V ) such that в = 1 on a neighbourhood o f U\V and в = 0 on a neighbourhood o f V\U. (This is a special case o f a well- known fact about existence o f partitions o f unity).

Now let со ë<#p(U п V ) with 3co = 0. We define <X] S ^ i U ) , p¡ s W p(V ) by:

a , ( z ) = f ( l - 0(z )) <o(z) ( z e u n v )l o (z EU\V)

P i ( z ) = f - 0(z ) co(z) ( z e u n v )I 0 ( z 6 V\U)

The fact that a ¡,P ¡ are smooth forms follows from the stated properties o f в ; clearly u) = al - f i , on U n V. H o w e v e r ,P¡ need not be closed forms.

Now observe that, on U П V,

0 = 3co = da1 - 3/3j

and we may thus uniquely define r¡ s W p + 1 (U u V ) by putting:

_ Эс*! on U

V [ dp1 on V

Moreover, br¡ = 0, being equal to d2 a l on U and to Э2 P, on V. By our assumption that Hp + 1 (é ’CU U V )) = 0, there is thus a form y EéFp(U и V) such that tj = dy. We now define a = a , - 7 0 n U and P = P, - y on V; then a - P = a l - p ¡ = со on U n V, while За = За, — dy = даг - n = 0 on U and, similarly, dp = 0 on V. The proof is thus complete.

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С. POLYNOMIALLY CONVEX DOMAINS IN Œ"

Recall that, for a compact set К с (Dn, its polynomial hull is

K ={zGŒn : |p(z)| < ||p||K (all polynomials p)}

and that К is polynomially convex if and only if К = К.We now define a polynomially convex domain to be an open subset U o f (Tn such that for

any compact К С U we have also K C U .An open polydisc is a set o f the form

D ={ze<E n : |zk — ak| < r k (k = l,...,n )}

where the centre a = (a1; an) G<En and the polyradiusj = ( r , , rn) is an n-tuple o f positive real numbers.

A polynomial polyhedron is a set o f the form

P={z<ED : |pk(z )| < l ( k = 1,...,«)}

where D is an open polydisc in (En and pb ..., p„ are finitely many polynomials. An open polydisc is thus a special kind o f polynomial polyhedron. It is a very simple exercise to show that any polynomial polyhedron is a polynomially convex domain.

An easy compactness argument proves the following:

Lemma 7. Let К be a compact polynomially convex subset o/(Tn and let U be open, U D K. Then there is a polynomial polyhedron P with КС P С U.

We shall need to know that Hp(<#(P)) = 0 for all p > 1, where P is any polynomial polyhedron. Although we only use the result in the case p = 1, the method o f proof involves proving the theorem simultaneously for p = 1 , 2 ,3,... .

Our starting point is the case o f an open polydisc; this result will be stated without proof; a proof may be found, e.g. in Ref. [3] (Chapter I, D5).

Theorem 8. I f D is an open polydisc in <En, then HP(# (D )) = 0 (p > 1).We shall now show how to extend this result to a general polynomial polyhedron; the method

is due to Oka and the key idea lies in the following lemma.Suppose that U is open in (P1, that ф e 0 (U ) and let Д = {z €(E : |z| < 1}. We define

U0 = {z S U : |<Kz)l<l}.

Lemma 8. I f Hp (<ÿ ( U X Д )) = 0 fo r all p > 1 then

(i) Н Р Ш и 0)) = О(а//р> 1);

(ii) fo r any f e é ? (U 0), there isa function V Ç.0 (U X Д) such that f(z ) = F(z, 0(z )) (z 6 U^).

Proof. Define ¡i : U -*■ <Tn + 1 by ju(z) = (z ,ф(z )); observe that = д~‘ (U X Д). Then д is certainly a holomorphic mapping and we may thus define the morphism o f graded differential groups 11* : é ’CU X Д) -*■ as in the section just before Lemma 6 . By hypothesis,HP(<^(U X Д )) = 0 (p > 1) and the lemma would thus be proved if we_could show that, for each p > 0, Н (д*) maps № ( Î (U X Д )) onto HP(# (U 0)). (Recall that H °(«# (V ))= 0 (V ) for any open set V, so that part (ii) o f the lemma is just the case p = 0). But it is then sufficient to show that, for each p > 0 and each p-form cj on with 3cj = 0, there exists S2 £ ^ P(U X Д) with 3Í2 = 0 and n * (i i ) = cj.

BANACH ALGEBRAS 15

Thus, let 0)€<^P(Uф), with doj = 0. Define A = X Д, В = {(z ,w )£ U X i : w f ф( z)}; then A, В are open subsets o f U X Д and А и В = U X Д. On A n В define the p-form tj by

r?(z, w) = (w - ф(z))~ l co(z)

Then, since (w - ф(г))-1 € 0 (k n B) and Э w = 0, we deduce that 3rj = 0 on A n B. But then, by Lemma 6, there exist a€<^P(A), /ЗЕ<^Р(В) with 9 « = 0, d/3 = 0 and r? = a - /3onAnB. Thus, on А п B,

co(z) - (w - 0 (z)) a(z,w) = - (w - ф( z )) (3(z,w)

There is thus a well-defined element Í2E (#P(U X Д) defined by

Í oj(z) - (w - 0 (z)) a(z, w) on A Ji(z,w) = -I

(w - ф(z)) (3(z,w) on В

Clearly 9Í2 = 0 and = oj. The proof is thus complete.

Corollary 1. Leí D be an open poly disc in 4a,let p ,,..., p„ be given polynomials and let

P = {z £ D : |pk(z)| < 1 (k = 1 ,...,»)}

Then (i) HP(^ (P )) = 0 (p > 1 ); (ii) fo r any f G 0 (P ) there is a function F E 0 (D X Ду) with

f(z ) = F (z ,P l(z ),...,PlXz)) (zE P )

Proof. This is a simple induction on v, using Lemma 8. The case v = 1 follows from Lemma 8 since HP(D X Д) = 0 (p > 1 ) by Theorem 8. We omit further detail.

Corollary 2. Let Qbe a polynomial polyhedron, let pb ..., p„ be given polynomials and let

P = {z E Q : |pk(z )| < l (k = l,..., i0>

Then fo r any f E 0 (Y ) there is a function F £ 0 (Q X A?) with

f(z ) = F (z ,p , (z ), ...,pr(z )) (zE P )

Proof. This is similar to Corollary 1, except that the start o f the induction uses Corollary 1 itself to give H1 (Q X Д) = 0 (for Q X Д is a polynomial polyhedron in <Cn + 1 ).

Corollary 3 (Oka - W e i l t h e o r e m ) .

Let P be a polynomial polyhedron in <Cn; then the set o f polynomials is dense in 0 (P ) (for the topology o f uniform convergence on compact subsets o f? ).

Proof. Let P be as in Corollary 1, let f E 0 (P ) and let F E 0 (D X Д") be such that

f(z ) = F (z ,p ,(z ),...,p „(z ))

Then substitution o f Pi (z ) , ..., p„(z) into the Taylor series o f F on D X Д” gives the desired sequence o f approximations to f — by polynomials in iz ^ ..., zn, P ] (z ) , ..., p„(z)}, hence by polynomials in {z , ,..., zn}.

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Corollary 4. With the notation o f Corollary 2, let F G & (Q X Av) be such that

F(z, p, ( z ) , p i , ( z ) ) = 0 (all z G P)

Then there exist H, , H„ 6 & (Q X Д") such that

v

F (z ,w )= Y Hk(z, w) (w k - pk(z ))

к = 1

for all (z,w ) = (z j,..., zn, W j , w,,) G Q X Ду.

Proof This is again by induction on v, but we shall give some details.

Case v = 1 (this case makes no use o f any cohomology vanishing): We have F E 0 (Q X Д) such that F (z,p (z )) = 0 for any z £ Q p = {z G Q : |p(z)| < 1}, and we wish to find H G 0 (Q X A ) such that F(z, w) = H(z, w) (w - p (z)) on Q X Д. Any such H is clearly unique on any open set (if such an H exists at all), so that it is sufficient to define an H on a neighbourhood o f each point o f Q X Д.

Thus, let (z0,w0) G Q X Д and suppose that w0 = p(z0) (for otherwise the definition o f H is trivial). Now the mapping 0(z,w) = (z,w - p(z) is (1 - 1 ) holomorphic o f Œn onto (En, the inverse mapping being 0_I (z,w) = (z,w+p(z)), which is clearly also holomorphic. Now в (z0,w0) = (zo,0), so that FoéT1 is holomorphic on the open neighbourhood 0(Q X A) o f (zo,0) and (Fo0-1 ) (z, w) = 0 whenever w = 0; thus the Taylor expansion o f FoO-1 at (z0,0) contains w as a factor, i.e. F(0-1 (z,w )) = wH'(z,w ) on some open W 9 (z 0,0), where H' G 0 (W ). Thus, for (z,w) G 0~l (w ) we have F(z,w) = (w - p (z)) H(z,w), where H = H'° в G 0 (в ~ ' (w )). Thus the result is proved for v = 1.

Case v > 1 : We shall use notation such as V(p i ,...,pJ t0 m ean{zG V : |pk (z)| < 1 (k = l,...,v )}.

We aré given F G 0 (Q X Av) with

F (z,p i(z ),...,p „(z)) = 0 (z G Q(Pb ...;Pv))

But Q(pi....P[;) = (QPl) (Pj,...,Pv); we define GG 0 (Q pi X A ”' 1) by

G(z,w2, ..., w ¡,) = F(z,pj(z), w2, ..., wv)

(for (z,w2,..., w^G Q pj X Д*'- 1 ); then we have

G(z,p2(z ),...,p y(z ) )= 0 (zG (Q p i)p2; : Pv)

Thus, by the induction hypothesis, we can find H2, ..., Wp in 0 (Q Pl X Av~ 1 such that

V

G(z,W2, ..., Wj;) = Hk (z,w2,..., w„) (wk - p k(z)), on QPl X Д " - 1

k - 2

By Corollary 2, there are Hk G 0 (Q X Av) (k = 2,..., v) such that

Hk(z,w2,..., wM) = H k iz .p jiz^w j.-.w ,,)

BANACH ALGEBRAS 17

fo r (z ,w 2,...,w,,) £ Q Pl Х Д *1 1. Thus

v

F(z,w 1; w„ ) - Y Hk(z ,w ,,...,w „)(w k - p k(z))

к = 2

is holomorphic on Q X Д" and vanishes whenever Wj = P! (z). Thus, by case у = 1 above, this function has the form Hj (z ,W !,..., wy) (w t - p j ( z ) ) for some H, S 0 (Q X Д"). The proof is thus complete.

D. HOLOMORPHIC FUNCTIONAL CALCULUS

We start by recalling Theorem 7; i f A is a commutative Banach algebra and x ,,...,xn E A , then for any open polydisc D Э SpA(x ) there is a unique continuous unital homomorphism 0 X : © (D ) -> A such that 0 x(zk) = xk. As explained in Section A, we aim to replace D by an arbitrary open neighbourhood o f SpA(x). The next step is as follows:

Lemma 9. Let P be a polynomial polyhedron P Э SpA(x). Then there is a unique continuous unital homomorphism 0 X : 0 (P ) -* A with 0 x(zk) = xk (k = 1,..., n). Moreover, fo r any f G © (Y ) and any ф 6 ФА we /iave~0(0x(D) = f(0 (x, ), ~., Ф(хп)).

Proof. Let P = {z E D : |pk(z) < 1 (k = 1,..., p)}, where D is an open polydisc in (Cn and p , ,..., p„ are polynomials. Define

p : © ф X Д") Ч- © (P)

by putting

p (F )(z ) = F(z,p, ( z ) , ..., pv(z )) (z € P)

Then p is clearly a continuous homomorphism and, by Lemma 8, Corollary 1, p is surjective. Thus, since 0 (D X Д"), 0 (P ) are Fréchet spaces, p is an open mapping (by the open mapping theorem).

Also we have defined, by Theorem 7, a continuous unital homomorphism:

®x,Pi(x),...,p„(x): © (D X Д ") -*• A

such that

|®х,р! (x),.... p„(x) (zk )= xk (^ = 1>—>n)

1.®x,Pi (x),...,p„(x )(wk) = Pk (*) (k =

where{z1; ...,zn, w 1;..., wv} are the co-ordinate projections on Œn + ’! Note that

V

SpA(x ,P i(x ),...,p „ (x )C S p A ( jp x П Pk(SpA (x )) Ç P X Д*1 Ç D X A"к = 1

But, by Lemma 8, Corollary 4, ker p is generated as an ideal by {wk - Pk(z) : к = 1,..., v}, and thus

ker p Ç ker 0xlPl(x),...,Plj(5)

18 ALLAN

It follows that there is a unique continuous unital homomorphism 0 X : © (P ) A such that

Thus we have 0 x(zk) = xk (k = 1 , n). The uniqueness o f 0 X follows since polynomials are dense in © (P ), by Lemma 8, Corollary 3. By the same Corollary we also deduce that for any f € © (P ) and 0 G ФА we have

0 (© x (f)) = f(0 (X !),...,0 (Xn))

since the result is clear when f is a polynomial.

Now we come to the functional calculus theorem itself. (It would be possible to strengthenthe statement to include a form o f uniqueness; see Ref. [9].)

Theorem 9. Let U be an open subset o f (Dn with SpA (x ) С U. Then there is a continuous unital homomorphism 0 X : © (U ) -*■ A such that:

(i) 0 x(zk) = xk (k = l,...,n );

(ii) fo r any f G 0 (U ) and ф G ФА, 0(©x(f>) = f(0 (x j), ...,0(xn)).

Proof. The method o f deducing Theorem 9 from Lemma 9 is usually known as “ the Arens-Calderón method” , since it is essentially contained in Ref. [5].

Let D = {z G Œn : |zk| < ||xk|| (k = 1,..., n)}; thus D is a compact polydisc with SpA(x ) Ç D (by Lemma 4). I f X_= (X ,,..., Xn) £ D\U, then X_^ SpA (x ) so that there are elements y l5..., yn £ A for which

It follows from Lemma 1 that, for all juin some open neighbourhood V\ o f X, the element

^ yk(MkJ - x k)

к = 1

is invertible in A and that the inverse lies in the closed subalgebra B^of A generated by {1, X j,..., xn, У i, Уп}; thus Va П SpB^(x1;..., Xn) = 0. Since D\U is compact it is covered by finitely many o f the neighbourhoods V^ (¡V G D\U) and we thus obtain a finitely generated subalgebra В o f A, generated, say, by {1, xb ...,xn, b j , ..., b„}such that

SpB(x ls ...,x„) С U

By Lemma 5, SpB(x , , ..., xn, b ,,..., bv) is polynomially convex and so, by Lemma 7, there is an open polynomial polyhedron P with

® x ° P - ® x , p , ( x ) ......p „ (x )

n

к = 1

n

SpB(x 1; ...,xn, b „ ... ,b v) C P C U X f

BANACH ALGEBRAS 19

By Lemma 9, there is a (unique) continuous unital homomorphism 0 X t, : <2HP) -► A such that

We define i : 0 (U ) - + 0 (P ) by “ ignoring co-ordinates” i (0 (z 1; zn + >;) = f(z1,...,zn) ( f G0(U)). Then clearly i is a continuous homomorphism. We now define

0 X : 0 (U ) -*■ A

©x = ©x.bOi

That ©x has the required properties now follows from Lemma 9.

As an immediate application we deduce the Shilov idempotent theorem [4], which provides a strong generalization o f the Corollary to Theorem 6 . Observe that the statement makes no mention o f complex variable theory; no more direct proof is known, however.

Corollary (Shilov idempotent theorem). Let E be an open-and-closed subset o f Фд. There is a unique idempotent element e € A such that ê = Xe ( the characteristic function o fE ).

Proof. Let F = Фд\Е; thus ФА = E u F, where E, F are disjoint compact subsets o f Фд. It is a simple consequence o f the definition o f the Gelfand topology that there are elements x lt ..., x„ G A such that x(E ) П x (F ) = 0 (where, for example, x(E ) = {(¡¿¡(ф ),..., xn(0) : ф G E}); clearly x(E), x(F ) are compact subsets o f (En.

Now let V, W be open neighbourhoods o f x(E), x (F ) respectively with V n W = 0 and set U = V U W. Define f G 0 (U ) by f(X ) = 1 (X~G V)~f(X ) = О (X G W); then P = f. Now let e = ©x(f); then e2 = e and, for any ф G ФА ,

so that ê = Xe-

For uniqueness, suppose d = d2 G A and that d = ê. Then (d - e)3 = d3- 3d2e + 3de2 - e3 = d3- e3 = d - e; thus (d - e) x = 0, where x = 1 - (d -e )2. But, for any ф G ФА, ф(х) = 1 - (0(d) - 0(e))2 = 1, so that x G A-1 (Corollary to Theorem 3) and thus d - e = 0. The proof is complete.

The importance o f the functional calculus theorem lies in the fact that it provides homomorphisms (i.e. the maps 0 X) from algebras o f holomorphic functions which have an intensively studied structure, into a commutative Banach algebra where, from many viewpoints, direct knowledge o f the structure is hard to obtain.

This was strikingly illustrated in the Shilov idempotent theorem (Corollary to Theorem 9) where a trivial fact about 0 (U ) (that it contains any locally constant function) led to the highly

(xk (k = 1 , n)

by

ê(0) = 0 (0x (f ) ) = f(0 (X j),..., 0(xn))

E. SOME APPLICATIONS OF FUNCTIONAL CALCULUS TO THE STUDY OF COMMUTATIVE BANACH ALGEBRAS

20 ALLAN

non-trivial fact that a Banach algebra with a disconnected character space ФА contains a nontrivial idempotent. Naturally, we hope that more subtle information about 0 (U ) may lead to further facts about A. We shall aim to illustrate that this is so.

Our first illustration is a remarkable result o f H. Rossi - the “ local peak set theorem” . First some terminology. Let A be a commutative Banach algebra with 1 and let ФА be the space o f characters in its Gelfand topology. A subset К С ФА is a peak set for A if and only i f there is an element x G A such that(i) * (0 ) =1 (Ф e K)(ii) |х(Ф)| < 1 (Ф e ФА\К)

Note that a peak set is necessarily compact. A closed subset К o f ФА will be called a local peak set if and only if there is an element x G A and an open neighbourhood U o f К in ФА such that

(i) х (0 )= 1 (ф G К)(ii) |x(0)| < 1 (0 GU\K)

Clearly any peak set is also a local peak set. The theorem o f Rossi gives the converse:

Theorem 10 (Rossi [6 ]) — Any local peak set is a peak set.

Proof. We reduce at once to the case in which К consists o f a single point — for let A^ be the closed subalgebra o f A, A K = {x G A : x/K is constant}; then it is almost trivial that ФА is the quotient space o f ФА obtained by identifying К to a point [K], say, and that{[K ]}is a local peak set for A K. It is also clear that proof o f the theorem for A^ would immediately imply its truth for A itself. \

Thus we suppose that К = {0 O} and that for some open neighbourhood U0 o f ф0 and element x0G A we have к0(ф0) = 1, |x„W>)l < 1 (0 £ Uo\{0o}-

By definition o f the Gelfand topology we may suppose that

U0 = {0 G ФА : |xk(0)| < 2 (k = 1,..., n)}

where xk G A, xk(0o) = 0 (k = 1,..., n).Now let x = (x0, ..., x„), x = (x 0, ..., xn),cr = <JA (x) = {£ (0 ) : 0 G ФА}; observe that

po =%(Фо) = (1,0,0,...,0). Let

V 0 = {z = (z0,...,zn) G(Tn + 1 : |zk| < 1 (k = l,...,n )}

Note that z0 = 1 at P0, but that |z0| < 1 whenever z £ V n o , z # P 0 (for, i f z G V 0 n a then z = x(0 ) for some 0 with |xk(0)| < 1 < 2 (k = 1,..., n) and so with 0 G U0).

Thus (bdy V0) П ais a compact subset o f <Cn+1 : |z0| < 1}; thus there exists an open set V, D(j\V0 such that V0 n V , C { z G I n + l : |z0|<1}. ThenV= V0 U V, is an open neighbourhood o f ain d n + 1.

Using the Arens-Calderón method (see proof o f Theorem 9) we now choose {x n+i,...,xj,} in A such that, i f В = âIgA{x 0, x , ,..., x¡,} then SpB(x ) С V. Now SpB(x0, ..., x„) is polynomially convex (Lemma 5) and so we may choose a polynomial polyhedron P in (C " + 1 such that

SpB(x 0, . . . , x „ ) C P C V X r ‘ n

Set P0 = (V 0 X (Г"~п) П p; Pj = (V , X <Г"-П) П P; then P = P0 U P ,. OnPo n p , we have |z0| < 1 so that we may define a holomorphic branch o f log (z0 - 1). By Lemma 6 , Corollary, and Lemma 8, Corollary 1, there are functions h0 G & (P0), ht € 0 (P ¡ ) with

(z0- 1Г1 log(z0- 1) = h j (z ) - h o (z ) on P0 ПР,

i.e. ( z 0 - 1) exp ( (z0 - 1) h0(z ) ) = exp ( (z0 - 1) h j ( z ) ) о п Р 0 П Р ]

BANACH ALGEBRAS 21

Thus there is h G 0 (P ) defined by

hf ч _ Г exP ((zo - 1 ) hi (z )) (z G Pi )Z l ( z 0 - l ) e x p ( ( z „ - l ) h 0(z ) ) ( z G P 0)

It is now an elementary geometrical argument (in the complex plane) to show that, for a sufficiently small e > 0, the function H, defined by

H(z) = - e(h(z) - e)"1 ( z £ P )

is holomorphic on a neighbourhood o f Spg(x0,...,xv) and peaks precisely (х0(Фо ),..., кр(ф0У) relative to SpB(x0, ..., x„). Thus, putting у = 0 (xo,...,x„)(H),we have that у(ф0) = 1, |ÿ(0)| < 1 (ф G ФА\{ф0}), i.e. {ф0} is a peak set, as required.

Our final application is the Arens-Royden theorem (see Refs [7], [8]). Suppose that К is any compact Hausdorff space; let С = C(K), the algebra o f all continuous complex-valued functions on K, let C ' 1 be the multiplicative group o f units and let exp С be the subgroup o f СГ1 consisting o f those elements having continuous logarithms defined on K. Then a theorem from topology gives that СГ1 /exp C s H 1 (X ,Z ). We shall show that the same result holds not merely for C, but for any commutative Banach algebra A with ФА = К. To avoid presupposing knowledge o f topology we shall simply prove that A-1 /exp A s C 1 /ехр C. First we need some definitions.

Let A be, as usual, a commutative Banach algebra with 1 and let A-1 be the group o f units (invertible elements). For any x G A we define

(the series is norm-convergent in A ) and let exp A = {exp x : x G A}. Then exp A is a subgroup o f A-1 (in fact exp A is the topological component containing 1). We write К = Фд and С = C(K) as above.

T h e o r e m 11 (A r e n s — R o y d e n )

exp A exp С

Proof. Let G : A -> С be the Gelfand representation o f A (i.e. G (x) = x (x G A )). Then clearly G (A -1 ) Ç C '1, G(exp A ) Ç exp C,so that there is a naturally induced homomorphism G : A-1 /exp A - »C _1/exp C, defined by G ([x ]) = [G (x)] (where, for x € A-1, [x] denotes the coset x-exp A, etc.).

We shall show that G effects the required isomorphism between A -1 /exp A and C"*/exp C. The proof falls into two parts:

(i) G и (1 -1 ): for this we must show that if x G A-1 and x G exp C, then x £ exp A.

(ii) G maps onto С-1 /ехр C: we must show that i f f G C" 1 then there exists x G A" 1 such that f/x G exp C.

Proof o f ( i) : Let x, G A-1 be such that Xj G exp C, i.e. there exists fG C with х 1(ф) = exp f (ф) (all ф G К).

OO

П = 0

А ' 1 С' 1

22 ALLAN

We shall show now that there exist elements x2,..., xn £ A such that whenever ф ¡ , ф2 £ К and xk(0 ! ) = xk(02) (k = 1, ...,n) then also f (0 ,) = f(02). Thus, suppose that ф° фф%\ then we can find a £ A with â(0?) Ф â(02) and hence also к(ф,)ф â(02) for all (фи ф2) in some neighbourhood W o f (0?, ф°) in К X К. For a diagonal point (ф°,ф°) G К X К we choose a neighbourhood V o f ф° in К such that |f(0) - f(0°)| < ir (all ф £ V); then i f (01; ф2) £ V X V and i f fc i($ i) = xi (02) then exp f (0 , ) = exp f(02) and If(0 j) - f(02)| < 2ir, so that f(0 i) = f(02). Thus for (0°,ф°) we take W = V X V , a = x1. Then for each (0?, 02) £ К X К we have found a neighbourhood W and an element a £ A such that if (0 1; 02) e W and â(0 i) = â(02) then f(0 i) = f(02). The compactness o fК X К now yields the required finite set o f elements x2, x n £ A.

Define a : К -> о = Эрд ( x , , xn) by а(ф) = ( ф С х Д 0(xn)); then there is a unique function F : a -*■ (E such that Foa = f and it is elementary that F is continuous. Also eF< = z , (7. 6 K).We shall now showjthat F may be extended to a holomorphic function F , on a neighbourhood U o f <7, such that eF z = on U.

For each z° £ о we choose S (z°) > 0 such that (if B(z°; 8(z0)) is the open ball, centre z° and radius 6(z0)) then

(a) |F(z) - F(z°)| < n (z £ an B(z°; 8(z0)))

(b ) there is a unique holomorphic function 0Z0 on B(z°; 6(z0)) such that both 0Z0 (z °) = F (z°)and exp(02o(z )) = z¡ on B(z°; 5(z0))

(c) |0zo ( z ) - 0 zo(z°)| < 7Г on B (z°; 6(z0))

Then (a) and (c) together imply that F (z) = 0Z0 (z ) for all z £ B(z°; S(z°)) n a.Define

U = U B(z°; i 6 (z0))

Then U is an open neighbourhood o f a. We claim that there is a well-defined F £ & (U ) defined by putting F (z ) = 0Z (z ) on B (z°;è 6(z0)) for each z° £ a. Assuming, for the moment, that F is well-defined, it is evident that F/a = F and that exp(F (z)) = z, on U. I f we now put У = © (xb ...,xn) (F ) we have that exp у = ©(xb ...,x„)(exp F ) = Xj and we have proved that X! £ exp A, as required.

It remains to show that F was well-defined on U. Thus, suppose that, for some z ° , z1 £ о we have B(z°; è 5(z0)) n B (z! ; i 5 (z ,)) Ф 0. Suppose, without loss o f generality, that 6(z , ) < 5(z0); then B(z, ^ ( Z j ) ) Ç B (z°;5 (z0)) and so, by the uniqueness assertion for B(z* ¿¡(z1)) (since ф?о (z 1) = F (z ' ) = фг i (z1)) we must have 0Zo(z) = фг i (z ) on B(z' 6( z ' ) ) and so, in particular, on B(z°;4 S(z0)) n B(z‘ ;J 8(z ') ) . Thus F was well-defined and the proof o f (i) is complete.

Proo f o f ( i i ) : Let f £ СГ1 and let e = inf{|f(0)| : 0 £ K }; then if g £ С with ||g - f||K < e we have g £ C" 1 and f/g £ exp С (for g r 1 = 1 + (g - f ) f_1, and ||(g - O f-1 IIk < !)•

In particular, by the Stone-Weierstrass theorem, we may find x , , x2n £ A such that, forn

k= 1

we have ||g - f]|K < € and thus fg 1 £ exp C.Define a :K ->-a = 8рд (х1; ...,x2n) by a (0) = (0(x , ),..., 0(x2n)) and define G on (C2n by

BANACH ALGEBRAS 23

Then G is a smooth function and G (z) Ф 0 on a and hence also G(z) Ф 0 for z in some open set U э a.

By the Arens-Calderón method we choose elements X2„ + i , •••, x„ in A such that, for В = algA{x 1;..., Xp} we have SpB(x ! , ..., X2n) С U. Then SpB(x 1;..., xv) is polynomially convex and so we may choose a polynomial polyhedron P with

SpB(x , .....хг) с Р с и х Г ' 2л

By an abuse o f notation we regard G as defined on P, by the formula

G(Zi , Zy) ■Iк = 1

zk zk + n

Then G is smooth on P and G(z) Ф 0 (z G P). Also, i f (3 : К -* SpA (x , ,..., x„) is defined by /3 (ф) = ( ф ( х ф ( х р) ) (<t> G К ) then Go|3 = g (note that SpA (x b ..., x„) Ç SpB(x b ..., x¡,)).

Define со G ^ 1 (P) by

9G -„ . -=— dzuG 1—* 3Zb.

к = 1

_Then a trivial calculation shows that Эсо = 0 and so since, by Lemma 8, Corollary 1,H1 (if(P ))j= 0, there exists k G C "(P ) such that со = Эк. Set h = e~k G GC” (P); then3h = e“k(3 G — G Эк) = 0, so that h G 0 (P ) and G h-1 = ek on P. But then if we put x = ®(Xl,..., x„)(h)we have x = hop and so

g/x = ek'^Gexp С

Thus f/x = (f/g) (g/x) G exp C, and the proof is complete.

REFERENCES

[1 ] A L L A N , G .R ., Em bedd ing the algebra o f fo rm a l pow er series in a Banach algebra, P roc. L on d on Math.

Soc. (3 ) 2 5 (1 9 7 2 ) 329 -340 .

[2 ] H A G H A N Y , G ., N orm in g fo rm a l pow er series in n variables, t o appear in Proc. L on d on Math. Soc.

[3 ] G U N N IN G , R .C ., R O SS I, H ., A n a ly tic Functions o f Several C om p lex Variables, P rentice-H all (1 9 6 5 ).

[4 ] S H IL O V , G .E ., O n the decom position o f a norm ed ring as a d irect sum o f ideals, Mat. Sb. 32 (1 9 5 4 )

3 5 3 -3 6 4 and A .M .S . Translations (2 ) 1 (1 9 5 5 ) 3 7 -4 8 .

[5 ] A R E N S , R ., C A L D E R O N , A .P ., A n a ly tic functions o f several Banach algebra elem ents, Ann . Math. 62

(1 9 5 5 ) 2 0 4 -2 1 6 .

[6 ] R O SS I, H ., T h e loca l m axim um m odulus p rincip le , A n n . M ath. 7 2 (1 9 6 0 ) 1 — 11.

[7 ] A R E N S , R ., T h e g roup o f invertib le elem ents o f a com m utative Banach algebra, Studia Math. (S eria Spec.)

Z . 1 (1 9 6 3 ) 2 1 -2 3 .

[8 ] R O Y D E N , H ., Fu nction algebras, Bull. A m . M ath. Soc. 69 (1 9 6 3 ) 2 8 1 —298.

[9 ] B O U R B A K I, N ., Th éories spectrales, Herm ann, Paris (1 9 6 7 ).

IAEA -SM R-18/101

ANALYTIC CONVEXITYSome comments on an example o f de Giorgi and Piccinini

A. ANDREOTTI, M. NACINOVICH Istituto Matematica,Université di Pisa,Pisa, Italy

Abstract

A N A L Y T IC C O N V E X IT Y : SO M E C O M M E N TS O N A N E X A M P L E O F DE G IO R G I A N D P IC C IN IN I.

1. In trodu ction . 2. C °° and analytical con vex ity fo r a com p lex o f d ifferen tia l equations w ith constant

coe ffic ien ts . 3. R edu ction o f the analytic con vex ity to the С ° ° con vex ity . 4. T h e exam ple o f de G iorgi-P iccin in i.

1. INTRODUCTION

De G io rg i was the f i r s t to make the fo llo w in g observa tion :

Consider the Laplace opera tor in two v a r ia b le s x , y

as op era tin g on fu n ction s o f th ree v a r ia b le s x , y , t , and consider the

equation

( 1 ) Au = f

fo r и and f fu nctions o f x , y , t .

Then we have the fo llo w in g fa c ts :

(a ) V f e C°°0R^) th ere e x is t и £ С (JR ) such th a t (1 ) ho lds;

(b ) V R and every e > 0 and fo r every f complex valued r e a l a n a ly t ic

in B(R) = { ( x , y , t ) e I + y ^ + t^ < R} we can f in d и complex va lu ed ,

r e a l a n a ly t ic in B(R - e ) such th at (1 ) ho lds;

3(c ) There e x is t some f r e a l a n a ly t ic in К such th at the equation

3( 1 ) does not admit any g lo b a l so lu t io n r e a l a n a ly t ic in К

Facts (a ) and (b ) can be proved d i r e c t ly by m eans,for instance, o f the Poisson

in t e g r a l.

25

The su rp r is in g phenomenon i s fa c t ( с ) and P ic c in in i has g iven an e x p l ic i t

example o f th is phenomenon.

Let us make f i r s t the fo llo w in g remark. I f we se t z = x + i y then

2 6 ANDREOTTI and NACINOVICH

3z 3z

Th ere fo re to v a l id a te fa c t (c ) i t is enough to show th at th ere

3e x is ts a complex valued r e a l a n a ly t ic fu n ction f on К such

th a t the equation

does not admit any g lo b a l complex valued r e a l a n a ly t ic so lu tion s .

(Indeed , i f Au = f could be so lved w ith a g lob a l r e a l a n a ly t ic

fu n ction u then v = 4 ^ would be g lo b a l r e a l a n a ly t ic and would

s a t is fy equation ( 2) ) .

Th is phenomenon is s im ila r to the phenomenon o f Hans Lewy and is

c lo s e ly r e la te d to the th eory o f con vex ity f o r d i f f e r e n t ia l operators

w ith constant c o e f f ic ie n t s .

2. C°° AND ANALYTIC CONVEXITY FOR A COMPLEX OF DIFFERENTIAL EQUATIONSWITH CONSTANT COEFFICIENTS

(a ) Let ti be an open se t in Rn , and l e t

P0 V D) h Ai (D) Ad - i (D) J d( 1 ) # ° (П ) —— -> . . . ■■ > <f (fl) - > 0

be a complex o f d i f f e r e n t ia l operators w ith constant c o e f f ic ie n t s . Here

(а ) 0 Щ ) denote the space o f C° fu n ction s on and

A0 (Í2) = x . . . xffîQ ) p^-tim es;

(g ) A^(D) = Сац ^ ( ° ) ) i s a m atrix o f type ( P i + i ’ P i ) w ith en tr ie s

d i f f e r e n t ia l opera tors w ith constant c o e f f i c ie n t s ;

(у ) The fa c t th a t the above sequence is a complex means that

A. + 1 (D) Ai (D) H 0 , V i

Example. Let AQ(C) be a (p^ ,pQ) m atrix w ith polynom ial e n tr ie s .

Consider the r in g &>= , . . . ,Ç ] o f polynom ials in n v a r ia b le s

and con sid er t AQ as a ^-homomorphism

&> 1 __о__ 0

Then by H i lb e r t ’ s theorem we can continue th is map by a f i n i t e sequence

o f maps

?d \ 1 (Ç) ^ P d - l a$2 4 ^о — ■>& — — — .. — =— >9&2 — I— >3я г — °— > g>° — >n - lo

(and d i n o r d = 2 i f n = 1 ) to g iv e an exact sequence.

tRep lacing the m atrices A^(Ç) by th e ir transposed A^(Ç) and the

v a r ia b le s Ç. by -5— we g e t a complex ( 1 ) o f d i f f e r e n t ia l opera tors w ithJ °Xj

constant c o e f f ic ie n t s which has moreover the p rop erty to be exact on e v e ry open

convex s e t fi (P o in ca re lemma).

The con d it ion th a t the complex (1 ) be exact on convex f i 's ch a ra c te r iz e

the complexes obta ined from a "H ilb e r t r e s o lu t io n " . Thus, in the sequel we

w i l l assume th a t (1 ) is a "H ilb e r t com plex".

We w i l l say th a t Q (no t n e c e s sa r ily convex) i s С convex f o r (1 )

i f f ( 1 ) i s a c y c l ic .

, OOI f £ 0 denotes the sheaf o f germs o f С so lu tion s o f AQ(D)u = 0,

00С -c o n ve x ity can be s ta ted by the con d ition s

ANALYTIC CONVEXITY 27

Hj (S2,£0) = 0 , V j > 1

(b ) Let us rep la ce the space o f C°° functions on Q by the

space o f complex valued r ea l a n a ly t ic fu n ction s . We g e t a complex

A (D) p, A CD) A (D) p . '( 1 ) . AC£2) > . . . ?- 1 ---- > 0 .

We w i l l say th a t Я is a n a ly t ic a l ly convex i f i-s a c y c l ic .

The f i r s t question th a t a r is e s is th e fo llo w in g . Let J& be the

sh ea f o f germs o f complex valued r e a l a n a ly t ic so lu tion s o f Aq (D)u = 0 .

Then is f i - a n a ly t ic a l ly convex fo r (1 ) equ iva len t to------------------- ---------

H3 = 0 , V j H ?

The answer to th is question is a f f irm a t iv e and i t i s a consequence o f the

fo l lo w in g two statem ents:

(a ) (a n a ly t ic Po in care lemma). Let denote the shea f o f germs

o f complex valued r e a l a n a ly t ic fu n ction s . Then the sequence


J* A / iA . . . . V4/d_ >0X „ X X

0 0 о

is exact f o r e v e T y xQ € Rn .

(B ) For every open se t S C S n we have

V?{ü,¡£) = 0 , V j S 1

The second o f these statem ents i s a consequence o f a theorem o f Grauert:

"e v e ry open s e t in Rn has a fundamental system o f neighborhoods in Œn

which are open se ts o f holomorphy".

The f i r s t o f th ese statem ents w i l l be proved la t e r ; i t i s a so rt o f

Cauchy-Kovalewska theorem fo r overdeterm ined systems w ith constant c o e f f i c i

en ts .

Assuming th ese fa c ts fo r the moment, l e t us go back to the example o f

de G io rg i and P ic c in in i . We take Í2 = R and th ere , w ith resp ec t to the

coord in a tes z = x + i y and t , we con sider the complexes

_э_, , , 3z ,

0 ------> < 5 r ) > é C R 3) — >¿0R ) ------> o

and_Э_

, , 3z0 — ч>¿£0¡ n ) ------- >¿?(R ) — > 0

We have

н1 OR3 , 4 ) = 0 » h V 3- ^ ) Í О

as we do have the exact cohomology sequences

Э z

0 — > ^ C R 3) — > ¿CR3) — > áOR3) — > H1 QR3 , ^ ) — 5- 0_Э_

3z0 — > ^ 0R3) — > # (R 3) ---- > ^CR3) ---- > t^CR3, ^ ) — •> 0

3and as the equation on К


is always s o lv a b le f o r f 6 ¿ (R 3) w ith u e ¿QR3) (as the complex is

H ilb e r t and R 3 i s convex) but is not always s o lva b le f o r f e ^ (R 3)

w ith u £ <^(R3) . T h ere fo re we do have complexes (H ilb e r t ) which are С

convex f o r some fi open in Rn but a re not a n a ly t ic a l ly convex.

3. REDUCTION OF THE ANALYTIC CONVEXITY TO THE C” CONVEXITY

a) Let

b Po Ao (-D-) ¿ P 1 V D) Ad l (D ) ,¿pd ¿ ( f i ) > ¿ > ( f i ) — — > . . . — > Á ( f i ) -------------------> 0

be a H ilb e r t complex d e fin ed f o r a l l open se ts fi С Rn . We con sider

Rn с (En as the r e a l pa rt o f Cn and we con sider f o r Ù open in Cn

the same complex o f d i f f e r e n t ia l op era tors as a c tin g on fu n ction s o f 2n

va r ia b le s the r e a l and im aginary part o f the compelx coord ina tes in Cn .

~ V D) ¿ г ~ Ai m Ad - i (D) ¿ Pd ~(H) : £ (£2) - 2— ><£ (f i) - Í — > . . . > <S d (fi) -------- > 0

where D. = тЛ— , z . = x. + iy . being the complex coord ina tes in Cn .i dx^ j j j

A lso on Œn we con sider the D olbeau lt complex

m : C ° ° (f i ) Co l (fi) —L » Con(fi) ------ >0

where Co s (fi) denote the space o f С form o f type (o , s ) in fi and

where Э i s the e x t e r io r d i f fe r e n t ia t io n w ith resp ec t to antiholom orphic

lo c a l coo rd in a tes .

From these two complexes we form a th ird complex; the tensor product

(h ) e m •

Th is is (by d e f in i t io n ) the simple complex assoc ia ted to the double

complex К = {K r ,s , Э,А^ } where


К = С (fi) 0£ (x£ (fi)

= (C0 r ( f i ) ) Ps

and where the d i f f e r e n t ia l opera tors are induced by Э and A* .

This complex i s a complex o f d i f f e r e n t ia l operators w ith constant

c o e f f ic ie n t s ;r a th e r la r g e . I t looks l ik e th is :

p C ° ° (f i ) 2^ C ° ° ( f i ) \ ^ S «

oo ~ Po g ® С (fi)

1 ) " o i m i -С э ® ~ PС * ( f i ), 02 (fi) о

Theorem 1 . (a ) The ÇgZ$-suspended comp i ex (h) 0 is a H ilb e r t complex.

In p a r t ic u la r i t i s a c y c l ic on an open convex se ts fi . (b ) I f fi is

an open se t o f holomorphy in Œn the cohomology o f th at complex is n a tu ra lly

isom orphic to the cohomology o f the complex

vrñ - Л Л, Р 1 v è Ad - l (è Л,Р d . „(2J Г (j¿ фу У Г (í¿ f fi) • ' > • • • - Г (o¿ y&j U

ANALYTIC CONVEXITY 3 1

where Г(Й,<9) denotes the space o f holomorphic fu nctions on fi .

O u tlin e o f p r o o f . Accord ing to the prev iou s remarks abaut H ilb e r t

complexes, statem ent (a ) reduces to the fo llo w in g a lg e b ra ic v e r i f i c a t io n .

Let

¿ (¿ ) 4° ^ ¿ б Л - > o

be the considered suspended complex where (D) are m atrices w ith e n tr ie s

polynom ial in ( Zj = Xj x ) . I f ? = ï [Ç . . . ,Ç ] i sj n+j J J J

the polynom ial r in g in 2n va r ia b les , statem ent (a ) i s equ iva len t to the

exactness o f the sequence o f ¿P-homomorphisms

Jo 4 Л € ) Q , 4 2 (? ) 4 ( « q, Ч ю ^ оо — >? e_1 1 » ... I >9 2 — ?---># 0

This is an a lg e b ra ic v e r i f i c a t io n and i t is l e f t to the rea der. Statement

(b ) i s an immediate consequence o f the sp ec tra l sequence o f the double complex

{K r ’ S , F , A , } .

In p a r t ic u la r i f fi i s open convex then Q i s a domain o f holomorphy

and thus the sequence ( 2 ) must be exact.

C o ro lla r y . The a n a ly t ic complex ( 1 ^

¿?P° ( f i ) f i ) — >-/d (fi) ------ > 0

admits the Poincare (a n a ly t ic ) lemma.

Indeed le t x 0 G Г2 and le t feJ3^Ps (w ) (s ê 1) w h ere w is a con n ected n e igh b o r

hood o f x 0. Then f extends to a h o lom orp h ic function F d e fin ed in som e con nected

n e ighborhood w o f w in Cn. A s su m e that A sf = 0 then A $F = 0. W e can find

a convex neighborhood V o f xQ in Cn w ith V С w . There, there

P ,_ ,e x is t G holomorphic in V, G 6 T(V,¿9) such th at As _^ G - F . Re

s t r i c t in g to v n R n we g e t f o r g =G|y nRn>A x g = f .

(b ) L e t fi С Rn and l e t fi be an open se t o f holomorphy in С w ith

fi П ]Rn = fi .

L e t ¿ denote the sh ea f o f germs o f s o lu tion s o f the f i r s t operator

o f the suspended complex. We do have

Hj (fi , f 0) « Hj (fi , Г (п ,<Я * , A J

and thus, by r e s t r ic t io n a na tu ra l map

Hj ( f t . i 0) - Hj (fi , y 0)

One can then p rove the fo llo w in g .

Theorem 2. We have a na tu ra l isomorphism

Hj (fi , S J = -Ü Ï- > Hj (fi , I ) = H] (fi , Ю 0 ñnRn=n 0

which shows th a t the a n a ly t ic con vex ity o f fi i s the l im it o f the C°°

con vex ity o f the suspended complex.

(c ) What we are in te re s te d in are the cohomology groups H-1 (fi , . In -

stead o f suspending the g iven complex from Rn to Cn v ia the Dolbeaul.t

com plex,one can suspend the complex from Rn to Rn+* v ia a complex g iven ,

fo r in s ta n ce , by a s in g le d i f f e r e n t ia l op era tor

(<£} : ¿ ( f i ) ------ > 0 ,

fi open in ]Rn+* . What is requ ired from the p rev iou s con s id era tion s are

the fo l lo w in g con d it ion s :


(1 ) (H) в (.£ ) should be a H ilb e r t complex. Th is amounts to a

" t r a n s v e r s a l i t y " con d ition o f JC w ith resp ec t to Rn in to Rn+* (Rn

should be n o n -ch a ra c te r is t ic fo r L in Rn+* ) .

(2 ) The f i r s t op era to r Bq o f the suspended complex (H) 9 («£)

should be e l l i p t i c , i . e . BQu = 0 => u r e a l a n a ly t ic . Then one ob ta ins

a theorem o f the fo llo w in g type

■ ~ cr k i lim HJ ($2 , <fQ) = ® H3 (fi , &0)

n dR n=fi 1

where к i s the " tra n sv e rsa l o rd er o f the opera tor L on Rn " .

4. THE EXAMPLE OF DE GIORGI-PICCININI

-.3


We con sider Ж , where (X j, Vp X2) a re ca r te s ia n coord inates,

4 ~ 1 = X1 + 1 ^1 » z 2 = x 2 + 2imbedded in Í = I where 2 , = x, +

are complex coo rd in a tes.

■zOn R we con sider the complex

ЭSz.

¿ i QR ) ------> SpQR. ) -----> 0

then , the shea f o f germs o f a n a ly t ic so lu tion s o f the homogeneous

equation = 0 i s the shea f o f germs o f fu nctions which are r e a l

a n a ly t ic , holomorphic in and thus adm itting power s e r ie s expansions

<2 .

I f & i s the sh ea f o f germs o f holomorphic functions in С , the

7

r e s t r ic t io n map on the p o in ts o f R g iv e s a homomorphism

7

O ' , ------ > $ , - to p o lo g ic a l r e s t r ic t io n o f (У to R )R -5 К

which i s s u r jë c t iv e and has zero k e rn e l. Thus an isomorphism; th e re fo re

н1^ 3 Ж ) = h V 3 )о

Th is is noth ing e ls e but the statement o f Theorem 2 when we suspend the

3 2complex from R to (C v ia the complexЭ

GO : < £ («) ¿ (В ) ------ > О

At th is p o in t the statem ent o f de G io r g i-P ic c in in i amounts to show th at

h V 3 , Л ) / 0к

Now one can prove d i r e c t ly the fo llo w in g more comprehensive statem ent.

Theorem 3 . We have dim*. H1 (R3 , СУŒ 1R

Th is a lread y g iv e s more p re c is io n to the statement o f de G io r g i-P ic c in in i

in the sense th a t not on ly i s i t true t h a t , fo r in f in i t e ly many a n a ly t ic

fu n ction s f 6 -^(R3) , the equation = f i s not a n a ly t ic a l ly so lvab le ,- 1

but i t is tru e a ls o th a t the same happens f o r i n f in i t e ly many cohomology

c la sses (each one con ta in in g i n f i t e l y many fu n c t io n s ).

O u tlin e o f the p ro o f o f Theorem 3 . (a ) The p ro o f i s based on the fo llo w in g

2Lemma. Consider in С a compact reg ion К w ith non-empty in t e r io r d e fin ed

by the fo llo w in g in e q u a l i t ie s :

y j H , y 2 > 0 , H (z x , z 2) ¡S 0

where H is a p i u r i - harmonic fu n ction .

Let S be the pa rt o f ЭК on which У ]У 2 = 0 •

оLet р ^ , . . . » р ^ be d is t in c t po in ts in К and l e t us con sider the (2 ,1 )

34 ANDREOTTI and NACINOVICH

Э-c lo s ed form

where

( z 1 - z 1 (p i))d z 2- ( z 2- z 2 (p i ) ) d z 1

К = --------------------ñ--------------------T ô d z 1p i Cl zx“ z 1 Cpi ) I + |z2- z 2 (p i )| )

I f in some neighborhood V o f S (not con ta in in g the po in ts p^ )

we have x = where y is a (2 ,0 ) C°° form on V , then c^ = 0 fo r

1 < i < k .

(g ) From the exact sequence

0 ------> , --------------------> & ----> £>", ---- > 0Œ -Ró R ¿

2we deduce th a t (as С is a domain o f holomorphy)

h V 3 > (5 \ ) = H2 (Œ2-R3 ,& )R

2 2 3where ф denotes the fa m ily o f c losed subsets o f С contained in (E -R

1 2 ( y ) We can con struct on the f i r s t quadrant o f Ж (У р У 2) a function

У2 = M y x )

having the fo llo w in g p ro p e r t ie s :

k ( 0) = 1 , 0 < к * 1

к ' < О

к ■+• 0 as y j -*• “

such th a t , f o r every £ > 0 , <5>0 one can construct a reg ion K (e ,5 )

as in the lemma con tained in

|z2 | < e , Ix 21 < e and K (e ,6 ) n R 2 (y 1y 2) э { е у 2 s k f e y ^ - ô }

2( 6 ) We s e le c t a sequence o f p o in ts e R (y ^ ,y 2) w ith the

p ro p erty th a t f o r any e > 0 , e y 2 (pN) < kCey^Cp^j)) i f N S N (e ) is

s u f f i c i e n t ly la rg e Çy^CPj D "*• +00) •

(e ) We remark th a t & - Í22 the sh ea f o f germs o f holom orphic 2-form s.

We choose с.. e (E , с ?! 0 , 11 с.. I < <*> and con sider the d is tr ib u t io n N N 1 N'

N Pn

as an element o f H2 (Œ2 - R 3 , Í22) .

1 Using elem entary properties o f c on form a i m appings in the p lane o f th e Z i variable.

We have

36 ANDREOTTI and NACINOVICH

E сы 6 = 5 £ c „ КN PN N pN

(up to a non-zero c o n s ta n t).

I f

? 2then in some neighborhood V o f R in Œ we must have

£ с К = Э 0 N %

fo r some С °° form 0 o f type (2 ,0 ) in V .

Combining th is fa c t w ith ( 6) and the lemma, we g e t a c o n tra d ic t io n

and indeed the p ro o f th a t the space generated by the c la sses { î cB S }PN

is a lread y in f in i t e d im en siona l.

Remark 1. Another more conceptual p ro o f can be obta ined along the

fo llo w in g l in e s . From the statement a t the end o f Section 3, which

in th is case amounts to

h V 3, # o) = h V 3, ¿ ? 3) = lim H :(ñ ,<?)

R ñajR

1 3we see th a t a s u f f ic ie n t con d ition f o r H QR » « ^ ) to vanish is th a t

3 2R admits a fundamental system o f neighborhoods in Œ which are open

se ts o f holomorphy. Now one can show th a t th is con d ition is a ls o necessary .

T h e re fo re , the fa c t th a t H^fR2 » ^ 0) i* 0 can be concluded from the fa c t

Ъ 2th a t R does not admit a fundamental system o f neighborhood in Œ which

are open se ts o f holomorphy. Th is la s t statem ent is indeed a not to o d i f f i c u l t

e x e rc is e .

Remark 2 . The argument g iven above cannot be extended to the case o f

more than 2 v a r ia b le s . Th is is not su rp r is in g as one can e a s i ly show by

d ir e c t argument (using fla b b y res o lu tio n s o f the sheaf ¿£0 ) th a t fo r Í2

convex in 3Rn one has always

Hj (í2 , ¿ o ) = Н3 (П ,& Q) = 0 i f j S 2

I t i s worth n o t ic in g th a t i f fî С ]Rn is not convex we may w e ll have

H3 (£2 i 0 f o r j > 2 as can be seen on examples.

F o r the ca s e o f a s im p le c o m p lex con s is tin g o f a s in g le d if fe r e n t ia l

o p e ra to r , H ô rm a n d er has ob ta ined n e c e s sa ry and su ffic ie n t con d itions

fo r s o lv a b ility . W e hope to h ave c la r i f ie d con d itions in the lig h t o f the

fo r e g o in g .

BIBLIOGRAPHY

D E G IO R G I, E . , “ So lu tions analytiques des équations aux dérivées partielles à coe ffic ien ts constants” , Sém inaire

G ou laou ic-Schwartz, E co le Po ly techn iqu e (1 971 -1972 ), exposé 29.

H Ô R M A N D E R , L ., O n the existence o f rea l ana lytic solu tions o f partial d ifferen tia l equations w ith constant

coe ffic ien ts , Inv. Math. 21 (1 9 7 3 ) 1 5 1 -1 8 2 .

P IC C IN IN I , L ., N on -su ijec tiv ity o f th e C auchy-R iem ann operator on th e space o f the analytic functions in IRn,

genera lization to parabolic operators, Bull. U .M .I. N o .4 , V o l. 7 (1 9 7 3 ) 12—28.


COMPLEX TORI AND JACOBIANS

IA E A -S M R -1 8/24

M. CORNALBA*Department of Mathematics,University of California,Berkeley, California,United States of America

Abstract

C O M P L E X T O R I A N D JA C O B IA N S .

1. Generalities. 2. C oh om o lo gy o f X . 3. L in e bundles on com p lex to ri. 4. T h e Jacobian o f a curve.

5. T h e R iem ann-R och theorem fo r com p lex tori. 6. P ro jective embeddings. 7. M erom orph ic functions. 8. G eom etry

o f the theta-division.

This paper is meant to be an introduction to the geometry o f complex tori and to the theory o f theta-vanishing in Jacobians. The treatment o f complex tori is close to the point o f view o f R e f [ 3]. Some o f the arguments in Section 8 are adapted from Lewittes’ paper [2].

1. GENERALITIES

One o f the main goa ls o f t h is paper i s the study o f compact

connected complex L ie groups, o r , as they are u su a lly c a lle d , complex

t o r i . These are compact, connected complex m anifolds X w ith h o lo

morphic maps

( x , y ) -*■ xyx ,y € X

X X

s a t is fy in g the usual group axioms. They are very easy t o describe

e x p l i c i t l y , as a consequence o f

P ro p os it ion 1 .1 . A compact, connected, complex L ie group is ab e lia n .

To see th is consider the function

A / \ - 1 - 1 фх (у ) = хух у

When x is th e id e n t i t y , Фх (у ) equal t o th e id e n t ity fo r every у

T h ere fo re , when x is c lose t o the id e n t ity maps X in to a co -

* Present address: Is titu to M atem ático , Università d i Pavia, Pavia, Ita ly .

39

ord in a te patch. V ector-va lued holomorphic mappings on compact m anifolds

are constant, hence Фх (у ) i s equal t o the id e n t it y fo r every y . Th is

proves th a t X is commutative. From now on the product in X w i l l Ъе

w r itten a d d it iv e ly . We are now in a p o s it io n to d e s c r e e X. Let

фX + X

be, th e u n iversa l coverin g map. X i s an ab e lian , simply connected L ie

Ngroup, and is th e re fo re isomorphic to а С* . ф i s a group homomorphism

and i t s kern e l is a d is c re te subgroup o f o f maximal rank (a l a t t i c e ) ,

a fr e e abe lian group on 2N gen era to rs .

Any complex to rus is thus isomorphic t o the quotient o f a "by

a l a t t i c e .

Example 1 .2 . When X has dimensions 1 , i t is th e quotien t o f

С by the subgroup Л generated by two complex numbers th at

are l in e a r ly independent over B .

40 CORNALBA

N otice th a t have been numbered so th at the angle between them

is le s s than тт. Any o ther b as is o f Л w ith the above p roperty is o f

the form

= аш + bùjg ~ сшз_ + ^ 2

i s an in te g r a l m atrix w ith determinant +1 .

Two la t t i c e s Л,Л ' in С (o r C^) are sa id to be equ iva len t i f

th ey correspond to each other under a С- l in e a r transform ation . This

H H iamounts t o saying th a t С /Л and € /Л are isomorphic t o r i .

R eturning t o our example, we. may d iv id e by t o norm alize our

b as is Th is g ive s us a new la t t i c e which is equ iva len t to Л

and i s generated by 1 and ш = As a consequence o f the

assumptions, ш has p o s it iv e imaginary p a r t. I t i s now c le a r th a t

( 1 ,0) ) and ( 1 ,0) ' ) generate equ iva len t la t t ic e s i f and on ly i f

ao) + b0) = -----——со) + d

/ .

where \ c 4 /

is an in te g r a l m atrix w ith determinant 1. Thus the isomorphism

classes o f 1 -d im ensional t o r i correspond t o poin ts in the u pper-h a lf

plane modulo l in e a r fra c t io n a l transform ations w ith in te g ra l en tr ie s

and determinant 1 .

Remark 1 .3 . The add ition law on X is determined by th e complex

s tru c tu re , up t o tra n s la t io n . In fa c t , l e t x ' be another complex

torus and f a holomorphic mapping o f X in to x ' ca rry in g the id e n t ity

o f X in to the id e n t ity o f x ’ . Then f is a group homomorphism. This

can be shown by th e same method used t o prove P rop os ition 1 .1 . Consider

the function

f (x + y ) - f ( x ) - f ( y ) = ф( x , y )

Ф(0 ,у ) is id e n t ic a l ly z e ro , so fo r x c lose t o zero and every y ,

ф (х ,у ) l i e s in a coord ina te patch and must be a constant function o f

y . But ф (х ,0 ) = 0, so ф (х ,у ) vanishes id e n t ic a l ly .

2. COHOMOLOGY OF X

NLet X be the compact to ru s С /Л . Choose a basis

u ^ , . . . , u f o r Л. Then, from th e r e a l po in t o f v iew , X is the

COMPLEX TORI AND JACOBIANS 41

product o f the 2Я 1-spheres Eu /ïu^, i = l , . . . , 2 1 î . As a conse

quence, th e fundamental group (o r the f i r s t homology group) o f X

can he id e n t i f ie d v i th Л, and th e f i r s t cohomology group o f X v i th

Hom(A,£ )

M oreover, i t fo llo w s from repeated ap p lica tion s o f th e Künneth formula

th a t the cohomology a lgeb ra H *(X ,S ) i s ju s t th e e x te r io r a lgebra on

H ^ X .Z ) = Hom (A,Z). I f we introduce r e a l coord inates x ^ , . . . ^ ^ in

C* by the requirement

z = 2 x.u .i i

then th e d i f fe r e n t ia ls dx^,. . . , ar e t r a n s la t io n - in v a r ia n t , hence

tire pu ll-backs o f d i f fe r e n t ia ls on X th a t we w i l l denote by th e same

symbols. I t is then apparent th a t , under the deRham isomorphism, the

cohomology c lasses o f dx^, . . . , d x ^ correspond t o th e dual b as is o f

u ^ , . . . ^ ^ ; in fa c t , th is simply means that

42 CORNALBA

U

dx = 6 . . j l j

M oreover, s ince cup-product and e x te r io r m u lt ip lic a t io n o f forms

correspond t o each other under the deRham isomorphism, H °(X ,C ) is

generated f r e e ly by th e in te g ra l c lasses

dx. ~ . . . Ad i , , i., < i_ < . . . < i X1 "q q

In other term s, н 'Ч х .С ) is isomorphic t o th e group o f t r a n s la

t io n - in v a r ia n t q-forms on C^.

So fa r we have not taken in to account the complex structure o f

. I f we do so, and introduce complex coord inates z ,

another b as is fo r the in va r ia n t ш-forms is g iven by forms o f th e kind:

d z . л л d z. л dz . л . . . л d z .X1 Xr j l Jq

r+q = m , < . . . < i r , < . . . < j

T h erefo re any in varian t m-form ф can be w r itten un iquely as

ф = E ф1-’*r+q=m

where фГ , ~ has pure type ( r , q ) . This means th a t th e 111th complex

cohomology group o f X has a d ire c t sum decomposition

rf"(x,c) = £ нг,((x)r+q=m

where H1"’ ” is the group o f th e cohomology c lasses o f c losed forms

o f pure type ( r , q ) , and as a consequence:

Hr ,q = ^qTF

This i s the s o -c a lle d Hodge decom position, which is v a l id fo r a l l

compact Kahler m an ifo ld s , not on ly complex t o r i [ 4 ] .

Г QTo complete the p ic tu re we on ly have t o id e n t i fy each H

w ith the cohomology group H^(X,fir ) , where fir stands fo r th e sheaf

o f holomorphic r-form s on X. The id e n t i f ic a t io n is made v ia the

Dolbeault isomorphism:

_q/ . r , _ 5 -c losed (r ,q )- fo rm s on X’ 3 -exact (r ,q )- fo rm s on X

I t i s c le a r th a t every c lass in viewed as an in va r ian t ( r , q ) -

form , g iv e s a c lass in н 'Чх.Я1' ) . That th is homomorphism is in je c t iv e

can be seen as fo llo w s . Choose a t r a n s la t io n - in v a r ia n t measure dp

on C* ; th is measure can be viewed as be in g induced by a measure on

X th a t w i l l be denoted by th e same symbol. We may assume th a t dji

has been norm alized so th a t X has volume one. Then t o each form ф

on X we may assoc ia te an in va r ia n t one, 1 (ф ), by averaging:


44 CORNALBA

1(ф) = / Т*(ф)<1у(е )гКГУ »

where is tra n s la t io n by g . The norm aliza tion assumption ensures

th at I С ф ) = ф when ф is in va r ia n t. I t i s a lso c le a r th a t I

conmutes w ith e x te r io r d i f fe r e n t ia t io n , and th e re fo re by averaging

a 5-exact form we get ze ro . But then two in va r ian t forms cannot

d i f f e r by a 5-exact one unless they are the same.

The s u r je c t iv i t y statement is s l ig h t ly more su b tle . F ir s t o f a l l

remark th a t , s ince X is a group, i t s tangent bundle T and a l l the

{ \assoc ia ted bundles, in p a r t ic u la r ( T* ) , are t r i v i a l . I t is th e re fo re

s u f f ic ie n t t o prove our s u r je c t iv i t y statement fo r Н°(Х,СЙ, where &

stands f o r the sheaf o f germs o f holomorphic functions on X. Let

a = E э2T1 3z . 3 z . 1 1 1

Nbe th e Laplace opera tor on С (o r X ). A operates on forms

X a. . . . dz. л . . . л dz. л dz^ . . . л dz.V " V l - 3q *1 Xr h ¿q

by opera ting on each c o e f f ic ie n t sepa ra te ly .

We cla im th a t , fo r any 5 -c losed (0,q.) form ф, th e form

1 ( Ф ) - Ф

is 5-exact o r , in o th er term s, th at a 5 -c losed form whose average

i s zero is 5 -exact. This w i l l fo llo w from th e a u x ilia ry

Lemma 2 .1 . Let £ 2H x 2H symmetric -positive m a tr ix .

a2W rite E = £c. -------i j Эх ^Эх

and l e t a t e a smooth function on X such th a t I ( a ) = 0. Then th ere

is a unique smooth 6 such th a t Eg = a , 1 ( g ) = 0.

We postpone the p ro o f o f (2 .1 ) t o a la t e r tim e and t r y t o conclude

from i t . We l e t

ф = E ф. . dz. л л dz.■ * i , . . . i l . i i , <. . .< i 1 q 1 a1 q

Ъе а Э-c lo s ed (0 ,q )- fo rm such th a t I (ф) = 0. Lemma 2.1 app lies t o

E = Д and we may w r ite ф = Дф fo r some (0 ,q )- fo rm


ф = Ф ■ • dz. » . . . d z .i ,< . . .< 1 1 с 1 q

1 1

ф is Э-c lo s e d ; in fa c t ДЭф = 5Дф = 0 and 1(Эф) = 0, hence Зф = 0

Ъу th e uniqueness part o f Lemma 2 .1 . That ф is Э-c lo sed means th a t

( 2 . 2 ) 0 = E ( - l )'5 -1

For any permutation a o f th e numbers l , 2 , . . . , r , and any form

2 a * d z . /ч . . . a dzJl " ‘ °r J1

se t a = Sgn(a) ao ( l ) * * o ( r ) 'Sl ‘ " Jr

where Sgn (a ) stands fo r th e s ign atu re o f a . Now de fin e

46 CORNALBA

i n . . . !1 q J i 1 j q

= г д - ц Д - 1- - { ! î

Э2i Ъг.Ъг. ^ i . . . i _ ^ i . . . i

i i 1 q 1 q

as fo llo w s from ( 2 . 2 ) .

I t remains to prove Lemma 2 .1 . a has a Fourier s e r ie s that

converges abso lu te ly to ge th er with a l l o f i t s term by term d e r iv a t iv e s :

2ir>^ï (S i .x . )i i i

a ~ 2 a, v, e 1 " • 2N

where the h are in te g e rs . The assumption l ( a ) = 0 simply means

that cIq . . . q = 0.. Consider another Fourier s e r ie s w ith undetermined

c o e f f ic ie n ts and no constant term:

2Tr/^(Sh.x. )

6 = 4 h eV • 2H

and compute E$ by fo rm a lly d i f fe r e n t ia t in g term by term . The r e su lt

is

2n/=l(2b x )E 6 = - U / E В h ( 2 с h h^)e

h , . . .h _ „ 1 ' ’ 2N J 31 2N

2 с h h. i s always p o s it iv e by hypoth esis , th e re fo re J,k

2ir/^l(2h.x. )

is the -unique formal Fourier series such that 1(6) = 0, ES = a

(fo rm a lly ). I t fo llow s e a s ily from the convergence properties o f a

that 6 converges absolutely together with a l l o f i t s term-by-term

d e r iva tiv es .

3. LINE BUNDLES ON COMPLEX TORI

Let L be a lin e bundle on an И-dimensional complex torus X = С^/Л.N * M

Denote by тг the quotient map С -+ X. ту L is e lin e bundle on С .

But:

Proposition 3 .1 . Any lin e bundle ( any vector bundle. fo r that

matter) on isomorphic to a t r i v i a l one.

This is an easy consequence o f Cousin’ s theorem [ i ] ;

COMPLEX TORI AND JACOBIANS 4 7

Proposition 3 .2 . hW . o ) 0 .

To deduce (3 .1 ) from (3 .2 ) choose a covering {и Л o f by

open Sets on which L is t r i v i a l ; we may and w i l l suppose that the

U^'s are b a lls (o r , i f we wanted to do the analogue o f th is on

■another complex manifold, we would want U. П U to be simply connected1 J

fo r any choice o f i and j ) . Let f be tran s ition functions fo r L.

Because o f the simply-connectedness o f U. П U. we may w riteJ

2 tt/^I y . .f = e ^i j

and the cocycle condition = f . , implies that1J 1 л

YiJ + YJk = * ik + nijk

where the n ... 's are integers and sa tis fy the cocycle condition l j K

n „ - п . . + n. . - n. = 0jke ike î j e i jk

Therefore {n. } determines an in tegra l cohomology class o f degree 2,i j k

_N 2 Nthe Chern class o f L . In the sp ec ific case o f С , H (C ,Z) = 0,

there fore {n . } is a coboundarv and we mav modify the Y . , 's inl j k ' i j

such a way that y . . is a cocvcle. But then Y . . is a coboundarv'iD i j

by (3 -2 ), i . e . we may w rite

Yi j = ' Si

Ztî/ ^ T Ç -2tî/^T Ç . f . . = e J e х

and L is t r i v i a l .

In more fancy language, the above argument means that on any

cam-plex manifold M there is an exact sequence o f sheaves:

48 CORNALBA

e 2тт/Л *0 -*■ z -*■ © ---------*■ 6 0

where S stands fo r the sheaf o f non-vanishing holomorphic functions,

whence a long exact sequence

. . . - * ■ Н1 (М,Ж) -+■ H1 (M,C?) -*■ ^ ( M . d ’ j - í Е ^ М Д )

1 *The group H (M,C ) is the group o f isomorphism classes o f holomorphic

lin e bundles on M, and the coboundary map с associates to each lin e

bundle L i t s Chern class c (L ). When M = , Н^(МД) = H (M,£?) = 0,

so Н'Чм,© ) = 1. In general, two lin e bundles that have the same

Chem class d i f fe r by an element o f H1 (M,6)/H1(M ,Z), that is , a lin e

bundle with transition functions f (with respect to a suitable

covering) o f the form

2 n / -i у. f . . = e lJ

Yi j + = Yik

For la te r app lications, i t is in teres tin g to find an e x p lic it formula

fo r the deRham representative o f the Chern class o f a l in e bundle.

Let M Ъе a conçlex manifold, and L a lin e bundle on M with

tran s it ion functions f . , with respect to the covering ÍU .}.10 1

Suppose that U. П и is simply connected and w rite i J

2ïïÆ l y . ,f . . = e

10


The Chern cocycle {n } is the Cech coboundary o f the cochain l j l t

{ y . , } ; applying ex te rio r d iffe ren tia t io n to the la t t e r g ives a 1 J

1-cocycle o f holomorphic 1-forms |2~ 7^Т * i j } * ^si nS a

p a rtit ion o f unity we may w rite

d l o S f ±J = - ш. in U, П u,J i i J

fo r some С 1-forms {шЛ on {lL } . A deRham representative o f c (L )

is the global 2-form which is equal to <3uk in IK fo r every i .

This makes sense since

du. = du in U. П u.i j i J

Suppose now that L has a hermitian metric. Let be the

squared length o f the section o f L on that corresponds to the

constant function 1 under the given loca l t r iv ia l iz a t io n s . Then

a . = I f . . I 2 a. j 1 iJ 1 i

How 2i b d lo« fiJ = Ш З ? lo* lfi /

“ а т / л lo g “ j - 3 1os a i^

hence a deRham representative o f c (L ) is the global ( l , l ) - fo r m with

lo c a l expression

5 Í7 7 5 s lo g a i

To conclude, remark that the cocycle j d log f _ | defines

a class in H ^ ÍM ,^ ), the so-ca lled Atiyah Chern class o f L. On a

Kahler manifold, under the Hodge decomposition th is corresponds to the

usual Chern class o f L.

How we return to our o rig in a l problem. We noticed that ir*L

Ф Nis t r i v i a l . Choose one sp ec ific isomorphism ti*L -+■ С x C. Any other

t r iv ia l iz a t io n o f ir*L is obtained by composing ф and an automorphism

No f C x С, i . e . m u ltip lication by a nowhere vanishing holomorphic

N Nfunction on С . L can be viewed as the quotient o f С x С modulo

the id en tifica tion s

z G (П

(z ,Ç ) = (z+ u ^ u(z )-Ç ) Ç e С

U e A

where the f are nowhere vanishing holomorphic functions sa tis fy in g

the cocycle conditions

(3 .3 ) f u(z + v )fv (z ) = f u+v(z )

M u ltip lication by a nowhere vanishing function g (z ) has the e ffe c t

o f rep lacing the cocycle í f Î with the new cocycle

f (z )g (z + u )g (z ) 1

We now want to describe e x p lic it ly a l l the lin e bundles on X, i . e .

we want to put the cocycle i n' standard form.

We f i r s t try functions f that are exponentials o f lin ear

functions

50 CORNALBA

where a is a lin ea r form and Ъ a nimber. The cocycle condition u u

(3 .3 ) translates in to

(3 .1») au^z ) + av z ) = au + v ^


(3 .5 ) a (v ) + Ъ + Ъ = Ъ (mod 7L)u u v u+v

(3.!*) enables us to extend ац( 2) Ъу lin ea r ity to a b ilin ea r form

A (z,w ) such that au( z ) = A (z ,u ). A (z,w ) is С- lin ea r in the f i r s t

variab le and B -lin ea r only in the second. Interchanging the ro les

o f u and v in ( 3 .5 ) implies th a t, fo r any u ,v £ Л,

( 3 .6 ) A (u ,v ) - A (v ,u ) £ 1

Set E (u ,v ) = A (u ,v ) - A (v ,u )

°u = \ - 2 A (u ’ u)

Then formula (3 .5 ) translates in to :

с + c = с . + 77 E(u ,v) (mod Z )u v u+v 2 ’

This shows that Im(c ) is add itive in u and therefore extends uN

to an В-lin ea r forE on С . Therefore, a fte r m u ltip lication by the

exponential o f a suitable С-lin ea r form, we may suppose that с is

rea l fo r any u. I f we set

2ïï/^E с

p(u ) = e u

the fo llow ing hold:

IP (u )j = 1

(3 .7 )

p (u )p (v ) = p(u+v) E(u>v ) _ +p(u+v )

In c id en ta lly , i t is c lear tha t, fo r any given antisymmetric in tegra l

E , numbers p(u ) v ith the above properties ex is t.

At th is point our cocycle { f^J has the form:

2 т г / = Г [A (z ,u)+^-A(u,u) ].f u = p (u )e

52 CORNALBA

UWe may s t i l l change the t r iv ia l iz a t io n o f С x с by m ultiplying by

/ \ 2 - n S - l £ ( z , z ) ( z ) = e *

where £ (z ,w ) is a symmetric C -bilinear form. The e f fe c t o f th is

is to replace f with:

M z i g U + u J g i z ) ' 1 = f u ( z ) ^ ^ [ a C i a . u ) * C ( u , u ) ]

Notice th a t, by (3 .6 ), E (z ,v ) is r e a l, and thèrefore the

imaginary part o f A (z,w ) is symmetric, hence:

E ( / ô . z , i ^ l w ) = A (/ ^ l z , / ^ lw ) - A ( v 1 w ,/^1z )

= / 3 [A (a ,/ 3 .v ) - A (w ,/^ lz )]

= i^ l[A (/ ^ lw ,z ) - A (/^ lz ,w )] = E(z,w )

I t fo llow s that E (z,*^ lw ) is a rea l symmetric form and that

H(z,w) = -E (z,/^lw ) + E (z,w )

is hermitian. Moreover

2^==-[H(z,v) - H (w,z) ] = E (z ,v )


Th ere fo re we may take £ t o Ъе

■£ (z ,w) = H (z,w ) - i A ( z ,w )

a n d r e d u c e o u r c o c y c l e fo r m

7TE(z , u ) + ^ a ( u , u )(3 .8 ) f = d ( u ) e

u

For any herm itian H whose imaginary part E is in te g ra l on

Л x Л and any p such th a t (3 .7 ) h o lds, we w i l l d e fin e L (H ,p ) t o

he the l in e bundle on X g iven Ъу the cocycle (3 .8 ) . Note th a t

L(H1 ,P1 ) ® L (H 2 ,p2 ) = Ь(НХ + H2 ,P1P2

The main r e s u lt o f th is sect ion i s the

Theorem 3 .9 . ( Appell-Humbert)

( i ) Any l in e bundle on X is o f the form L (H ,p ).

( i i ) L (H ^,p^) and L(H2 ,p2 ) are isomorphic i f and on ly i f

= h2 , P l = p2 .

( i i i ) E = Im (H) is_ th e Chern c lass o f L (H ,p ) .

H otice th at th e statement o f (3 .9 ) can Ъе d iv id ed in two p a rts :

( i i i ) and the fa c t that th e group H^fXjCO/H^iX^Z) is isomorphic

fo r the group o f characters

Л S {z € C j zI = 1 }

Y/e w lU prove them in th is order.

NA m etric on С x С th a t is in va rian t under th e action

ttH ( z , u )+ j H (u , u )( z ,£ ) -*■ (z+ u ,p (u ) e О

is g iven Ъу

| (z ,Ç)|2 = e^ H ( z , 2 )| ç |2

Th is m etric induces one on L (H ,p ) which can Ъе used t o compute

c (L (H ,p ) ) .

I f we w r ite H (z,w ) = Ih ^ z^w ^ , the Chern form o f th e above

m etric is

1 * „ . - ttH ( z , z ) 1 ... .Э Э lo g e = ¿ j - Ih . .dz. . dZj

which corresponds e x a c tly t o E.

We now prove th e remaining part o f th e theorem o f Appell-Humbert.

Let о) Ъе a tra n s la t io n - in v a r ia n t (0 , l ) - fo r m on C^. ш corresponds

to a c lass in which in turn determines a l in e bundle on X

w ith zero Chern c la ss . We claim th at th is l in e bundle is L (0 ,p ) ,

where

(3 .10 ) p (u ) = e2lr>/ " (a) + 5 )

Th is is a s tra igh tfo rw ard ap p lica tion o f the e x p l ic i t formulae th at

g iv e the Dolbeault isomorphism and is l e f t to the reader. That p (u )

is a character fo llo w s from the fa c t th a t w i s t r a n s la t io n - in v a r ia n t .

Granting (3 .1 0 ), i t remains to prove ( i i ) when H = 0. But i t w i l l

be shown in Section 5 th a t i f p ф 1 , then L (0 ,p ) has no nonzero

section s and th e re fo re is not t r i v i a l . This concludes th e p ro o f.

The section s o f L (H ,p ) correspond t o functions on С which

s a t is fy the fu n ction a l equation

ttH(z , u )+^H(u , u )6 (z+u ) = p (u ) e e ( z )

54 CORNALBA

These are c a l le d th e ta -fu n c tio n s r e la t iv e to th e herm itian form H

and the m u lt ip lic a to r p. Among our goals in the next section w i l l

Ъе th a t o f determ ining the number o f l in e a r ly independent th eta -fu n ction s

r e la t iv e t o a f ix e d quadratic form and m u lt ip lic a to r ( Riemann-Roch

p rob lem ).

Remark 3.11. We want t o e x p l i c i t l y describe the l in e bundle

T *L (H ,p )

where T denotes t ra n s la t io n by a on X. I t is c le a r th a t i f 0 a

is a sec t ion o f L (H ,p ) (we are assuming here th a t L (H ,p ) has

s e c t io n s , but the argument is ju s t th e same i f th is is not the c a s e ),

viewed as a th e ta - fu n c tio n , then 0 (z+ a ) represen ts a section o f

T^LÍH jp) and s a t is f ie s the fu n ction a l equation

ttH (z , u ) + | h ( u , u ) + irH (a ,u )6 (z+a+u) = p (u )e 0 (z+a )

We are allow ed to m u ltip ly 8 (z+a ) by a never vanish ing fu n ction , fo r

example g^HÍZja)^ т^еп B (z ) = e11 2 , a 0 (z+a ) represen ts a section £

o f TaL (H ,p ) and s a t is f ie s th e fu n c tio n a l equation


О„ Г Т f t \ тгК ( z ,u)+^H (u ,u )A r\ t \ / \ 2тгv— 1 E ( a , u ) 2 a t \0 (z+u ) = p ( u ) e ’ e 8 ( z )

T h erefo re i t i s a th e ta function fo r th e herm itian form H and the

. . . . . . 2тг/^1 E (a ,u )m u lt ip lic a to r p (u j e .

Let us n o tic e th at i f a ,b are po in ts o f X then

T *L (H ,p ) ® T *L (H ,p ) = T *+tiL (H ,p ) ® L (H ,p )

This fo llo w s by comparing herm itian forms and m u lt ip l ic a to r s . I t is

c le a r th at the herm itian forms o f th e two sides are th e same. As fo r

m u lt ip lic a to rs , on the le ft-h a n d s id e we have

, v 2-k/^ Ï E (a ,u ) / v 2 ttÆ E (b ,u ) , >2 2тг/ЛГ E (a+b,u ) p lu ) e p (u ) e = p (u ) e

56 CORNALBA

as on th e righ t-hand s id e . The above formula means th at the mapping

a -*• T * L ( H , o ) ® L ( - H , p - 1 )

o f X in to the group o f l in e bundles on X w ith zero Chem c lass is

a group homomorphism. The kern el o f th is homomorphism is th e set o f

the c lasses modulo Л o f a l l those a such th at E (a ,u ) G Z fo r

any u G Л. I f H is non-degenerate (p o s i t iv e , fo r exam ple), i . e .

i f E is non-degenerate, th e kern el is f i n i t e and the mapping

a + H ( H , p ) K l l - H . p ' 1 )

is on to , fo r dimension reasons; in fa c t the se t o f isomorphism classes

o f l in e bundles w ith zero Chern c lass is th e complex torus H^iX.SJ/H^tX,

which has the same dimension as X.

4. THE JACOBIAN OF A CURVE

Let С be a smooth compact Riemann surface (a cu rve )■ Let g

be th e genus o f С. . К = Кc w i l l denote the canonical bundle o f C,

i . e . the l in e bundle whose associa ted sheaf is th e sheaf o f

holomorphic 1-forms on C. In what fo llo w s we w i l l f r e e ly use the

language o f d iv is o rs and l in e bundles associa ted to them; more d e ta ils

on th is formalism w i l l be g iven in Section 7. We w i l l w r ite hx { , )

fo r the dimension o f Нх ( , ) , and (ÿ (L ) fo r th e sheaf o f holomorphic

section s o f th e l in e bundle L . Let D = Sn.p. be a d iv is o r on С1 * 1

(p^ e C ). We w i l l denote by L^ th e l in e bundle assoc ia ted to D and

se t

deg( Lp ) = deg(D ) =

Th is degree depends on ly on th e l in e a r equ iva lence c lass o f D. From

th e th eo ry o f curves we w i l l assume (a t le a s t ) th e fo llo w in g fa c ts :

(a ) Every l in e bundle on С is o f th e form fo r some d iv is o r D.

(b ) (Riemann-Roch fo r curves)

( i ) h ° (C ,L ) - hX( С,L ) = X (C ,L ) = d eg (L ) + 1 - g

( i i ) (D u a lity ) IT*"(C,L ^ ® K) is dual t o H ^ (C ,L ), th e p a ir in g

being th e cup product : *

*

H ^ C .L - 1 ® K ) x H °(C ,L ) -*• H ^C .K ) = С

(с ) The to p o lo g ica l s tructu re o f С as a sphere w ith g handles

attached.

N otice th a t from th e short exact sequence

a -j0 C -*($->- -+ 0

th ere fo llo w s a short exact sequence

0 ■+ H °(C ,K ) ■+■ H ^C .C ) -*• ff^C.C?) 0

which n a tu ra lly s p l i t s (h1 ( c ,c5) can be id e n t i f ie d w ith H^(C ,K ).

Th is is th e sim plest ease o f th e Hodge decomposition.

I f d eg (L ) < 0 , L has no s e c t io n s , because th e number o f

zeroes o f a section o f a l in e bundle is equal to the degree o f the

l in e bundle i t s e l f . T h erefo re Riemann-Roch t e l l s us th e dimension

o f H ^ C .L ). D ually , H ^ C .L ) = 0 when d eg (L ) > deg(K ) = 2g-2.

So we know h ° (C ,L ) (o r , which is th e same, h ^ C .L ) ) , when d eg (L )

i s not between 0 and 2g-2 . When d eg (L ) = 0, two cases are p o s s ib le ;

e ith e r L i s t r i v i a l o r i t has no section s ( f o r a n o n - t r iv ia l section

o f L cannot have z e r o e s ). D ually , when d eg (L ) = 2g-2 , e ith e r L = К

or h ^ C .L ) = 0 ( i . e . h ° (C ,L ) = g - l ) .

There remains the range o f values 0 < d eg (L ) < 2g-2 .

F ir s t o f a l l , n o tice th a t i f r is a poin t o f C, then

58 CORNALBA

h°(C,Lj^t r ) < h ° (C ,L D) + 1

This fo llo w s from the cohomology exact sequence o f

о - с к у - е о ^ г) - cr -o

where С stands fo r the sheaf which is concentrated at r and r

has s ta lk С at r . How assume th at th ere i s on С a l in e bundle

o f degree 1 w ith two l in e a r ly independent sections Sq ,s^ ; each one

o f th ese has on ly one ze ro . Moreover, never vanish sim ultan

eou s ly ; in fa c t , i f th is were th e case, s } / s 0 would Ъе a holomorphic

function on C, hence a constant. I t fo llo w s th a t sq »sx d e fin e a

mapping o f С in to P 1 (th e "meromorphic fu n ction " s^/s^ ). More

d e ta i ls on th is construction can Ъе found in Section 6 . The ass'umption

d eg (L ) = 1 im p lies th a t th is mapping is 1 -1 , hence an isomorphism. Th is

shows th a t g = 0 .

Pu ttin g th e above remarks to g e th e r , and using d u a lity , we see th a t

fo r a curve o f p o s it iv e genus th e p oss ib le dimensions fo r H^(C ,L)

are as shown in Fig. 1.

We w i l l see la t e r th a t "g e n e r ic a l ly " th e dimension o f H °(C ,L ) i s

minimum p o s s ib le , g iven th e degree o f L.

From now on С w i l l Ъе a curve o f p o s it iv e genus. We can

now in troduce th e Jacobian v a r ie t y o f C, w r it te n J (C ) ; we w i l l

do th is in two, equ iva len t ways. Set

J (C ) = group o f (equ iva lence c lasses o f ) l in e bundles o f

degree 0 on С = ffL(C ,6 )/H1 (C ,Z ).


F IG . 1. Possib le d im ensions f o r f o r a curve o f

p os itive genus.

Since H^ÍCjZ) i s a l a t t i c e in H ^(C ,© ), J (C ) is a complex torus

Choose a base p o in t q £ C. Every po in t o f J (C ) i s o f th e form

c lass o f (L r l + . . . + r -gq

as fo llo w s from the ta b le o f F igure 1. There is a holomorphic

mapping (o r ф, fo r sh o rt) o f С in to J (C ), de fin ed as

fo llo w s :

ф ( r ) = c lass o f (L ) q r - q

Remark ^ ■1. Choose a b as is fo r th e holomorphic

d i f fe r e n t ia ls on C. Then J (C ) = IT*'( С, P )/Н"*-( С ,Z ) i s isomorphic

Rt o С modulo th e l a t t i c e con s is tin g o f a l l the vectors o f th e form

........ /Y“ g ) Y S H ^ C .Z )

the isomorphism being induced by

Н1 ( С , 0 ) Э ш ’-*-(/ Ш ш л ш )с 1 c g

We w i l l need th e fo llo w in g vers ion o f A b e l 's theorem .

60 CORNALBA

Theorem к .2. With th e above id e n t i f ic a t io n

ф ( r ) = c lass o f ( / V , — , /гш ) g •'q 1 * ' q g '

(independent o f th e path used to jo in r and q ) , o r , e q u iv a le n t ly ,

the isomorphism between j ( c ) and the torus constructed above i s :

c lass o f (L ) - » • . . .r i + . . . + r g-g ?

r l r 2 r i r o...->■ class o f ( f 0), + . . .+/ o)./ o) + . . .+/ то ); q 1 ' q 1 J q g 'о q

P ro o f: Consider a l in e bundle o f the form L . For s im n lic ity ,-------- r -q

we s h a ll assume th a t r and q are the po in ts 1 and 0 in a s u it

ab le coord inate patch P w ith coord inate z . Consider a covering

1L = {U ,V } o f С con s is tin g o f th e complement U o f th e segment

jo in in g r and q , and- o f a small neighborhood V o f 0 . R e la t iv e

t o J i, a t r a n s it io n function f o r L is /— . This function hasr -q \z-l/

a s in g le -va lu ed logarithm in U l"l V; upon crossin g a from th e low er

h a lf-p la n e to th e upper h a lf-p la n e , log^ -^ j- j increases by 2tt/^1 .

A Eolbeault rep resen ta tiv e o f th e cohomology c lass o f th e 1 -cocyc le

I o S ^ y ) Can 136 ог> а1пей as fo llo w s . Let X be a smooth function

which is equal t o 1 in a neighborhood o f V in P and vanishes out

s ide a neighborhood o f V in P. Then th e cohomology c lass o f

2тг/Д l o e ( r i ) is represen ted by

2i k 3(x logf e ) ) = “Now

/ c “ ” “ i = S S T S T I с 4 х l o s f â K )

= 2^ r [ / o l0 g f ë j “ i - /о lo g f t ) “ i l = / о Чlow er

where "upper" and "lo w e r" denote upper and low er determ ination o f

lo g along a . This concludes th e p ro o f o f (I t .2 ).


Remark it.3 . Ф is an embedding. To begin w ith , l e t \ls show th a t i t

i s 1 -1 . I f not, th ere are two d is t in c t poin t r ]_»r 2 such th at

r^ -q are l in e a r ly eq u iva len t, i . e . r^ and r^ are l in e a r ly equ iva len t.

But th is means th a t h^(C ,L ) = 2, which is absurd. I t remains to showr l

th a t ф has non-zero d i f f e r e n t ia l everywhere, i . e . th a t g iven a poin t

r o f C, th ere is a g lo b a l holomorphic d i f f e r e n t ia l th a t does not vanish

at r . I f th is is not tru e , the mapping

H°(C,K ® L_r ) - H °(C ,K )

is an isomorphism, o r , d u a lly ,

Н^С.вО -*■ H ^ C .L )

is an isomorphism. By Riemann-Roch t h is im p lies th a t h °(C ,L ) = 2 ,Г

absurd.

Remark U.i*. J (c ) has one more p ie ce o f s tru ctu re . On H^(C,©)

th ere is a p o s it iv e d e f in it e herm itian form:

whose imaginary part r e s t r ic t s t o th e n ega tive o f th e in te rs e c t io n

p a ir in g on H ^ iC .Z ). For any m u lt ip lic a to r p fo r H, L (H ,p ) is a

l in e bundle on J (C ). We sh a ll see la t e r th at the v e c to r space o f i t s

section s is one-dim ensional and we w i l l e x p l i c i t l y w r ite down a gen era tor,

"th e " Riemann th e ta function o f С (w e ll d e fin ed up t o t r a n s la t io n ).

5. THE RIEMANN-ROCH THEOREM FOR COMPLEX TORI

ИLet X = € /Л Ъе a complex to ru s . Suppose th a t th ere e x is ts a

p o s it iv e d e f in it e herm itian form H on whose imaginary part 5

is in t e g r a l on A * Л ("u s u a lly " th ere is no_ such form, as w i l l Ъе made

c le a r l a t e r ) . Let p Ъе any m u lt ip lic a to r r e la t iv e t o H. The datum

o f such a p o s it iv e H and a m u lt ip lic a to r p is c a lle d a p o la r iza t io n

o f X. We want t o compute th e dimension o f th e space o f sections o f

L (H ,p ).

We v i 11 use th e fo llo w in g

Lemma 5*1 » I t i s p oss ib le to choose & bas is

fo r Л such th a t

62 CORN ALBA

E(u ,u , ) = E (u .,u ) = 01 J 1 J

-E (u . ,ù . ) = E(Ùr u . ) = d .ô . j

where th e d^ are p o s it iv e in tege rs and d^|d^|. . . |d . In o ther term s,

th e m atrix o f t r e la t iv e to th e above basis can be w r itten in b lock

form as

О -A

Д 0

P ro o f . Choose a b as is fo r Л and l e t Q = (q . . . ) be th e m atrix” ~ 1J

o f E r e la t iv e t o th is b as is . Q is skew-symmetric. A change o f

bas is changes Q in to MQ^M, where M is a unimodular in te g ra l

m atrix . MQ*"M is sa id t o be unimodularly congruent to Q. We must

show th a t any non-degenerate, skew-symmetric Q is unimodularly

congruent t o a m atrix o f the form Elementary unimodular

congruences are:

( i ) In terchanging two rows and th e two corresponding columns.

This is accomplished Ъу choosing an M o f th e form


( i i ) Changing the sign o f a row and the corresponding column.

M has t o be o f th e form

( i i i ) Replacing a row by th a t same row plus к tim es another

(k any in te g e r ) then doing th e same fo r columns. The m atrix M is

We w i l l show th a t Q is congruent t o a matrix:

Repeated ap p lica tion s o f ( i ) then show th at th is m atrix is congruent

to

С ' )

The p ro o f is by induction on N. The case N = 2 is c le a r .

Repeated a p p lica tion s o f ( i ) enable us t o assume th a t is not zero

and is th e sm allest in absolute va lue among the non-zero e n tr ie s o f

m atrices congrujent t o Q. Repeated ap p lica tion s o f ( i i i ) then

enable us t o rep lace q ^ s • - • v i th th e ir residues modulo q^2 -

These are zero by the very cho ice o f Ч .^ - ® is now o f ^опп

64 CORNALBA

Applying ( i ) t o the f i r s t tv o rows and columns g ive s a Q o f the

form

Repteated a p p lica tion s o f ( i i i ) enable us to assume th a t

a lso vanish . Applying ru le ( i i ) , i f necessary, we

may assume th a t q g is n ega tive and set d^ = -д 1?. How inductic

proves (5 -1 ) except fo r one p o in t. We must show th a t d^ d iv id es

dg. Suppose th a t th is is not the case. Using ( i i i ) , Q is congruent

to


0 -dl0 -d.

*10 0 0

0 0 0 -d.

d20

d20

0 -d_

d3 ° .

Another a p p lica tio n o f ( i i i ) enables us to rep lace d^ by i t s

res idu e mod d^, which is not ze ro . This con tra d ic ts th e m in im ality

o f a1 = -q 12.

Remark 5 .2 . I t should be n o ticed th a t we have not used the

p o s i t i v i t y o f H, but on ly th e fa c t th a t H (o r E ) is non

degenerate. M oreover, th e lemma is obvious fo r th e Jacobian o f

a curve. A b as is Y . . .T0 o f H (C ,Z ) whose in te rs e c t io n m atrix -L ¿ g j .

о ■ I( i (y - >y J ) =

1 J \ - I о

i s shown as :

N o tice th a t w ith th e n o ta tion s o f Lemma 5 .1 , u ^ .- .- .u ^

con stitu te a b as is o f ^ . In fa c t , denote by U th e v e c to r space

over E th a t th ey generate. U / - í U is a complex v e c to r space

on which E, and hence H, vanish. But H is p o s i t iv e , so

U П ^ 1 U = {0 } , U + /^1 U = CN. We also set

66 CORNALBA

Ù. = St u , T = ( t . . ) , h = H(u. , u . )i Ji j i j i j i j

h . . i s a r e a l number, so B ( 2 e . u . ,2 m , u . ) = 2 i . , z . v . i j i l * j J l j i j

is a C -h ilin ea r , symmetric form which is r e a l on U. Moreover,

H (z ,u ) = B (z ,u )

whenever u belongs t o U, Ъу l in e a r i t y .

We are con sidering th eta -fu n ction s 0 th a t s a t is fy the

fu n ction a l equation

irH (z,u ) + ^H (u,u)( 5 . 3 ) e (z+ u ) = p (u ) e e ( z ) u е л

I t is e a s i ly checked th a t the function

- ® B (z ,z )0 ( z ) = e 6 ( z )

s a t is f ie s the r e la te d fu n c tio n a l equation

w [H (z ,u )-B (z ,u ) + § - (H (u ,u )-B (u ,u ))]_0 (z+u ) = p (u ) e 0 ( z )

In p a r t ic u la r

9 ( z +u . ) = p( u . ) 0( z )

(5 .i t)- 2 tti^ 1 2 z . y. d .—it/ - 1 Z y .d .t у

è ( z + u ) = p ( u ) e 1 1 1 1 0 ( z )

where we are w r it in g z = Z z.u , ù = 2 y ù , and y is ал1 J- J J J

in teg e r fo r any index j . (N o tic e th a t H (z ,ù ) - B (z ,u ) =J J

2 v^ ï 2z. E (u . ,u . ) ) . On U, p is a homomorphism; th e re fo re fo r 1 i j

any u G U ve may v r i t e


p (u ) = e2^ X^

Nwhere X is a С- l in e a r form on С which is r e a l on U. Then

e-2u»/, lX (z )g £ z0 i s p e r io d ic w ith respect to U, and 0 (z ) has a

Fou rier s e r ie s expansion:

2 ir/ 3 (X (z ) + Z ru zJ

ni-..! 6( 5 . 5 ) e ( z ) = Sa e 1 1

The fu n c tio n a l equation (5 *3 ) im p lies some r e s t r ic t io n s on

the c o e f f ic ie n t s a , namelyV " nN

( 5 ' 6 ) ami+kxdl ’ - - - ,mH + V h

r ^ r - - r-w~\ тг/Л S k .d .t.,k . + йгуСТ 2m t . , k .- -2tt/-1X (u ) i i i J J i i j jp (u ) e e a.

' V - h

where we have w r itten u = -S k .u .. Therefo re the Fo'arier c o e f f ic ie n tsi i

a are uniouely determined by those such th a t 0 < n. < d. ,nl ' ' -nN ‘ ______ N 1 1

i = 1 , . . . , N (th e re are /d it- Q = TT d. o f them ). We w i l l now shoVi= l 1

th a t fo r any choice o f the c o e f f ic ie n t s , subject t o th e above r e s t r i c t

io n s , th e s e r ie s ( 5 -^ ) converges abso lu te ly on compact s e ts , to ge th er

w ith a l l o f i t s term -by-term d e r iv a t iv e s . We may f i x one multi-index

m ^,...,m jj such th a t 0 < m < d^ and assume th a t

a = 1ш1 ‘ ‘ "mN

a = 0 unless m. = n . ( d . ) i = 1 , . . . ,N“l - - 1 1 1 1

The dominant term in th e right-hand s ide o f formula 5.6 is тг/^ï 2k. d . t . k.

e » a°d t o énsure convergence, i t s u ff ic e s t o show

th at Im ДТ is symmetric -positive d e f in it e . To do so, n o tic e th a t

68 CORNALBA

V j k = E (v V = Im H (v V

= 2 Im t . h . ' . l j lk

In o ther term s, i f we w r ite H = (h ., )lk

t Im T-H = Д

So, 1т(ДТ) = ^Im T В Im I

which is p o s it iv e d e f in it e because H is (and Im T is in v e r t ib le ) .

In conclusion,

P ro p os it ion 5 -7 . The space o f th e ta -fu n c tio n s r e la t iv e to

p o la r iz a t io n H, p has dimension

/det E = d ^ .. . djj

where E is_ th e im aginary part o f H and th e determinant is

computed r e la t iv e t o any bas is o f Л.

Example 5 .8 . Let X = J (C ) be th e Jacobian o f a curve С o f genus

g . R eca ll th at

J (C ) = H1 (C,e)/H 1 (C ,Z )

and th a t th ere is a p o s it iv e d e f in it e herm itian form H on H^(С ,C0

whose im aginary pa rt E r e s t r ic t s t o minus th e in te rs e c t io n p a ir in g

on C,Z). M oreover, we have seen th a t we can choose a b as is

u ^ . . -и u ^ .- .u o f H '^C.E ) such th at


E iu ^ u j) = E (ui5 i i j ) = 0

E' v V = 6ij

Choose th e m u lt ip lic a to r p fo r H such th at

p (u ) = p(u ) = 1 , i = l , . . . , g

Then P ro p os ition 5-6 says th a t the space o f sections o f L (H ,p ) is

one-dim ensional. M oreover, th e d iscussion preced ing (5 .6 ) enables us

t o e x p l i c i t l y w r ite down a th e ta -fu n c tio n 0 on which represents

a gen erator o f H^(x ,L {H ,p ) ) .

For any z £ H^iCjC) we w r ite z = ^Z^u^. Then a gen erator

fo r H ° (X ,L (H ,p )) is

S b ( z , z ) 2 n » ^ l ( 2 n . z . )0 ( z ) = e Sa e 1 1

where a.Q Q = 1 and

iti^ T 2 b .t . .n1 lJ J

n , . . .n1 g

In conclusion

?• Sh. , z . z . 2ttÆ l Sn .z. + тг/з: 2 b .t . n.6 { z ) = e 2 ^ 1 J ( V e 1 1 1 ^ J )

n, . . .П t z g1 g

9 (z ) , or ra th er i t s quotien t by th e nowhere van ish ing fa c to r

5 ^ i1 Zi Z1e “ " , i s c a lle d th e th e ta -fu n c tio n o f C. I t s zero locus

is a p e r io d ic d iv is o r on C^, which is the p u ll-Ъаск o f a d iv is o r on

J (C ). Any one o f th ese two equ iva len t ob je c ts is c a lle d the th e ta -

d iv is o r o f J (C ) . One o f our goa ls in the next sections w i l l be t o

exp lore th e re la t io n sh ip between th e geometry o f th e th e ta -d iv is o r

and th e geometry o f th e curve C.

Now l e t us return to the problem o f computing th e dimension o f

th e space o f th e ta -fu n c tio n s r e la t iv e t o the g iven herm itian form H

and a g iven m u lt ip lic a to r p .

Suppose th a t H is ze ro . Two cases are p o s s ib le . E ith e r p is

t r i v i a l o r i t i s not. I f p is t r i v i a l the on ly th e ta -fu n c tion s are

the constants. I f p is n o n - t r iv ia l , I claim th a t th ere are no non

zero th e ta -fu n c tio n s . In fa c t the fu n ction a l equation reduces t o

0 (z + u) = p (u )6 ( z )

This im plies th a t 0 is bounded, hence constant, so 0 must vanish

id e n t ic a l ly i f p (u ) / 1 fo r some u. Now suppose th a t H is p o s it iv e

Nsem i-d e f in it e . Let W be th e complex v ec to r subspace o f С c o n s is t

ing o f a l l th e v ec to rs z such th a t H (z,w ) = 0 fo r any w. W П Л can

be described as th e se t o f a l l u € Л such th at E (u ,v ) = 0 fo r any

v € Л , and i s th e r e fo r e a l a t t i c e in W. I f p is t r i v i a l on W П Л,

any th e ta -fu n c tio n 0 is constant on W/W П Л . I f p is n o n - t r iv ia l ,

N0 vanishes on W/W П Л . Moreover,- i f w 6 W, u £ W П Л and z 6 С ,

the fu n ction a l equation fo r 0 im plies th a t

0 (w + u + z ) = p (u )0(w + z )

so w -*■ 8 (v + z ) i s a th e ta -fu n c tio n on W/V! П Л and i s constant or

vanishes id e n t ic a l ly according t o whether p is t r i v i a l on W П Д or

n o t. In conclusion two cases are p o ss ib le :

(a ) p is n o n - t r iv ia l on W П Л: no non-zero th e ta -fu n c tio n s .

(b ) p is t r i v i a l on W П Л. Then 0 is constant on each coset

W+z. M oreover, i t is e a s i ly seen th a t H is t ie pullback o f a p o s it iv e

^ O CORNALBA

d e f in it e herm itian H on cV w whose imaginary part i s in te g r a l on

Л/Л П W, and th a t p is th e pullback o f a m u lt ip lic a to r p fo r H

In th is case any th e ta -fu n c tio n on X is th e pullback o f a th e ta -

function on (C^/W^A/A П W), r e la t i v e t o H' and p ' . The dimension

o f th e l a t t e r v e c to r space is g iven by P ro p os ition 5 .7 .

The la s t case t o be considered is th e one when H is not p o s it iv e

s em id e fin ite . We cla im th a t , i f th is i s th e case, where are no non

ze ro th e ta -fu n c tio n s r e la t iv e t o H and p. There is a subspace W

No f Œ on which H is n ega tive d e f in i t ê . Suppose th a t th ere is a

non-zero th e ta -fu n c tio n 6 . As we showed in Section 3, th e function


* ( z ) = e - ^ ’ ^ l e i z ) ! 2

His bounded on С . I t i s c le a r th a t the r e s t r ic t io n o f ijj t o any one

o f th e tra n s la te s o f W is plurisubharmonic ( i . e . th e m atrix o f second2

p a r t ia le ( ¡Hr ÿw ) P ^ ^ i v e s em id e fin ite , where the w^'s are coor-

/ Э \dinates on a t ra n s la te o f W ), and s t r i c t l y so ( i . e . I g_J j is d e f in i t e )

\ v i w j /where 9 is not zero . The same is tru e o f

-e £ | Z i |2 ф (г) = e ф (г)

fo r small e ; ф vanishes at in f in i t y , so i t has s maximum. At th is

poin t f 7^ Jji J is p o s i t iv e , which is absurd.V w i j /

Remark 5• 9 . We have shown th at ДТ has symmetric imaginary p a r t.

I t s r e a l p a rt is a lso symmet r i с . In f a c t , we know th a t

(5 .1 0 ) H Im T = Д

M oreover, i f we w r ite

where A = ( a^ j ) and В = (b^ ) are r e a l m atr ices , we conclude th a t

A + Re T ’ B = 0

Im T 'B = I

Th erefo re , Re T = -A Im T.

On the other hand,

Г Re t 4 hik = Re = -E (ù j S vÇÎ

72 CORNALBA

= - S aik E ( ^ , u . ) = - £ d .aJk

In other term s, '''Re T*H = -ДА .

Th erefo re ДА Im T = ^ R e T H Im T = - S e ТД

Д Re T = t Re T Д

i . e . Д Re T i s symmetric, and, in conclusion:

ДТ i s a symmetric m atrix

w ith p o s it iv e imaginary part

This statement u su a lly goes under th e name o f "Riemann b i l in e a r

r e la t io n s " . We may g iv e a more in t r in s ic vers ion o f i t as fo llo w s .

Choose á b as is v , , . . . , v „ , fo r Л and a b as is b , . . .b„, f o r C^.1 2ÏÏ I N

Set

v . = b .2 i j i

П = t o y )

4 ¡ = E ÍV i .V j ) ; « = ( ^ )


With our choice o f bases

- a ~:)

n = ( I , T )

The symmetry o f ДТ is equ iva len t to

(5 .11 ) fiQ- 1 = 0

and th e p o s i t i v i t y o f Im ДТ is equ iva len t to

(5.12) / 3 ОД-1 > 0

The above statements are c le a r ly in va r ian t under change o f bases

fo r Л and C®. They go under th e names o f f i r s t and second Riemann

b i l in e a r r e la t io n s .

6. PROJECTIVE EMBEDDINGS

Let X be any compact complex v a r ie ty , and L a l in e bundle

on X th a t has at le a s t one n o n - t r iv ia l s ect ion . Choose a b as is

Sq . - . s^ fo r the v e c to r space H ^ (X ,L ). Let x be a po in t on X

where one among s g . . . s does not van ish , and Ç a f ib r e coord inate

fo r L in a neighborhood o f x. Then th e p o in t

[s ( x ) : . . . : s ( x ) ] = [ Ç( s ( x ) ) : . . . : Ç( s ( x ) ) ]O n U n

in p r o je c t iv e n-space depends ho lom orph ica lly on x and does not

depend on th e choice o f Ç. T h erefo re i f L has at le a s t one non

zero s e c t io n , th e formula

ф (х ) = [s ( x ) : . . . : s ( x ) ]0 n

g ives a mapping (d e fin ed everywhere excep t, p o s s ib ly , on a proper

su bvarie ty o f X)

ф: X P 11

Changing th e bas is Sq . . . s has th e e f fe c t o f m odifying ф by a

p r o je c t iv e transform ation o f P n . T h ere fo re ф is e s s e n t ia l ly d e te r

mined by L. ф i s everywhere defin ed i f fo r any po in t x o f X th ere

i s a sect ion o f L th a t does not vanish at x. ф separates po in ts i f ,

g iven two poin ts o f X, x and y , th ere i s a sect ion o f L th at

vanishes at x but not at y , and v ic e versa , Ф is a lo c a l

embedding i f fo r any poin t x o f X and any tangent v e c to r a at

x , th ere is a sec t ion s o f L such th at

o ( ç ( s ) ) ^ 0 at x ; s ( x ) = 0

where Ç is a f ix e d f ib r e coord inate near x.

I f ф has the th ree p ro p ertie s described above, ф is an

isomorphism o f X onto a smooth subvariety o f p ro je c t iv e n-space.

We want t o apply th is to complex t o r i .

P ro p os ition 6 .1 . Let X be th e complex to ru s С^/Л. Suppose

th ere i s a l in e bundle L (H ,p ) on X such th at H is p o s it iv eО

d e f in i t e . Let L be L(3H,p ) . Then ф i_s an isomorphism o f X

onto & smooth su bvarie ty o f p ro je c t iv e space.

P r o o f : L (H ,p ) has at le a s t one sec t io n . Let 0 be a non-zero

th e ta function fo r H, p. Then, as we saw at the end o f §3, fo r

any cho ice o f a , b in C^,

0 (x + a )0 (x + b )0 (x -a -b ) = £ (x )

represen ts a sect ion o f L. I f we choose a and b in such a way

th a t 0 does not vanish at z+a, z+b, z -a -b , then Ç does not

vanish at z.

74 CORNALBA

Now suppose th ere are two poin ts а, Ъ which are not separated

Ъу ф and such th at а-Ъ £ Л . This im p lies th a t th ere is a non

zero constant К such th a t fo r any th e ta function 0 r e la t iv e t o

3H,p3 ,

0 (a ) = K0(b)

In p a r t ic u la r , fo r any choice o f x , y ,

0 (x + a )6 (y + a )0 (a -x -y ) = К8 (х+Ъ )0 (у+Ъ )0 (Ъ -х-у)

Keep у f ix e d in th e above id e n t ity and take loga rith m ic d e r iv a t iv e s

w ith respect t o x. The re su lt is

d lo g 0 (x+a ) - d lo g e (a -x -y ) = d lo g в (х+Ъ ) - d lo g в (Ъ -х -у )


which in p a r t ic u la r im p lies th a t

d lo g = d l ° g 6 (x+a ) - d lo g 6 (x-Kb)

is a t r a n s la t io n - in v a r ia n t d i f f e r e n t ia l 2 a .d z .. The conclusion isi i

th a t

S a x .8 (х+а ) = К e 1 1 0(х+Ъ)

or

- 2 a.b . Sa.x.(6 .2 ) 0(х+а-Ъ ) = К e 1 1 e 1 20 (x )

This in ç l ie s th at T^ ^L (H ,p ) = L (H ,p ), i . e . th a t E (a-b , u) is an

in teg e r fo r any u in Л. In p a rt icu la r , the group generated by Л ■

and a-Ъ is a l a t t i c e A' s in ce E i s non-degenerate. There are

a f in i t e number o f extensions p ’ o f p by h ' and each o f them

defin es a l in e bundle L (H ,p ') on X which p u lls back to

L (H ,p ) on X. Moreover (6 .2 ) shows th at any section o f L (H ,p ) is

th e pu llback o f a se c t io n o f L (H ,p ') fo r some p\ But th is is

absurd s in ce , by Riemann-Roch, th e dimension o f H °(X ’ .L ÍH jp ') ) is

s t r i c t l y le s s than the dimension o f H^(X, L (H ,p ) ) .

There remains to show th at ф is a lo c a l embedding. Suppose

Nth is is not th e case at some poin t a £ С . Then th ere are a tangent

gv e c to r v = 2 c . r — and a constant К such th a t i 3 z.

i

К 6 (a ) = v 6 (a ) , i . e . К = v lo g 6 (a )

3fo r any th e ta function 6 r e la t iv e t o 3H,p . In p a r t ic u la r , fo r

Nany x , y G С and any th e ta -fu n c tio n 6 r e la t iv e t o H, p,

(6 .3 ) f (a + x ) + f (a + y ) + f (a - x - y ) = К

where f = v ( l o g 6 ) . Then f is a f f in e . In fa c t (6 .3 ) im plies that

f (Ç ) + f (n ) = К - f(3 a -Ç -n )

S e tt in g n = 0 g ives

f (Ç ) = К ' - f(3 a -Ç )

hence f (Ç ) + f ( n ) = f (Ç + n) + К "

and f (Ç ) - К ” i s l in e a r . Th ere fo re , i f we w r ite с = (c^ , . . . ,Cjj) ,

f o r a su ita b le constant k and any complex number w , we have

2a ! V f ( z ) + kW -, >6 (z+wc) = e 6 ( z )

7 6 CORNALBA

As b e fo re , i t f o l lo v s th a t , fo r any w and any u G Л, E (wc,u ) = 0 ,

which i s absurd s ince E i s non-degenerate.

We have seen th a t i f a complex torus admits a p o la r iz a t io n , i t

i s isom orphic t o a smooth subvariety o f p ro je c t iv e space. These

complex t o r i are c a lle d abelian v a r ie t i e s . The datum o f an abelian

v a r ie t y and a p o la r iz a t io n is c a lle d a p o la r iz e d abe lian v a r ie t y .

What can we say o f a complex torus' X w ith a l in e bundle L (H ,p )

such th a t H i s p o s it iv e sem id e fin ite and has, say, p p o s it iv e

e igenva lues? We have seen in §5 th a t a quotient o f X is a

p-dim ensional abelian v a r ie t y ; th e re fo re th ere is a mapping o f X

onto e p-dim ensional smooth subvariety o f p r o je c t iv e space whose

d i f f e r e n t ia l has rank p everywhere.

Remark 6 . it. With th e no tations o f P ropos ition 6 .1 , i t can also

be proved th a t ф(Х) is an a lgeb ra ic subvariety o f p ro je c t iv e space,

i . e . i s th e se t o f common zeros o f a f i n i t e number o f homogeneous

polynom ia ls , In f a c t , i t is a gen era l r e s u lt (Chow's theorem) th a t

a c losed a n a ly t ic subset S o f p r o je c t iv e space i s a lg eb ra ic . With

th e a d d ition a l assumption th at S be smooth, th is w i l l a c tu a lly be

proved in §7 .

As an a p p lica tio n we see th a t 'any compact Riemann su rface С is

a lg e b ra ic ; in fa c t th e hypotheses o f P rop os ition 6.1 are c e r ta in ly

s a t is f ie d f o r J ( c ) , and С is embedded in i t s Jacobian. When th e

genus o f С i s one,* i . e . when С i s a one-dim ensional conqplex

to ru s , -Wiis means th a t th ere i s always a p o la r iz a t io n L (H ,p ) on C.

Such an H is easy t o produce. With th e no tations o f Example 1 .2 ,

a good H is


The s itu a t io n is e n t ir e ly d i f fe r e n t fo r two-dim ensional (o r h igh er

dim ensional) t o r i , as we w i l l see in th e fo llo w in g sect ion .

7. MEROMORPHIC FUNCTIONS

Let p , a be two holomorphic functions in a neighborhood o f

0 S c®. We say that p , q are r e la t iv e ly prime at a po in t x i f

t h e ir power s e r ie s at x are r e la t i v e ly prime in th e r in g o f convergent

power s e r ie s . We w i l l need th e fo llo w in g :

Fact (See [ 1 ] , fo r exam ple): I f p and q are r e la t i v e ly prime

at 0 , th ey are r e la t i v e ly prime at any poin t in a neighborhood o f 0 .

A meromorphic fra c t io n on an open set U o f i s a fra c t io n

p/q, where p and a are holomorphic and r e la t i v e ly prime at every

p o in t o f U. A meromorphic fra c t io n uniquely determines p and q

up to m u lt ip lic a t io n by th e same u n it.

Let X be a connected complex m anifo ld . A meromorphic function

on X is th e datum o f a covering o f X by coord inate patches

and a c o l le c t io n { f . } o f meromorphic fra c t io n s on th e U . 's which i i

agree on th e over laps П и . Two meromorphic functions ( { l h } , { f ^ } ) ,

( t y . i g j j ) are regarded as th e same i f f^ and agree on

U. П V fo r every choice o f i and j .i J

A d iv is o r on X i s a formal lo c a l ly f i n i t e l in e a r combination

w ith in te g ra l c o e f f ic ie n t s D = Sn.D. o f ir re d u c ib le codimension 1i i

a n a ly t ic subsets D. . D is c a lle d e f f e c t i v e i f a l l th e m. are i --------------- l

non -n egative and at le a s t one o f them is p o s it iv e .

The •polar locus and zero locus o f a meromorphic fra c t io n p/q

( i . e . th e sets o f po in ts where p and q van ish , r e s p e c t iv e ly ) are not

changed when p and q are m u lt ip lie d by th e same u n it. Therefo re

i t makes sense t o speak about th e zero locu s and p o la r locus o f a mero

morphic fu n ction f ; th ey are codimension 1 a n a ly t ic subsets o f X.

Let Y be an ir re d u c ib le component o f , say , th e ze ro locus o f f .

The m u lt ip l ic it y to which f vanishes along Y a ls o has in t r in s ic

7 8 CORNALBA

meaning; th e same is tru e fo r m u lt ip l ic it y o f p o le along components

o f th e p o la r lo c u s . Therefo re we may d e fin e th e p o la r d iv is o r P

and zero d iv is o r Z^ o f f as fo llo w s

= Zf = V

where P (re s p . Z) runs through the ir re d u c ib le components o f th e p o la r locus

(resp . ze ro lo cu s ) o f f , and nip (resp . n^) stands fo r th e order o f

p o le (resp . z e ro ) o f f along P (resp . Z ). The d iv is o r o f f i s

( f ) = P f - Zf

Two d iv is o rs are sa id t o be l in e a r ly equ iva len t i f th e ir d iffe r e n c e

is th e d iv is o r o f a meromorphic fu nction . Let D be an e f fe c t iv e

d iv is o r . Cover X w ith s u f f ic ie n t ly small open sets and l e t

f^ be a lo c a l d e fin in g equation fo r D in and l e t f be a

lo c a l d e fin in g equation fo r D in U^,' i . e . a holomorphic function

on th a t van ishes, w ith the appropriate m u lt ip l ic i t ie s , exa c tly

on D П и . The l in e bundle assoc ia ted w ith D, denoted L^, is

de fin ed as th e l in e bundle whose t r a n s it io n fu n ction s , r e la t iv e to£

the coverin g (и Л , are i /f ^ . The sections o f L^ can be viewed

as meromorphic functions having p o les at most along D. I f D is not

e f f e c t i v e , i t may be decomposed as D = E - F, where E, F are

e f f e c t i v e , and L^ is d e fin ed as LE . I t is a s tra igh tfo rw ard

consequence o f the d e fin it io n s th a t i s t r i v i a l and th a t ,

con verse ly , two d iv is o rs are l in e a r ly equ iva len t i f and on ly i f th e

assoc ia ted l in e bundles are isom orphic.

With th is language, one o f th e consequences o f Theorem 3.9 is

th a t : On a complex to ru s X, any meromorphic function is th e quotient

o f two th e ta -fu n c tio n s ( r e la t i v e to the same herm itian form and m u lti

p l i e r ) . In fac t, l e t f be a meromorphic function cm X; f can be


viewed as a section o f Lp . Another sec t ion o f Lp " i s " the functionf f

1 . f and 1 , viewed as section s o f 1 , are represen ted Ъу th e taf

functions 0^ and 0^. C lea r ly f = •

Meromorphic functions on X can be added and m u lt ip lied and are

e a s i ly seen t o form a f i e l d K ( x ) , the s o -c a llè d function f i e l d o f X.

The two main re s u lts that we wish to prove are:

Theorem 7 -1 . Let X be_ a_ compact connected complex m anifold o f

dimension N. Then K(X) i ¿ a f i n i t e l y generated extension o f С o f

transcendence degree at most К .

P ro p os ition 7 -2 :* Let X = /Л be a complex to ru s . The

fo llo w in g are eou iva len t :

( i ) The transcendence degree o f K (X) over С is at le a s t p;

( i i ) There is a herm itian form H on which is p o s it iv e

semi d e f in it e w ith at le a s t p p o s it iv e e igenva lues and

whose im aginary part is in t e g r a l on Л.

We w i l l prove (7 .2 ) (using 7 .1 ) b e fo re (7 -1 ). We have seen th a t i f

( i i ) i s s a t is f ie d , then X maps onto a smooth a lg eb ra ic v a r ie t y S

o f dimension at le a s t p in p ro je c t iv e space F 0 . The homogeneous

coord ina te r in g P o f S is the quotien t o f th e r in g C fx^ , . . . .x^ ]

by the homogeneous id e a l generated by th e forms th a t vanish on S. The

f i e l d o f meromorphic functions on S i s (we on ly need "con ta in s " which

is p re t ty obvious) the f i e l d o f quotien ts o f homogeneous elements o f

th e same degree in P. K ( S ) has transcendence degree equal to th e

dimension o f S ; on th e other hand, K (S ) maps in to K (X ) by p u ll

back so ( i ) ho lds.

Conversely l e t f ^ , . . . , f be p a lg e b ra ic a l ly independent

meromorphic functions on X. For each i l e t be th e p o la r

d iv is o r o f f^ . Set L = L^ + +D . L is isomorphic t o some

80 CORNALBA

L (B ,p ). H i s p o s it iv e s em i-d e fin ite s ince L has sec t ion s . Suppose

H has le s s than p p o s it iv e e igen va lues. Then, as we showed in §6 ,

th ere are an abe lian v a r ie t y Y o f dimension le s s than p and a l in e

bundle M on i t which p u lls back t o L (we use again th e fa c t th a t L

has s e c t io n s ). On the other hand, as was shown b e fo re , f^ i s a

quotien t 6^/6 o f th e ta -fu n c tio n s r e la t iv e t o H and p. As we

showed in §6 , 0 and 0 are pu llbacks o f section s o f M, and

th e f^ are pu llbacks o f a lg e b ra ic a l ly independent meromorphic

functions on Y. Th is con trad ic ts Theorem 7.1.

We now turn t o th e p ro o f o f (7 .1 ) , which i s e n t ir e ly elementary

and is based on the fo llo w in g form o f th e Schwarz lemma.

Lemma 7 .3 . L e t f be si function which is holomorphic in ¿

neighborhood o f th e p o ly c y lin d e r P = { ( z ^ , . . . ,zn )/ |z^| < 1 , i = 1 , . . . ,n }

and vanishes o f order h at_ 0. L e t M denote th e maximum o f |f| on_ p.

Thai I f ( z , . . . , z n )[ < M max |z |h , [ z i | < 1 , i = 1 , . . . ,n.i

P r o o f . I t s u f f ic e s to prove th e statement on each l in e through

the o r ig in in Cn , i . e . we may assume th a t n = 1 . But then f ( z )/ z ^

i s holomorphic and i t s modulus is bounded by M, by th e maximum p r in

c ip le . The lemma fo llo w s .

P ro o f o f 7 .1 - Let f ^ , . . . , f j j +^ be holomorphic functions on X.

Using th e compactness o f X we may fin d th ree f i n i t e coverings o f X

by coord inate open s e ts , U 3D Vx 3D W^, where x runs through a

f in i t e set o f r p o in ts in X, w ith the fo llo w in g p ro p er t ie s :

P.( i ) in U , f . = pp1— where P. and Q. axe r e la t i v e ly

X 1 W. 1 %X 1 «X1 ,X * *

prime at each p o in t;

( i i ) x e wx

( i i i ) R e la t iv e to a su itab le system o f coord inates

(z ) centered at x in и , V is the open.L X 1< 9 X X X *

p o lv c y lin d er |z^| < 1 , i = 1 , . . . , Я , and is th e open

p o ly cy lin d e r |z^| ^ , i = 1 , . . . ,E.

Condition ( i ) im plies th a t

V. = Q. /3. i , x , y i ,x / i , y

is holomorphic and novhere vanish ing in Ux ^ U^, and bounded in

V n v . We set x y

çn = "ТГ 05.^ x ,y t ^ i , x , y

82 CORNALBA

С = max max |cp Ix ,y V x ,y

x у

С is not le s s than 1 s in ce V ц = 1. We must f in d a polynom ialx y ’ yx

F o f degree £ such that

W riteRx

F ( f i , . . • , f N+1) “T " in Vx where Qx = TT Q. >x>

Sc 1

The T>roof v i l l be in two s tep s . F ir s t ve w i l l see th a t R canx

be made t o vanish t o any order th a t we wish at x . Then we w i l l show

th a t i f each Rx vanishes t o a s u f f ic ie n t ly high degree at x then

a l l th e Rx are id e n t ic a l ly zero .

To sav th a t R vanishes t o order h at x means th a t я.П the x

d e r iv a t iv e s o f R o f order le s s than h vanish at x. These are x

a l l l in e a r cond itions on th e c o e f f ic ie n ts o f F and th ere are

h (h + l). . J[h+N-l) N!

o f them: in a l l _ h . . . (h+N-l). con d ition s .r N!

On the o ther hand th e number o f polynom ials o f degree < Í in

H+l va r ia b les is

U + l ) . . . U +N + l)(N+1 ) 1

which is la r g e r than r h '-‘ ~~ i f & i s la rge enough. Set

M = max max | R |X V х

x

Since E = <e В we conclude, using Schwarz's lemma, th a t x * x ,y у


£ hWe can choose an I such th a t С < 2 ( i . e . £ lo g 2 С < h ) and

U + l ) . . . ( l + N + l ) ÿ h . . . (h +N -l)(N+1 ) ! Г N!

s ince th e le ft-h a n d s ide has degree N+l in Í and the righ t-hand

side has degree N in h. So M = 0 , i . e . F ( f ^ , . . . = °-

S im ila r arguments would a lso show th a t i f i f ^ , . . . , f ^ } contains

a transcendence bas is fo r K (X ) over C, then th e degree o f F can

be bounded independently o f thus showing th a t K (x ) i s f i n i t e l y

generated over Œ.

Remark 7 • • One o f th e consequences o f Theorem 7.1 i s Chow's

Theorem fo r p r o je c t iv e m an ifo ld s . Let V be an n-dim ensional

connected sub-manifold o f p ro je c t iv e space. We want to show th at V

is a lg eb ra ic . In fa c t, l e t Y be th e sm allest a lg eb ra ic subset o f

p r o je c t iv e space which contains V ( i . e . th e in te rs e c t io n o f a l l the

a lg eb ra ic subsets o f p ro je c t iv e space th a t contain V ). We must show

th a t V = Y. Y is ir re d u c ib le . In fa c t i f th e product PQ o f

two homogeneous polynom ials vanishes on V, then e ith e r P or Q

vanishes on an open subset o f V, hence on a l l o f V. Every ra t io n a l

function on Y r e s t r ic t s to a meroniorphic function on V. By

Theorem 7.1 the f i e l d o f r a t io n a l functions on Y cannot have tra n s

cendence degree g rea te r than n over С ; th e re fo re Y has dimension

n. But now Y is an ir re d u c ib le (a lg e b r a ic a l ly , but i t can be seen

th a t th is im p lies a n a ly t ic i r r e d u c ib i l i t y ) complex space o f dimension

n con ta in ing an n-dim ensional V and must th e re fo re co in cid e w ith V.

Example 7 .5 . Let V be a r e a l v e c to r space o f dimension 2N and

A a l a t t i c e in V. By a complex stru ctu re on V we s h a ll mean an

2automorphism J o f V such th a t J = - I . V becomes a complex

v e c to r space by s e t t in g : (a + /^ ï b )v = av + bJv t a , b r e a l.

Th is makes V/A in to a complex to ru s .

Let E be a skew-symmetric b i l in e a r form on V which is in te g ra l

on A. Let J be a f ix e d complex stru ctu re on V. : As we saw b e fo re ,

a necessary and s u f f ic ie n t con d ition fo r E t o be th e Chern c lass o f

a l in e bundle on V/A, i . e . fo r E t o be the imaginary part o f a

herm itian H, is th a t

(7 .6 ) E (Ju ,Jv) = E (u ,v )

fo r every choice o f u and v .

I f n ^ 2 and E 4 0 , con d ition 7.6 i s not s a t is f ie d fo r J

in a dense open subset o f th e space o f a l l complex s tructu res on V.

On the other hand,there is on ly a denumerable in f in i t y o f forms E

84 CORNALBA

which are in te g r a l on Л, so the B a ire category theorem im plies

th a t on "most" complex t o r i o f dimension two or more th ere are no

l in e bundles w ith non-zero Chern c la ss . I t fo llo w s th a t on "most"

complex t o r i o f dimension two or more the on ly - lin e bundle w ith

section s is the t r i v i a l one, o r , a lt e r n a t iv e ly , th ere are no e f fe c t i v e

d iv is o r s , hence no non-constant meromorphic fu nctions .

So fa r we have seen th at fo r t o r i o f dimension g rea te r than one,

at le a s t two extreme cases can occur. K ( x ) can have transcendence

degree H or ze ro . I t can be e a s i ly seen th a t a l l th e in term ediate

cases can a lso occur. We w i l l do th is when N = 2.

2Let Л be th e l a t t i c e in С generated by

( 1 , 0 ) , ( / 3 , 0 ) , ( 0 ,1 ) , (a ,B )

where 6 i s not r e a l . There is a s u r je c t iv e mapping

С /д=Х 56 ► c/E + ^

go tten by sending (z ,w ) onto w. Therefo re K (x ) has transcendence

degree at le a s t one. We w i l l see th a t , fo r a "g e n e r ic " choice o f cc

and B, K (X ) has transcendence degree ex a c tly equal t o one. We w i l l

do th is by showing th a t , in g en e ra l, the l in e a r subspace {w = 0 }

2has no complementary subspace W in ® such th a t W П Л is a

l a t t i c e in W. This would be tru e in an abelian v a r ie t y , where

we could take as W the orthogonal complement o f {w = 0 } w ith

respect t o a p o s it iv e d e f in it e herm itian form whose imaginary part

i s in t e g r a l on Л.

I f a W w ith th e requ ired p roperty e x is t s , then W is generated

by a v e c to r

a ( l , 0 ) + b (v Q !,0 ) + c ( 0 , l ) + d (a ,B ) = (a + / ^ ï b + da, с + dg) = v


where а , Ъ, с , d are in te g e rs , and th ere i s a non-rea l X such that

Xv belongs t o A. This im p lies th a t , f i r s t o f a l l

X € { ¡ (g )

and secondly th a t

a e Q (/0 !,X ) c ® (/ Т , 6 )

This is u su a lly not th e case (e .g . В = a = Æ ).

In th e course o f th is d iscussion we have g iven an in d ica tio n

o f th e p ro o f o f th e Poincare complete r e d u c ib i l i t y theorem . Le t X = С*1/

be an abe lian v a r ie t y , A an abelian su bvarie ty . Then th ere is

an abelian su bvarie ty Б o f X such th a t the natural mapping

A x В -» X i s an isogenv ( i . e . has f i n i t e kern el and is s u r je c t iv e ) .

This can be x>roved as fo llo w s . A — V/ V ■"'1 A where V is a l in e a r

subspace o f С1* such th at V n A i s a su b la tt ic e . Choose any p o s it iv e

N *d e f in i t e herm itian form on С which has in te g ra l imaginary part on A

and se t В = V^/v*" !”'1 Л where V"*" i s th e orthogonal complement o f

V (V1 f"1 Л is au tom atica lly a l a t t i c e in V ) .

8. GEOMETRY OF THE THETA-DIVISOR

Let С be a curve o f genus g > 2. Let X = J (C ) be the

Jacobian o f С. H w i l l be the standard p o la r iz a t io n o f X:

Н ( ш , и ) ’ ) = J 0) /ч c o '

As usual, E w i l l denote th e imaginary part o f H. Choose a bas is

i l , . . . , u , { L , . . . , ù fo r Н 'Ч с .г ) such th a tJ- о В

86 CORNALBA


We a lso w r i t e , as usual,

u. = 2 t . ,u. J . i 3 i

The Riemann th eta -fu n c tio n o f С is

_ 2tt/^1 2h.z. + тг/^Г 2n .t. ,n.6 ( z ) = £ » 1 1 1 ^ J

(nr ..n

N otice th a t 0 is even , i . e . 6 ( - z ) = 0 (z ) .

Choose a hase po in t a e C. Then th ere is d e fin ed г mapping

( Г )ф^: С ■+■ X. Le t С denote th e r - fo ld symmetric product o f С

w ith i t s e l f ( i . e . th e quotien t o f СГ Ъу the action o f the f u l l

( r )symmetric group, a smooth v a r ie ty in our c a s e ). Po in ts in С

w i l l Ъе w r itte n as form al sums p -+ . . .+ p . ф can Ъе extended

to a mapping

ф( г ) : С(Г ) - Xq

d efin ed as fo llo w s :

♦Jr ) ( P l+ . . . + P r ) = V Pl )+- '-+ V Pr }

( Г ) ( Г )ф , ф o r ф w i l l Ъе used in s tead o f ф when no confusionq q

i s l ik e ly to occur. We beg in w ith two simple remarks.

Lemma 8.1." Let u. . . . . Ъе a b as is fo r th e holomorphic d i f f e r ---- -- » g ------------------------------e n t ia ls on C. Le t p ^ , . . . , p r be d is t in c t po in ts o f C, and

m ^,. . . ,mr p o s it iv e in te g e r s . Le t be a lo c a l coord inate in ¿

neighborhood o f p^. W rite = o i^d t^ near p^. Then

88 CORNALBA

= g - rank

(m - l )

• ’ V (p r }

(m -1)

{v ) gr - r

vhere stands fo r th e h-th d e r iv a t iv e o f a. . v i th resuect------ i j -------------------------------------- U ------------ K---to T . .

0

P ro o f. By d u a lity :

1 1 1*1

I t fo llo w s from the exact sequence

0 H°(C,K BL ^ ) ■+ H°(C,K) ° H°(c,2 с x) -*■ ...-Sn.p. '11 1

th at

lip-. )■ ■dim ct(H^(C ,K )) = g - rank

This concludes th e p ro o f.

Remark 8 .2 . Choose a basis <o^,...,co fo r the holomorphic

( r )d i f f e r e n t ia ls on C. Then the mapping is g iven by

I " ) / p ? \ф * (тз +. . .+TD ) = c lass o f f / ш + . . .+ / ru> u + . . .+ / Гш )

q ' I -2 V q l ' q 1 ; q g ; q g /

I f p, , . . . ,p are a l l d is t in c t and т . i s a lo c a l coord inate near

p . , then t , , . . . , ! are lo c a l coord inates near p + . . .+ P on С * i 1 * 9 r ‘ 1 r( I* )

(i*)R e la t iv e to th ese coo rd in a tes, the Jacobi an m atrix o f ф is

obv iou slv ( a . . ( p . ) ) . There is a vers ion o f th is statement th a t aup lies i j

to the case when two or more o f th e co in c id e , hut we w on 't go in to

d e ta i ls .

( r )A consequence o f (8 .1 ) i s th a t fo r r < g , ф is g e n e r ic a lly


ф (в ) (с ( б ) ) = Xj so ф(г ) ( с ( г ) ) is1-1 onto i t s im age. In fa c t

( Г )r-d im ensional and g e n e r ic a lly the Jacobian m atrix o f ф has maximal

rank. Hence, fo r a gen eric Ç €E , h ° (C ,L ^ ) = 1 by (8 .1 ) and

Riemann-Roch. But ф( Ç) = ф (? ’ ) i f and on ly i f Ç and ç ’ are

l in e a r ly equ iva len t. S ince h^(C ,L^) = 1, th is means th a t ç ’ = Ç.

Now we return to the main goa l o f our in v e s t ig a t io n . Let 0

denote th e th e ta -d iv is o r o f C, i . e . th e zero d iv is o r o f 6 ( z ) . We

want t o show th a t 9 i s a t ra n s la te o f ф(С^6 ^ ) .

Consider a po in t e = ф(р + . . .+ p ) 6 X. Le t T denote t r a n s ía --L g 6

t io n by e. In te rs e c t Te (0 ) and ф(С) - and assume th a t ф (С ) is not

contained in T (0 ) . T (0 ) cuts out on С a d iv is o r Ç .. I f th is е е e

were th e b est o f a l l p o s s ib le worlds would be l in e a r ly equ iva len t

t o p ^ + ...+ p . -This is not qu ite tru e , but:

Lemma 8.3. E ith e r T (0 ) 3 ф ( с ) o r deg(c ) = К and th e l in e a r— ■ " s © ~

equ iva lence c lass o f Ç - ( р . + . . . + p ,) i s independent o f e (but depends e x g

on the base poin t q ) . In other terms

ф(с ) + X = e (X independent o f e ) e q q

I f th ere is no danger o f con fusion we w i l l u su a lly w r ite 'K.

fo r J i . Assume, fo r the tim e b e in g , th a t (8 .3 ) has been proved.

Then choose d is t in c t po in ts p^, — ,р^ such th a t h (C ,L^ ) = 1 ,

where Ç = S p^ Set e = ф (?) + i i . I f Tg (0 ) Э ф ( с ) , then

0 э ^ (P g ) - e = iJj( ç ) - e - T p ip ^ -. .+Pg_1)

= - X - lliipj + . . .+ р )

and фСр^- • •+Pg_;L) + X 6 0

because 0 = -0 .

I f , on the o ther hand, Tg (0 ) in te rs e c ts ф(С) in a d iv is o r

o f g po in ts çe , then çg = ç , by th e choice o f Ç, and i t

fo llo w s again th a t

lpCpj+.-.+p + X e 6

In conclu sion , fo r a gen eric choice o f p ^ , . . . , p i|i(p.j + . . .+ p ^)

belongs t o T v (0 ) , hence“X

ф(С(б - 1 ) ) C T (0 )

90 CORNALBA

To show th a t the converse in c lu s ion i s t ru e , we f i r s t have t o show

th a t the in te rs e c t io n number o f ф(С) and Т ^ ^ (ф (С ^ ® ^ ) i s equal

( a )t o g f o r gen eric t £ С .

We must f in d po in ts p , p. , . . . , p such th a t p is l in e a r ly± g -x

equ iva len t to

P l+ --* tP g _1 + t - (2 g -2 )q = Д - p^- . . . - р ^ + t - (2 g -2 )q

where Д = p, + . . .+ p , + p '+ . . .+ р ' _ is the d iv is o r o f an abelianK1 g-1 1 g-1

d i f fe r e n t ia l (a canonical d iv is o r ) . The ex is ten ce o f Д is guaranteed

by the fa c t that

1 < h ° (C ,L 2 , ) = h°(C,K ® L_2 < ) P¿ Pj

Ъу Riemann-Roch. W rite

ф(Л + t ) = ф (г + . . .+ r )J- g

Then vhat ve are look in g fo r is po in ts p , p ^ .- .- .p ^ ^ such th a t

p + Sp i s l in e a r ly equ iva len t t o Гг^. For a gen eric choice o f

r , , . . . , r ( i . e . o f t ) th is can happen on ly i f p is one o f th e r 's1 g

and th e Р ^ 'Б a re the remaining ones. This proves th a t the in t e r

section number o f ф(С) and ф ( С ^ ) is g.

What ve know so fa r i s th a t T v (0 ) = шф(С^® ^ ) + Y . I f ve■yt

in te r s e c t both s ides w ith ф( С) , we can conclude th at

m = 1 , У*ф(С) = 0

where the dot denotes in te r s e c t io n . A l l we must show is th a t Y = 0.

Suppose n o t. Choose a smooth po in t p o f Y. The tangent l in e t o

ф(С) at any p o in t i s p a r a l le l t o the tangent space to Y a t p

(o th erw ise ф(С) and a tra n s la te o f Y could be made t o meet tra n s -

v e r s a l ly , c on tra d ic tin g У>ф(С) = 0 ) . Th is is im possib le s ince ф( С)

generates X (e v e ry po in t o f X is th e sum o f g po in ts o f ф (С )).

We have thus proved


P ro p os ition 8 .h . 0 = Т^(ф (с^8 -1^) ) •

Now choose an e f f e c t i v e d iv is o r Ç o f degree g-1 and set

e = ф (?) + №

Then 0 (e ) = 6 ( - e ) = 0, so

-e = ф (с ') + X

Adding the above equations g ives

Ф(С + z ' ) = -2 К

On th e other hand Ç + Ç contains g-1 a rb itra ry p o in ts , so

92 CORN ALBA

h °(C ,L ?+¡;, ) = g

i - e - L?+¡_, = К , so

ф(К) = -2 &

In p a r t ic u la r we see th a t th e symmetry o f 0 corresponds, on ф (С ^ - '*'^),

t o th e operation o f tak in g the res id u a l d iv is o r w ith respect t o a

canonical one, i . e . passing from Ç = p + . . .+ p .. t o a d iv is o r-L g -L

= pi + . . .+ р * . (g e n e r ic a l ly unique) such th a t Ç + Ç1 is the ± g~J-

d iv is o r o f a holomorphic d i f f e r e n t ia l .

We now want t o exp lo re the re la t io n sh ip between s in g u la r it ie s o f

0 and sp ec ia l d iv is o r s , i . e . d iv is o rs Ç = p ^ + ...+ p r such that

h ° ( C , L ç ) > 1

The f i r s t r e s u lt i s

P ro p os it ion 8 .5 . Let Ç Ъе_ a. sp ec ia l e f fe c t i v e d iv is o r o f

degree g -1 . Then e = ip( Ç ) + Ц is_ a s in gu lar p o in t o f 0. More

p r e c is e ly , th e m u lt ip l ic it y o f e is at le a s t equal to h ° (C ,L ^ ).

P r o o f . Let d + 1 Ъе an in teg e r not g rea te r than h ^ (C ,L^ ).

We w i l l p rove, in d u c tiv e ly on d, th a t th e m u lt ip l ic it y o f ijj(ç ) + &

is at le a s t equal t o d + 1 , th e case d = 0 be ing c le a r . The

fu nction :

( p ^ . - . j p ^ ) h- 0 (^ (2 ^ ) + ¡X.)

vanishes id e n t ic a l ly . D i f fe r e n t ia t in g w ith respect to p ^ , . . . , p ^ ,

we get the id e n t ity :

i . ? . i . 3z. 5. . 6.3z. ' {^ i ] + 3 i4 (p d) +1 d i . i . 1 d1 d

+ Q(p1 , . . . , p g_1 ) = 0

where Q in vo lves d e r iv a t iv e s o f 6 o f order le s s than d. By

induction hypothesis:

E . dz. ( * ( ç ) +Ю ш i 1 (p1 ) - - - “ i J Pà } = °V ^ d *1 Xd I d

i f Spi i s l in e a r ly equ iva len t t o Ç. The assumption th a t

h ° (C ,L ç ) > d + 1

im p lies th a t p , __ ,р^ can be chosen a r b i t r a r i ly w ithout changing

M z ) , so

< Ф ( С ) + Ю = 03z. . . . 3z.X1 1d

fo r every cho ice o f Th is concludes th e p ro o f.

We now prove a p a r t ia l converse o f (8 .5 ) .

Lemma 8.6 . The s in gu la r set o f 0 con sis ts p r e c is e ly o f the

■points ф (с ) + , where Ç is an e f fe c t i v e sp ec ia l d iv is o r o f

degree g -1 .

P r o o f . Suppose ф (с ) + JÍ i s a s ingu lar p o in t o f 0. We must g-1

show th a t ç = £ p . i s s p e c ia l. These two assertion s c le a r ly do i = l 1

not depend on th e choice o f th e hase p o in t ; we can thus assume th at

q У p^, i = Consider th e curve

Г = W Ü - ф(р) + x l p e c }

I f Ç i s not s p e c ia l, Г does not l i e in 0, and meets i t at

ф (? ) + Hi = ф( Ç) - ^ (q ) + 3 i w ith m u lt ip l ic it y one, showing th a t

,4>(Ç) + 3Í i s not a s in gu la r po in t o f 0.

Remark 8 .7 . Another p ro o f o f (8 .6 ) can be obtained by combining

(8 .1 ) and (8 .2 ) . Taken to g e th e r , (8 .1 ) and (8 .2 ) say th a t , when

p , . . . , p are d is t in c t -points o f С such th a t Ç = Ep. i s non-J. g —i- 1

s p e c ia l, the Jacobian m atrix o f ф at Ç has rank g -1 , and th ere

fo re ф (? ) + 14 is a smooth po in t o f 0 (n o tic e th a t i f ф (? ’ ) = ф( Ç ) ,

then ç = ç ' because ç is not s p e c ia l ) . With a l i t t l e more work, th is

method app lies a lso t o th e case when p ^ , . . . ,p ^ are not a l l d is t in c t .

We are now in a p o s it io n t o prove th e f u l l converse o f (8 .5 ) . This

r e s u lt i s e s s e n t ia l ly due to Riemann, and can be fo rm a lly s ta ted as

fo llo w s :

Theorem 8.8. Let Ç be_ e f f e c t iv e d iv isor o f degree g-1 on_

С. The m u lt j-p lic ity o f ф (с) + К i£. Q is. p re c is e ly equal t o

h0 (C ,L ^ ).

P r o o f . Set d+1 = h ^ (C ,L ^ ). A l l we have t o do is show th at

th e m u lt ip l ic it y o f ф (Ç) + К i s not g rea te r than d+1. We w i l l

use induction on d. The case d = 0 is Lemma 8 .6 . For any choice

o f p ^ , . . . ,p ^ we may w r ite

94 CORNALBA

Ф(с) = ф(д1+...+де_а_1 + 5x+...+5d)

where depend on For gen eric p = (p ^ ,.

h ^ C . L j ^ ) = 1

Now, fo r f ix e d p , set

v(p iS . • • ,Pd »v ) = - Ф М + X

p ^ , . . . , p d , v € C. Rote th a t

y (p ,v ) = ф(?) - \p (v) + J Í E 0

and moreover i f Ç + a is l in e a r ly equ iva len t to ç ' + v , and q

(the base p o in t ) and v are g en e ric ,

h° (C ,LÇ+q) = d+1

(8 .9 )

h0 (C ,L^ ) = d

This means, by induction hypoth esis , th a t fo r gen eric v th ere is

d e r iv a t iv e o f order d o f 0 th at does not vanish at

p (p ,v ) = ip(Ç ' ) + In gen era l h ^ (C ,L ^ i) > d. Now we claim th a t

0( p (p ^ ,. . . ,Pd , v ) ) , as a function o f p , . . . , p a lone, vanishes at

p o f order at most d-1 , fo r gen eric p ,v . I f th is were not the

case, in fa c t , we would have:

0 - Ц г ~ ъ — w c ' ) + x > “ i (p a )i x xd I d


fo r gen eric p and v , so th a t 0 vanishes o f order at le a s t d

at ф(с ' ) + !K * c on tra d ic tin g ( 8 .9 ) and the induction hypothesis.

We can thus f in d a smooth curve ( p ^ t ) , . . . »Pd( t ) ) such that

96 CORN ALBA

p i(°) = P ± , i = 1 , . . . ,d

2q + 2b .( t ) + q is non-special fo r generic q and t Ф 0 ,0

at t=0

This im p lies th a t , fo r each f ix e d t ,

t e (y (p ( t ) , . . . , p d( t ) , v ) )

is not id e n t ic a l ly ze ro , in fa c t vanishes p re c is e ly at

q ,q ^ , . . . ,q d_ l5 p ^ ( t ) , . . . ,Pd ( t ) . In p a r t ic u la r , fo r generic

— de (u (p 1 ( t ) ........ P d (t),v)) |t=Q3t

has a simple zero at q , corresponding t o y (p ,q ) = ф(Ç) + .

Hence 6 cannot vanish at iji(ç) + ÍK. o f order g rea te r than d.

Th is concludes th e p ro o f o f Theorem 8.8.

The la s t m issing p ie c e is

P ro o f o f Lemma 8 .3 - We f i r s t e s ta b lish a few n ota tion s .

Consider a polygon S :

t io n (meaning th a t the i n i t i a l po in t o f v i l l Ъе id e n t i f ie d v i th

the end poin t o f a1113- v ic e v e rs a ), the r e s u lt i s an o r ien ta b le

compact two-m anifold o f genus g . Moreover the 1 -c yc le s corresponding

t o a . . . a ,b . . .b generate th e f i r s t homology o f th is m anifo ld and,1 S I 6

r e la t iv e t o a su ita b le o r ie n ta t io n the in te rs e c t io n numbers o f the a. 'si

and the b . 's are0

l ( a . , a . ) = l ( b . ,b . ) = 0 1 0 i J


I ( a . ,b . ) = 6. .i j i j

T h erefo re we can put a complex s tru ctu re on S such th a t С is

obtained from th e Riemann su rface w ith boundary S by id e n t i fy in g

s id es . Choose a b as is fo r the holomorphic d i f fe r e n t ia ls

on С such th a t

/ ш. = 6 ..S 1 10

Set u. = i th standard bas is v e c to r fo r C®. We a lso set l

/ to. = t

Vu. = St. u.

0 i j i

The Jacobian o f С is the quotient o f C® by the l a t t i c e Л generated,

by the u and the u . Relative to the standard po larizat ion o f J(C):

E (u .,u ) = E (ù . ,u . ) = 01 J 1 J

E (u .,u ) = 6. .1 j IJ

so the n o ta tion s th a t we have ju s t es tab lish ed are consisten t w ith those

introduced at th e beginn ing o f th e sec t io n .

98 CORNALBA

The th e ta -fu n c tio n o f С s a t is f ie s the fu n ction a l equations

6(z+u ) = 6 (z )

-2тт»^Г z . - тг/^ï t . . 0(z+u. ) = e 1 i : l9 ( z )

I f we se t cp(z) = 8 ( z - e ) , then 6 s a t is f ie s fu n ction a l equations:

cpU+UjJ = 4>(z)

2 e. -2ïïÆ l z . - тг/Л t . . Cp(z+ûi. ) = e 1 e 1 11 cp(z)

Let f Ъе a m u ltivalued function on C. Choose a s in g le -va lu ed

determ ination o f f on S ( in the ap p lica tion i t w on 't m atter which)

and denote Ъу f + the va lue o f f on ,Ъ^ and Ъу f~ th e value

o f f on , ЪкХ , к = l , . . . , g . A ty p ic a l example w i l l Ъе the

th Êi ‘ coord ina te o f ф viewed as a m u ltivalued mapping С -► С .

On

ф7 = фТ + t r i i ik

whereas on Ъ, к

i|). = ф. + б.. r i Ti lk

Another example is « ’оф , o r cp, fo r sh ort. One has

Cf+ = cp on

27,^1 ek -2 ttS=Î +Cf = e e cp on a ^

I t fo llo w s th at

d lo g cp = d lo g <j> on

d lo g cp" = d lo g - 2-n/^l on

Assume th a t 0 (z - e ) does not vanish id e n t ic a l ly on ф (С ). We want

to show th a t cf has e x a c tly g zeros . We may assume th a t a l l the

zeros o f occur in the in t e r io r o f S, in which case th e number o f

zeros o f Cp is

¿ 7 7 2 U + A> (d los ^ ■ a lo g {?~ )к к k


This proves th e f i r s t part o f Lemma 8. 3. To prove th e second part

o f (8 .3 ) , l e t p , . .. ,p Ъе the zeros o f ч> in S. We must eva luate1 S

2ф (р .) . We w i l l do th is component Ъу component. Using th e residueitheorem again , we g e t:

£ 4Jh(p i ) = s / + /ъ d lo g - ФГ d lo g cp“ )i k T t k

/ъ • • • = 6hk L à lo g k k

••• = ^ фь “к + 2^ 4 * - Ч к ^ a lo g *

Denote Ъу th e i n i t i a l and end po in ts o f aad

Ъ^, r e s p e c t iv e ly . The fu n ction a l equation fo r q> im p lies th at

100 CORNALBA

/ a lo g 4 = lo g 4 l o s = v fc

/ъ a lo g v+ = lo g if (S^) - lo g <p+ ( ^ ) =

= 2 i r / 3 e fc - 2 tt/ - 1 Ф к ( ^ ) - чт/^î tk k + 2 tt/ ^ Î

where v^ , are in teg e rs . Pu ttin g everyth in g to ge th er , we

obtain

£ ф(р^) = e - ¡C (mod Л)

where

К c le a r ly does not depend on e. With th is th e p ro o f o f (8 .3 ) , and

hence o f Theorem 8.8 is complete.

REFERENCES

[1 ] G U N N IN G , R ., R O SS I, H ., A n a ly tic Functions o f Several C om p lex Variables, Prentice-Hall (1 9 6 5 ).

[2 ] LE W IT T E S , J., R iem ann surfaces and the theta function , A c ta Math. I l l (1 9 6 4 ) 37—61.

[3 ] M U M F O R D , D., A belian Varieties, O x fo rd Univ. Press ( 1970).

[4 ] W E IL , A ., V ariétés Kah leriennes, Herm ann, Paris (1 9 5 8 ).

INTRODUCTION TO COMPLEX TORI

IAEA-SMR-18/13

P. DE LA HARPE Institut de mathématiques.Université de Lausanne,Dorigny,Switzerland

Abstract

IN T R O D U C T IO N T O C O M P L E X T O R I.

I. R ea l to ri: c losed subgroups and lattices in R n; C onnected abelian real L ie groups and real to r i; A com p lex

exam ple; Vocabu lary o f L ie groups. II. C om p lex to r i — generalities and abelian curves: Isom orphism s o f com p lex

to ri; C lassification o f com p lex to r i o f dim ension 1 ; One-dim ensional to r i as non-singular cubics. I I I . Th e group

o f periods o f a m erom orph ic fu nction : M erom orph ic functions on C " ; G roups o f periods; S tatem ent o f a

fundam ental reduction theorem . IV . P eriod relations: La ttices in C n and m atrices; F roben ius relations; A com p lex

torus w ith ou t non-constánt m erom orph ic functions.

“M ach ines can w rite theorem s and p ro o fs , and read them . The pu rpose o f m a them atica l exp os ition f o r peop le

is to com m u n ica te ideas, n o t theorem s and p roo fs . E xp e rien ce shows tha t a lm os t always the best way to

com m u n ica te a m a them atica l idea is to ta lk a b ou t co n cre te exam ples and unsolved p rob lem s".

(P .R . Halm os in “T en problem s in H ilbert space” , Bull. A m . Math. Soc. 76 (1 9 7 0 ) 887—9 33 )

INTRO D UCTIO N AND REFERENCES

Real and complex tori are nice examples o f two kinds o f objects: Lie groups and (in some cases) algebraic manifolds. We shall emphasize here the first aspect, and actually hope that these examples could motivate a more comprehensive study. The few proofs where general facts about Lie groups are used (mostly in Section 1.2 and II. 1) can safely be jumped over in a first reading.

For a possible introduction to Lie groups, read first Chapter I in

[1 ] C H E V A L L E Y , C., T h eo ry o f L ie G roups, P rin ceton Univ. Press (1 9 4 6 ).

Then Chapters 1 and 2 in

[2 ] A D A M S , J .F ., Lectures on L ie Groups, Benjam in (1 9 6 9 ).

And then one o f the many classical books on this subject, such as (besides the two above)

[3 ] H O C H S C H IL D , G ., T h e Structure o f L ie G roups, H o lden -D ay (1 9 6 5 ).

Part o f the very fundamental work o f E. Cartan on Lie groups is contained in

[ 4 ] H E L G A S O N , S., D iffe ren tia l G eom etry and S ym m etric Spaces, A cadem ic Press (1 9 6 2 ).

Later, enrich your collection o f examples by practising the exercises in (you may start with those

o f Chapter 3)

101

102 DE LA HARPE

[5 ] B O U R B A K I, N ., G roupes e t algêbres de L ie ; so far published by Herm ann, C h .l (1 9 6 0 ), C h .2 and 3 (1 9 7 2 ),

Ch. 4, 5 and 6 (1 9 6 8 ), Ch. 7 and 8 (1 9 7 5 ).

Some o f the words most often used are listed in our Section 1.4.

Chapter I is a discussion o f real tori and other abelian real Lie groups, which have a particularly simple structure. An easy example given in Section 1.3 will show that this simplicity does not carry over to the complex case. The material o f this chapter can be found scattered among numerous textbooks on various subjects; we have used those o f N. Bourbaki and J. Dieudonné.

There are many exercises in this chapter and in the next one, usually given with an almost complete solution. The reader will probably be able to solve many o f them in his head. Other exercises may, on the other hand, take much effort; they are intended to indicate developments or connections with other areas o f study.

In Chapter II we first look at some general properties o f complex tori o f dimension n (but read n = 1 all through i f you wish) and then consider in more detail the classification problem in complex dimension one. The third and last section should connect both with a traditional chapter o f many, function-theory lecture courses ( “elliptic functions” ) and with elementary algebraic geometry (in particular, Exercise 35 may be taken as a justification for the expressions “ complex torus o f dimension one” and “ abelian curve” to be synonymous). Our main sources have been the two following:

[6 ] R O B E R T , A ., E llip tic Curves, Springer Lecture N o tes in M athem atics 326 (1 9 7 3 ).

[7 ] S E R R E , J.P., Cours d ’arithm étique, Presses Universitaires de F rance (1 9 7 0 ) (an English translation has

been published by Springer).

Material on elliptic functions may be found in (among many others):

[8 ] C A R T A N , H ., Th éo rie élém entaire des fonctions analytiques d ’une ou ¡plusieurs variables com plexes,

Herm ann (1 9 6 1 ) (ex ists in English ).

[9 ] M A R K U S H E V IC H , A .I . , T h eo ry o f Functions o f a C om p lex Variab le, 3 vols, P rentice-H all (1965 and 1967).

[1 0 ] S A K S , S., Z Y G M U N D , A ., A n a ly tic Functions, 3rd Edn, PW N , W arsaw (1 9 7 1 ).

[1 1 ] S W IN N E R T O N -D Y E R , H .P .F ., A n a ly tic T h eo ry o f A be lian Varieties, L on don Math. Soc. Lectu re N o te

Series 14, C am bridge U n iv. Press (1 9 7 4 ).

[1 2 ] W H IT T A K E R , E .T ., W A T S O N , G .N ., A Course o f M od em Analysis, 4th Edn, Cam bridge U n iv. Press (1 9 2 7 ).

There are a few flippant mentions o f some books mostly concerned with algebraic geometry — the following:

[1 3 ] S H A F A R E V IC H , I.R ., Basic A lgeb ra ic G eom etry , Springer (1 9 7 4 ).

[1 4 ] M U M F O R D , D ., A be lian Varieties, O x fo rd Un iv. Press (1 9 7 0 ).

Chapter III contains important statements without proofs. They describe a convenient way for our purpose to write down a meromorphic function on a torus, namely as a quotient o f two theta functions. Besides the often-quoted paper by M. Field (these Proceedings), we call on one o f the basic references:

[1 5 ] G U N N IN G , R .C ., R O S S I, H ., A n a ly tic Functions o f Several C om p lex Variab les, Prentice-Hall (1 9 6 5 ).

Three very classical textbooks on complex tori that we have used frequently are:

[1 6 ] C O N F O R T O , F ., A belsche Funktionen und algebraische G eom etrie , Springer (1 9 5 6 ).

[1 7 ] S IE G E L , C .L ., A n a ly tic Functions o f Several C om p lex Variables, Inst. A d v . S tudy, Princeton (1 9 5 0 ).

[1 8 ] S IE G E L , C .L ., T op ic s in C om p lex Function T h eo ry , 3 V o ls , W iley-In terscience (1 9 6 9 , 1971 and 1973).

COMPLEX TORI 103

One way to approach the voluminous [ 18] is to start with:

[1 9 ] B A IL Y , W .L ., Jr., R ev iew o f C .L . S iegel’ s T op ics in C om p lex Function T h eo ry , Bull. A m . M ath. Soc. 81

(1 9 7 5 ) 5 2 8 -5 3 6 .

The final (and long) exercise in Chapter III is in fact an alternative to Chapter IV and sketches a section in:

[2 0 ] W E IL , A ., V ariétés kâhlériennes, Herm ann (1 9 7 1 ).

Chapter IV is in fact a small part o f [16] and [17]. There are strong conditions for the existence o f “many” meromorphic functions on a complex torus o f complex dimension at least two, which are known as Frobenius relations. Modulo the important statements o f Chapter III (unproved there), we show in Section IV .2 that these conditions are necessary. This is good enough to write down in Section IV .3 a numerical example o f a torus without any non-constant

meromorphic function.Chapter V o f the lectures given during the Summer Seminar Course (not included in these

Proceedings) was essentially the beginning of:

[2 1 ] S IE G E L , C .L ., S ym p lectic geom etry , A m . J. M ath. 65 (1 9 4 3 ) 1—86.

with some motivations from the classification problem for algebraic complex tori that the reader will find in [ 17]. But as we have not introduced the notion o f polarization, we rather give references only.

We have not mentioned examples o f either current research or unsolved problems.The reader interested in the historical importance o f complex tori for algebraic geometry

should read Ch. V II, §7, in:

[2 2 ] D IE U D O N N E , J., “ Cours de géom étrie a lgébrique, 1: aperçu h istorique sur le développem en t de la géom étrie

a lgébrique” , Presses Universitaires de France ( 1974).

See also the historical sketch in [13].Though not mentioned in this text, we finally cite two books studying the analogues o f

complex tori, in some sense, for other ground fields than Ç:

[2 3 ] W E IL , A ., Variétés abéliennes e t courbes algébriques, Hermann (1 9 4 8 ).

[2 4 ] L A N G , S., A be lian Varieties, Interscience (1 9 5 9 ).

Out o f these many references, we would advise from personal taste to start further reading in Robert [ 6 ], Serre [7 ], Swinnerton-Dyer [11] and Siegel [17].

Digressions and exercises refer also to the following books and papers:

[2 5 ] (B O R E L , A ., M O S TO W , G .D ., E ds),“ A lgeb ra ic groups and discontinuous subgroups” , P roc . S ym p. Pure

• M athem atics IX , A m . Math. Soc., 1966.

[2 6 ] A U S L A N D E R , L ., M A C K E N Z IE , R .E ., In trodu ction to D ifferen tiab le M an ifo lds, M cG raw H ill (1 9 6 3 ).

[2 7 ] B E R G E R , М ., G A U D U C H O N , P ., M A Z E T , E-, L e spectre d ’une variété riem annienne, Springer Lecture

N o tes in M athem atics 1 9 4 (1 9 7 1 ).

[2 8 ] B O R E L , A ., “ C om pact C liffo rd -K le in fo rm s o f sym m etric space” , T o p o lo g y 2 (1 9 6 3 ) 111 — 122.

[2 9 ] B O R E L , A ., In trodu ction aux groupes arithm étiques, Herm ann (1 9 6 9 ).

[3 0 ] B O R E L , A ., C H O W L A , S., H E R Z , C.S., IW A S A W A , K ., S E R R E , J.P., Sem inar on C om p lex M u ltip lica tion ,

Springer Lectu re N o tes in M athem atics 21 (1 9 6 6 ).

[31 ] B O R N , М ., H U A N G , K ., D ynam ica l T h eo ry o f C rystal Lattices, O x fo rd Un iv. Press ( 1968).

[3 2 ] C O X E T E R , In trodu ction to G eom etry , W iley (1 9 6 1 ). See also, by the same au thor:“ L ’œ uvre

d ’Escher et les m athém atiques” , in L e m onde de M .C . Escher (L O C H E R , J .L ., E d .), Ed itions du Chêne,

Paris (1 9 7 2 ).

[3 3 ] D IE U D O N N E , J., Fondem ents de l ’analyse m oderne, G auth ier-V illars.(1963 ). Th is is the first vo lum e

o f “ E lém ents d ’analyse” , w ith six volum es published up to July 1975.

[3 4 ] Th e G raph ic W ork o f M .C . Escher (n ew , revised and expanded ed it ion ) O ldbou m e (1 9 6 7 ).

[3 5 ] G O D E M E N T , R ., Cours d ’a lgèbre, Herm ann (1 9 6 3 ).

[3 6 ] H IR Z E B R U C H , F ., N E U M A N N , W .D ., K O H , S.S., D ifferen tiab le M an ifo lds and Quadratic Form s, D ekker (1 9 7 1 ).

[3 7 ] K O B A Y A S H I, S., N O M IZ U , K ., Foundations o f D iffe ren tia l G eom etry I I , Interscience (1 9 6 9 ).

[3 8 ] K U R O S H , A .G ., Th e T h eo ry o f Groups 2, 2nd English Edn, Chelsea (1 9 6 0 ).

[3 9 ] M O STO W , G .D ., S trong R ig id ity o f L o ca lly S ym m etric Spaces, Princeton Univ. Press ( 1973 ).

[4 0 ] N A R A S IM H A N , R ., Several C om p lex Variables, Ch icago U n iv. Press (1 9 7 1 ).

[4 1 ] Sém inaire H en ri Cartan, 10e année, 1957/1958: Fonctions autom orphes, Secrétariat m athém atique,

11 rue P ierre Curie, Paris 5e (1 9 5 8 ).

[4 2 ] S E R R E , J.P., C oh om o log ie des groupes discrets, in Annals o f M athem atics Studies 70, Princeton Univ.

Press (1 9 7 1 ) 7 7 -1 6 9 .

[4 3 ] S E R R E , J.P., A rbres, amalgames e t S L a , to appear in Springer Lecture N o tes in M athem atics.

[4 4 ] D U V A L , P ., E llip tic Functions and E llip tic Curves, L on d on Math. Soc. Lectu re N o te Series 9, Cam bridge

Univ. Press (1 9 7 3 ).

[4 5 ] W A N G , H .C ., C om p lex parallelizab le m anifo lds, Proc. A m . M ath. Soc. 5 (1 9 5 4 ) 771—776.

[4 6 ] W H IT N E Y , H ., C om p lex A n a ly tic Varieties, Add ison-W esley (1 9 7 2 ).

104 DE LA HARPE

SOME N O TAT IO N

N = {0 , 1, 2 , . . . } is the set o f natural integers, zero included.Ж is the group o f rational integers.Q is the field o f rational numbers.R is the field o f real numbers.R_i_ is the set o f real numbers > 0.R * is the set o f non-zero real numbers.С is the field o f complex numbers.I f z = x + iy with x, y G R , R e (z ) is x and Im (z ) is y.Ç * is the set o f non-zero complex numbers.¡51 — { z E C| |z| — 1 } is the unit circle in Ç.The sets R , R + , R * , Ç, Ç* and $_! are always furnished with their usual topology.7Г denotes sometimes a canonical projection, sometimes the real number 3 ,14 ....

COMPLEX TORI 105

Chapter I

R E A L T O R I

1. CLOSED SUBGROUPS AN D LATTICES IN R n

We start with two easy exercises. The first should be kept in mind in the whole o f the present section, and the second provides the idea o f the proof o f Propositions 1 and 2.

Exercise 1. Check that the map from R to (^defined by x ехр(д2тгх) induces an isomorphism o f topological groups R /Z -»• $ '. State and check the analogous fact for R 2 /Ж2 -*• SJ X SJ.

Exercise 2. Classify all closed subgroups o f the additive group R.[H in t: i f Г is such a subgroup, Г Ц 0 } is either empty or has elements o f minimal absolute value, or has elements o f arbitrarily small absolute value.] Give a few examples o f other subgroups.

Let now n, v, m be three natural integers with 1 < n + v < m. We shall identify the abelian group Z n © R " with the subgroup consisting o f those vectors (\[, ..., Xn, Xj,..., x„, 0 ,..., 0 ) for which the Xj’s are rational integers and the xk's are real numbers. In particular,Zn is a discrete subgroup o f R m and Z n ® R v is a closed subgroup o f R m ; we want to show that there are essentially no others. (O f course R m is furnished with its standard topology.)

Proposition 1. Let Г be a discrete subgroup o f R m , with m > 1 and Г Ф {0 }. Then there exist an integer n with 1 < n < m and an invertible linear transformation ф: R m -*■ R m such that

ф( Г ) = Ж".

Proof. Let .^ b e the set o f all subspaces P o f R m such that P П Г is a free Ж-module generated

by a basis o f P over R. Choose a maximal element P 0 in ¿^(which is ordered by inclusion) and l e t { e ] , ..., en}b e a basis o f P 0 over R which generates P0 П Г. We want to show that Г С P0.

Suppose it is not true, and choose a e Г with a Í P 0 • Let С be the subset o f R m consisting o f points o f the form

where r and the rk's are in the closed interval [0,1 ] o f R. Then С is compact and С П Г is finite because it is both compact and discrete. Let now D = {c e С П Г|с $ P0} , which is a finite nonempty set (a S D ), and choose a point d0 = s0a + I s kek in D with the component s0 positive and smallest possible (0 < s0 < 1). (In this proof, summation signs without indicated limits are summations over к from 1 to n.)

We claim that (R d 0 ® P0) П Г = Z d 0 ffi (P 0 n Г ); the non-trivial part o f the claim is that the left-hand side is included in the right-hand side, and we check this now. Let x = t0a ++ 2 tkek G (R d 0 ® P0) П Г. Subtracting from x first a convenient integral multiple o f d0 and then convenient integral multiples o f the ek's, we can obtain a vector y = u0a + Xukek in Г with0 < u0, U [, ..., un < 1, namely a vector у € С П Г. Now u0 is an integral multiple o f s0 [otherwise

there would exist q e N with qs0 < u0< (q + 1 )s0 ; by subtracting from y-qd0 convenient integral multiples o f the ek's, one would find z = (u 0-qs0) a + Xvkek G С П Г with 0 < u0 -q s 0 < s0, and that would contradict our choice o f d0 ]. Hence t0 is also an integral multiple o f s0. It follows that x has integral components t0, t 1 ;..., tn, so that the claim is proved.

П

к =

106 DE LA HARPE

But now the claim contradicts the fact that P 0 is maximal in 3P, so that we were not allowed to choose a as above; consequently Г = P0 П Г = {x = S tkek G P0 I t j , tn G Ж}. Finally, let ф be an invertible linear transformation o f R m such that { 0 (e j), 0 (e2) , 0 ( e n) } are the first n vectors o f the canonical basis in R m . Then ф(Г) = Ж п . ■

Remark. Clearly, there are no maximal discrete subgroups o f R m . But it follows from Proposition 1 that any discrete subgroup o f R m is contained in a (non-unique) subgroup isomorphic to 2£m , i.e. in a subgroup o f maximal rank.

Exercise 3. Classify all discrete subgroups o f the direct product R m X A where A is a finite abelian group. [ A n s w e r : Let Г be a discrete subgroup o f R m X A ; then there exists a subgroup

Д o f R m X A such that the projection R m X A - * R m induces an isomorphism o f Д onto a discrete

subgroup o f R m , and there exists a subgroup В o f A , such that Г = Д В is isomorphic to the direct product Д X B, hence to Z n X В for some n.[ C o m m e n t : Let G be a connected compact real Lie group; then the centre o f the universal covering G o f G is o f the form R m X A , and any real Lie group which is locally isomorphic to G is o f the form G /Г, where Г is a discrete subgroup o f R m X A .]

Corollary. Let Г be a subgroup o f a real vector space V o f dimension n. Then the following conditions are equivalent:

( i ) Г is discrete and V /Г is compact;(ii ) Г is discrete and generates V as a real vector space;(iii) there exists a basis { e j ,..., en} o f V over R which is a basis o f Г over Ж.

Definition. A lattice in V is a subgroup o f V which satisfies the conditions o f the corollary.

Exercise 4. Let Г be a lattice in R n and let II II denote a norm on R n . Show that the series 2 ' ( U r s converges absolutely for all complex numbers s with Re(s) > n; in this exercise, the sign 2 ' indicates a summation over all elements in Г , except the origin.[ S k e t c h : Choose a basis { e j ,..., en} o f Rn over R which is a basis o f Г over Ж. A ll norms being equivalent on R n, there is no loss o f generality i f one chooses that defined by

П 2 П

where the right-hand side summation is over all n-tuples (X !,..., Xn) with X j,..., Xn £ N and X, + ... + Xn > 1. Group those terms for which X! + ... + Xn is a given strictly positive integer X. Show by induction on n that there are

Then

o f them, where с is a constant independent o f X.

COMPLEX TORI 107

(O r even better: get the estimate without computing the actual number, knowing that the surface o f a sphere o f radius X in R n is Xn_l up to a multiplicative constant.) It follows that the terms for which Xj + ... + Xn = X contribute to the series by a factor like Xn '“ 1 “ s According to classical criteria, the series converges i f R e (—n + 1 + s) > 1.]

Digression. More generally, i f G is any connected real Lie group, a lattice in G is a discrete subgroup Г o f G such that G /Г is compact (some authors consider as lattices in G more general objects than these). For example, it is known that any connected semi-simple Lie group has lattices (see Borel [28]). The deformation theory o f lattices has important applications in geometry

(see the papers by Garland in [25], or Mostow [39]). For a Lie group without lattice, see Bourbaki [5 ], Ch. I l l , §9, Exercise 30.

Proposition 2. Let S be a closed subgroup o f R m , with m > 1 and S Ф { 0 } . Then there exist two positive integers n, v with 1 < n + v < m and an invertible linear transformation ф: R m -*■ R m such that 0 (S ) = Z n © R ".

ProofStep one. Let us first show that i f S is not discrete then S contains a real line.

We furnish R m with some norm II II. By hypothesis, there exists a sequence (ap je n o f non-zero vectors in S which converges towards the origin. Define bj = aj/ llaj II for each j G N. As the unit sphere in R m is compact, there is a subsequence (bJk) keN o f (b j)j which converges towards some point b o f the unit sphere o f R m .

Let x > 0 be a real number, fixed for a while. For each к in N , let xk be the integer (> 0 ) defined by

xkl!ajk II < x < ( x k + 1 )Hajk II

Then xkajt G S and

llxkajk - xb|| = ||xk||ajkH(bjk- b) + (x k||ajj| — x)b||

< x k ||ajk l| Ilbjk - b | | + Ц kII ||b ||< x ||b jk - b|| + l la l l i ||b||

for all k e N , so that

lim ||xkajk— xb|| = 0 к

As S is closed, it follows that xb S S.As this holds for any x 6 R + , one has Rb С S and step one is proved.

Step two. Let V be the subspace o f R m which is the union o f all the lines in S; then V is the largest subgroup o f S which is a subspace o f R m . Let W be any supplement o f V in R m : one has then S = V ® (W П S) with W n S discrete. [Otherwise W П S contains a line by step one, which contradicts our choice for V .] We know from Proposition 1 what W f i S can be like, and Proposition 2 follows. ■

Exercise 5. Describe closed subgroups o f closed subgroups o f R m.We give one solution. Let S and T be closed subgroups o f R m with S C T . Let U and V be

the maximal vector subgroup o f S and T, respectively; then U is a subspace o f V . Choose a supplement X o f U in V. By Proposition 2, one has V П S = U ® (X П S).

108 DE LA HARPE

For any supplement W o f V in R m, let pw :R m -* W be the parallel projection along the direction o f V. Then pw (S ) is a discrete subgroup o f W; indeed pw (S ) С pw (T ) = W П T , which is discrete. It follows from Proposition 1 that there are R-linearly independent vectors w „

in W which generate pw (S ) over Ж. Choosey t , ..., y k in S such that P\y(Yi ) = w i . •••! Pw (Ук) = wk ,

and let Y be a supplement o f V in R m which contains the yj's. Clearly S = (V П S) ® (Y П S) and one has:

S = U ® (X П S) ® (Y П S) U ® X = VПT = V ® (Y П T ) V ® Y = R m

The two follow ing exercises go beyond our subject, but they should make it clear that lattices show up in numerous contexts; see also the historical note after Ch. V II o f Bourbaki's “Topologie générale” (Les groupes additifs R n).

Exercise 6. Let { в ! , ..., en} be the canonical basis o f R n where n > 2, let (I ) denote the standard scalar product on R n, and let Г 0 be the lattice in R n generated by the ej's.

( i ) Check that {a G Г| (a la ) = 1 } = { ± e , , ..., ±en} ;(ii ) Check that {a G Г| (a la ) = 2 } = { ±ej±ek G Г| j < к } .It is known that {a G Г| (a la ) is 1 or 2 } is a root system o f type Bn ; see Bourbaki [5] Ch.VI,

§4, N o .5.

Exercise 7. The notation being as in Exercise 6 , let Г ] = { a G Г 01 (a|a) is even} and let Г be the subgroup o f R n generated by Г ] and i ( e, + .... + en).

( i ) Check thatn

Г i = { ( x b xn) G Г 0 j 2 xj is even } is a lattice in R n and that the quotient group Г 0 /Г ! j = 1

has two elements. [Hint: x 2 = x (m od.2) for all x G Ж.]( i i ) Suppose n = 0 (mod.4). Check that Г is a lattice in R n, that the quotient group Г/Г,

has two elements, and that

■2xj G Ж j = 1, ..., n

( X i , -, xn)G R n x j — x k e ^ к = 1,..., n

nr—\

) Xj G 2Ж

. j = 1

[Observe that 2e G Г ] and that e ф T j .](iii) I f n = 0 (mod.4), show that (a||3) G Ж for all a, (3 G Г. [Consider the three cases:

a and |3 in Г , ; a in and /3 = e; a = /3 = e.]

(iv) I f n = 0 (mod.8), (a|a) is even for all a in Г.C om m en t: I f n = 8, the set {a G r|(a|a) = 2} is known to contain 240 elements and to underlinea root system o f type Eg; the form

r x r - » z

(a, 0) w (a|j3)

COMPLEX TORI 109

happens to play a crucial role in the theory o f integral quadratic forms (see Serre [7 ], Ch.V), and hence in algebraic topology (see Hirzebruch, Neumann and Koh [36]).

Lattices occur naturally in physics, and in particular in crystallography: open any book on classical solid-state physics. As a more specific example, let us mention that the evaluation o f sums like those o f Exercise 4 is important for knowledge o f the potential energy in various crystal systems; indications o f computational methods are given, e.g., in Appendix III to Bom and Huang [31 ].

2. CONNECTED AB E LIA N R E A L LIE GROUPS AN D R E A L TO R I

Let V be a real vector space (o f finite dimension n) and let Г be a lattice in V. The quotient group T = V /Г has a unique structure o f a real analytic manifold which makes the canonical projection tt:V -*■ T a covering. Furnished with this, T is a real Lie group known as a real torus. Notice that T is connected (because V is connected and ir is continuous) and compact (see the definition o f a lattice). From the point o f view o f Lie group theory, V is the Lie algebra o f T and 7Г is its exponential map. The next result will allow us to speak about the real n-torus.

Proposition 3. Between any two n-dimensional real tori, there exists a group isomorphism which is also an analytic diffeomorphism.

Proof. Let Vj be an n-dimensional real vector space, let Tj be a lattice in V j, let Tj = Vj/Tj be the associated torus and let {e J1; ..., eJn} be a basis o f Vj over R which is a basis o f Tj over Ж 0 = 1 , 2). Let ф'.У i -* V 2 be the isomorphism o f vector spaces defined by 0 (e j.) = e£ for к = 1, ..., n. Then ф induces a quotient map <p:T, -*■ T 2 which has the desired properties. ■

Proposition 3 expresses the fact that two real n-tori are isomorphic as real Lie groups. In this sense, any п-torus can be looked at as $' X I 1 X ... X S1 (n times), see Exercise 1.

Exercise 8. Check that any continuous group-homomorphism <p:Rm -*• T n can be written as 7Г 0 0 , where ф is a linear map from Rm to Rn and where 7r:Rn -* T n is the canonical projection. [S o lu t io n : ip lifts to R m ->■ R n because Rm is simply connected.] What can you then say about continuous homomorphisms from Tm to T n?

Exercise 9. Let Tn = R n/Zn and let

R n ---- ► T n

(Xi , ..., xn) « - [ x , . ..., xn]

be the canonical projection. For any n-tuple (X , , ..., Xn) o f rational integers, let p (X j, ..., Xn) be the map which sends [ x j , ..., xn ] in T n onto exp{i2n,(X 1x 1 + ... + Xnxn) } in S1. Show that these maps are pairwise distinct continuous homomorphisms from T n to S1, and that any continuous homomorphism from T n to S' is one o f them.C o m m e n t : these maps are the irreducible complex representations o f T n (see Adams [2], Ch.3, for more details).

Proposition 4. Let G be a connected abelian real Lie group o f dimension m. Then there exists a natural integer n < m such that G is isomorphic to the direct product T n X R m _n . In particular, any connected compact abelian real Lie group is a torus, and any simply-connected abelian real Lie group is a vector group.

n o DE LA HARPE

Proof. We use some general facts about Lie groups. Let V be the underlying vector space o f the Lie algebra o f G. As G is abelian, exp :V -> G is a group homomorphism. As exp is a local diffeomorphism, its kernel Г is a discrete subgroup o f V. The proposition follows as G = V /Г. ■

Exercise 10. Show that there exist non-abelian real Lie groups which are diffeomorphic to manifolds o f the form T n X Rm _ n .C o m m e n t : It is known, e.g., that any connected real Lie group G which is solvable is diffeomorphic to some Tn X Rm -n ; but i f G is moreover compact, then G must be a torus. See the proof in Hochschild [3 ], Theorem 2.2 o f Ch. X II and Theorem 1.3 o f Ch. X III.

Corollary. Let S be a connected closed subgroup o f a torus T. Then S and T/S are both real tori and T is isomorphic to the direct product S X (T/S).

Proof. Both S and T/S are connected compact abelian real Lie groups, hence they are tori. As T and S X (T/S) are tori o f the same dimension, they are isomorphic. ■

Exercise 11. Show that there are subgroups o f tori which are not closed, or not connected, or both.

Exercise 12. Let P be the polynomial function

-R

; = (x ,, . . . , x2n)-> _ ( x 2k - 1 + * 2k2 - l )2

к = 1

and let be the submanifold ( x £ R 2n |P(x) = 0 } o f R 2n. In sophisticated terms, -» is an affine algebraic submanifold o f R 2n. Let

даХ-да -* да

(x ,y ) z

be the algebraic map defined by

Zl = X , y 1 - X 2y 2 Z2 = X i y 2 + X 2y,

z 2 n — 1 ~ x 2n — 1 У 2 п — 1 “ х 2пУ 2п z 2n — x 2n — 1 У2п х 2пУ 2п — 1

Then fi endows with a structure o f real Lie group. Check that -да is isomorphic to T n, so that any real torus is algebraic in this sense.

[Hint: i f X! = cos01 ; = cos i^ ; x 2 = sinO,; y 2 = sini^j, then z¡ = cos(91 + ip¡) andz2 = sin(0 j + cp,).]

The last proposition o f this section is a generalization o f the corollary above.

Proposition S. Let S be a connected closed subgroup o f a connected abelian real L ie group A.

Then A is isomorphic to the direct product S X (A/S).

COMPLEX TORI 111

Proof. We identify the underlying subspace o f the Lie algebra o f A with Rm and we denote by 7r:Rm ->■ A the natural projection (in other words it = expA ). Let Г be the kernel o f tt and let

T = 7Г-1 (S ), which is a closed subgroup o f R m containing Г. Let V be the maximal vector subgroup o f T and let Y be a supplement o f V in R m such that (see Exercise 5)

Г = (V П Г ) ® (Y П Г )T = V © (Y П T )

As Г is also the kernel o f the restriction o f 7r to T , one has S = T /Г = { V/(V П Г ) } © { ( Y П T )/ (Y П Г ) } . But S is connected; this implies Y П T = Y О Г and so S = V/(V П Г ). On the other hand,A/S s (R m/D/(T/D = R m/Т = (V ® Y )/T = Y/ (Y Л T ) = Y/ (Y П Г ). It follows that S X (A/S) s { V/(V П Г ) } ffl {Y / (Y n Г ) } = (V ® Y )/ r as A. ■

Exercise 13 (Alternative p roof o f Proposition 5, o f which we keep the notation ). By Proposition 4, there exist natural integers m, n, p, a, b, с such that A = T m X R a, S = T n X R b and A/S = T p X R c. Show that m = n + p and a = b + c. [Sketch: Counting dimensions makes it sufficient to check only one o f these equalities. Now m (or n, p, n + p ) is the dimension o f a (in fact the) maximal compaçt subgroup o f A (o r S, A/S, S X (A/S), respectively). Consider maximal compact subgroups in the middle term o f the sequence S -» A -»• A/S and check that m = n + p.]

3. A COM PLEX EXAM PLE

The statements o f Section 1.2 do not carry over to the complex cáse. The aim o f this section is to show in particular that Proposition 5 does not hold for connected1 abelian complex Lie groups. (Neither does the corollary to Proposition 4, but examples to show this are more complicated; we shall give one in Chapter IV .) |

Let Ж2 be canonically embedded in C2, and let A = С 2/Ж2 = (С/Ж) X (С/Ж) be considered as a complex Lie group; we denote by -n:C2 -*■ A the canonical projection. Let S be the image o f С by the injective homomorphism

C-+ A

V IZ (z, iz ) I

Then S is a closed subgroup o f A , and also a complex submanifold o f A ; in other words,S is a sub complex!Lie group o f A . Let us show that A and S X (A/S) are not isomorphic as complex manifolds (a fortiori not isomorphic as complex Lie groups). >

We claim that A/S is cómpact. Indeed, let ÿ:ÇR/Z) X (R /Ж) -*■ A/S be the continuous map obtained by composition o f the inclusion (R /Ж) X (R /Ж) -*• A and o f the canonical projection A -*• A/S. As the domain o f ф is compact, it is sufficient to prove that ф is onto. But let a be an arbitrary point in A . Choose (u,v) e C 2 with 7r(u,v) = a, and define z =j Im (v ) + ilm (u ) € C.Then (x ,y ) = (u,v) - (z ,iz ) is clearly in R 2, and so defines an element [x ,y ] in (R /Ж) X (R /Ж).In other words [x ,y ] = a(mod.S), so that ф is onto. [

Then we claim that any holomorphic map f : S X (A/S) -*■ A is constant on the “ fibres” { s } X ( A / S ) . Indeed, the map I

С -> C * !

z >-► exp (i2îrz) ^

112 DE LA HARPE

defines an isomorphism С / Ж -> С *, so that A and С * X С * are holomorphically isomorphic. Now for each s £ S, the map f induces two holomorphic maps:

« Pij{ s } X ( A / S ) - » A - » C * X C * ^ C J ^ C 0 = 1 , 2 )

where pr, and pr2 are the coordinate projections; these two maps are constant by the maximum principle, and the claim is proved.

It follows that S X (A/S) and A are not isomorphic as complex manifolds, as stated above.

4. V O C A B U LA R Y OF LIE GROUPS

This section is not intended to be a crash introduction to Lie groups, but merely a recall o fsome words used here and there in these notes. The letter К denotes either the real field R orthe complex field C.

A Lie algebra over К is a (finite-dimensional) vector space g_ over К furnished with a bilinear mapping (X , Y)*-> [X , Y ] from g X g to g such that [X , X ] = 0 for all X S g and such that [X , [Y , Z ]] = [[X , Y ] , Z ] + [ Y j X , Z ]] for all X, Y , Z 6 g . An abelian Lie algebra over К is a vector space V over К furnished with the bracket (or product) defined by [X , Y ] = 0 for all X, Y € V.

A Lie group over К is a group furnished with a structure o f analytic manifold over К which makes the group operations analytic. (The implicit function theorem makes it sufficient to require that the multiplication G X G -*■ G o f the group is analytic.) I f a Lie group is denoted by a capital roman letter such as G, its Lie algebra will often be denoted by the corresponding underlined small letter g ; it is by definition the tangent space to G at the identity, which is identified with the space o f left-invariant vector fields on G, and so is furnished with a bracket which makes it a Lie algebra (see Robertson’s paper in these Proceedings). The exponential map

is written exp:g -» G and assigns to X G g the element <РхО) e G, where v?x is the integral curve R -*■ G defined by X and by the initial value ipx (0 ) = e = identity in G. (The map yjx is a group homomorphism from R to G.)

The group o f all invertible linear transformations o f a vector space V over K , known as the general linear group o f V, is a Lie group over К which is denoted by G L (V ). Its Lie algebra gS (V ) is the space o f all linear endomorphisms o f V together with the bracket defined by [X , Y ] = X Y — Y X , where X Y is the usual composition o f the maps Y and X. Lie subgroups o f G L (V ) are called linear Lie groups and provide many examples o f Lie groups; however, not any Lie group is isomorphic to a linear one (see Hochschild [3 ], Ch. X V III).

Exercise 14. Write down a subset o f the space o f complex (n X n)-matrices which is a complex Lie group for the usual topology and the usual product for matrices, and which contains no invertible matrix. [Answer: the group G o f matrices (| zt) where z is a non-zero complex number, which is isomorphic to С * by

C * G

]

COMPLEX TORI 113

Let G be a Lie group over К and let g € G . The inner automorphism

G - + G

h 1-* ghg-1

is denoted by In t(g ), and its derivative at the identity o f G by Ad (g ). The adjoint representation o f G is the analytic homomorphism

Ad:G -*■ G L (g )

g ^ A d fe )

Its derivative at the identity o f G is the adjoint representation o f g and is known to be given by

-*■ g®(g)ad:

X ~ ( Y - * [ X , Y ] )

Then the formula exp (ad (X )) = A d (exp (X )) holds for all X € g , where the left-hand exp is given by the traditional power series and where the right-hand exp is that defined for G. It follows among other things that g is an abelian L ie algebra i f and only i f the connected component G 0 o f G is an abelian group.

I

Let G be a complex Lie group; the real Lie group obtained by restricting the scalars is denoted by (G )R. Similarly, i f g is a complex Lie algebra, the real Lié algebra obtained by restricting the scalars is denoted by (g )R . If, moreover, g is the Lie algebra o f G, then (g )R is

that o f (G )r . Clearly, i f G and H are isomorphic as complex Lie groups, then (G )R and (H )R arc isomorphic as real Lie groups; Section 1.3 shows that the converse does not hold.

I f G is a real Lie group, i f there exists a complex Lie algebra h such that (h ) R = g_, and i f G is connected, then there exists a complex Lie group H such that (H )R = G. Giving H or h is by definition giving a complex structure on G or £ , respectively.

Exercise 15. It is known that any continuous homomorphism o f a real Lie group into another is real analytic (see e.g. Hochschild [3 ], Ch.VII, Theorem 4.2). Show that the analogous statement for complex Lie groups does not hold. [ S o lu t io n : consider complex conjugation in ÇJ

Exercise 16. It is known that any closed subgroup o f a real Lie group is a sub Lie group

(Hochschild [3], Ch. V III, § 1). Show that the analogous statement for complex Lie groups does not hold. [S o lu t io n : consider R С С .] !

Exercise 17. Let G 0 be the group o f all real (2 X 2)-matrices o f the form ^ ^ ^ , with x 2 + у 2 Ф 0.

/ 1 0 \ ( xy \Let j be the matrix { I and let G i be the set o f all real (2 X 2)-matrices o f the form I J,

\ 0 -1 / \ y -x /

with x2 + у 2 Ф 0. Show that G = G 0 U G , is a linear real L ie group ;with two connected components

114 DE LA HARPE

G 0 and G ! . Show that G 0 (hence g ) has a complex structure, but that G has no complex structure. [Hint: G 0 is isomorphic to C*, and the isomorphism carries

G „^ G oto

g ’- ’-jgj' 1

Comment: As pointed out by K. Honda and S. Yamaguchi (A note on non-connected complex L ie groups, Mem. Faculty Sci. Kyushu Univ. Ser. A , 27 (1973) 65—67), this implies that there is a misprint in many standard text books, such as Helgason [4 ], p. 323.]

Let G be a real Lie group with identity e and let M be a smooth manifold. A smooth action o f G on M (say from the le ft) is a smooth map ip.G X M - » M such that </>(e,m) = m for all m £ M and ip(g,ip(h,m)) = cp(gh,m) for all g , h £ G and m £ M; one often writes gm instead o f <p(g,m);G is said to act on M (in these notes, i f a Lie group acts on a smooth manifold, the action will always be smooth). The action is said to be:

effective i f gm = m for all m £ M implies g = e;almost effective i f there is a discrete subgroup D in G such that gm = m for all m £ M

implies g £ D;transitive i f for any pair (m t , ma) o f points in M there exists g £ G with gm , = m 2.

I f G acts on M, the isotropy subgroup o f G at a point m £ M is Is(m ) = {g £ Glgm = m } and is a closed subgroup o f G. I f the action is transitive and i f m, and m2 are points in M, it is easy to check that Is (m ,) = g_1 Is(m2)g for any g in G such that gm, = m2; otherwise said, the isotropy subgroups o f G are conjugate by inner automorphisms.

A real Lie group G acting on a complex manifold M (i.e. acting in the above sense on the real smooth manifold underlying M ) acts by holomorphic transformation i f the map т<-> gm from M to M is holomorphic for each g £ G.

I f Г is an abstract group acting on a set S, the orbit o f a point s in S under Г is the subset { t £ S I there exists 7 £ Г with t = 7 s}.

The set o f orbits is then the quotient set o f S by the action o f Г , and is denoted either by Г \ S

or by S/Г.

Chapter II

COMPLEX T O R I - G ENERALITIES AND AB E LIA N CURVES

1. ISOMORPHISMS OF COMPLEX TO RI

Now let V be a complex vector space o f finite complex dimension, say n. By definition, a lattice in V is a lattice in the underlying real vector space (V )R = R 2n o f V. The quotient group T = V /Г o f V by the lattice Г has a unique structure o f a complex analytic manifold which makes the canonical projection тт: V -+ T a holomorphic covering. Furnished with this, T is a complex Lie group known as a complex torus. Notice that T is a connected compact complex Lie group. Keeping in mind Proposition 4, we now state the converse (it will not be used in what follows, and you may jump over the p roof i f you feel uncomfortable with the matter o f Section 1.4).

COMPLEX TORI 115

Proposition 6. Any connected compact complex Lie group is a complex torus and, in particular, is an abelian group.

Proof. Let G be a connected compact complex Lie group, let g be its L ie algebra, and consider the adjoint representation Ad :G -+ G L (g ). As this is a holomorphic map and as G is compact,Ad is locally constant (maximum principle); as G is connected, Ad is constant. It follows that

its derivative ad:g -*■ g8(g) is the zero map, i.e. that g is abelian. Hence G is abelian (see Section 1.4) and the result follows from Proposition 4. ■ I

Exercise 18. Show that any complex torus can be given a Kâhler structure.[Sketch: Furnish Cn with the standard Kâhler structure defined by thè form

dZb л dz i

к = 1

Show that со is invariant by any translation. I f Г is any lattice in C n, it follows that (+> induces a KShler structure defined by a form cop on the torus С п/Г (see Robertson’s paper, Section 10, these Proceedings).]

Digression. A complex manifold M o f complex dimension n is said to be complex parallelizable i f there exist n holomorphic vector fields on M which are C-linearly independent at each point o f M. Observe that, for any complex Lie group G and any discrete subgroup D o f G, the quotient space G/D is complex parallelizable. It is a result due to Wang [45] that all connected compact complex parallelizable manifolds are o f this type; among these, complex tori are the only ones which admit Kâhler metrics.

Proposition 7. Let Г , and Г 2 be two lattices in C n, and let <р:Сп/Г, - j С "/ Г2 be a holomorphic

map. Then C n and a vector b £ C n such that the map

C n -> C n C n — ;— ► C nmakes the diagram i, | ^ 4-

zi->A z + b C n/ T , p С п/Г2

commutative. ’In particular, when n = 1, tp is a diffeomorphism i f and only i f it is not a constant; and ip

is a group homomorphism if and only i f it maps the identity in С п/Г, onto the identity in С п/Г2.1

Proof (written for n = l ) : The map ¡p being given, let ф'.С -*■ С be any lifting o f ip to the universal coverings (С/Г[ ) = С and (С/Г2 ) = С. Let со,, oj2 £ С be such that F j = Za>t ® Zcj2 (со, and cj2 are linearly independent over R, i.e. Im(co¡/co2) Ф 0). The images o f the maps z 0 (z + cj ! ) - ф(z ) and z «■* ф ( z + co2 ) - ф(z ) must be in Г 2, hence theU maps are constant by continuity. Taking derivatives:

ф '(z + со,) - ф\г) = ф'(т. + cj2) - ф\ z ) = 0

for all z £ С. It follows that ф ' is a bounded holomorphic map from Q to C , hence is a constant, so that ф must be o f the form z<->- az + b, where a and b are complex constants.

116 DE LA HARPE

The same proof works for n > 1 : consider then the 0j's, where 0:(z , , z n)

(0 i (z i , zn), 0n(z 1, z n)). One may also consider left-invariant holomorphic 1-forms on С п/Г2 and their pull-back by ifi. ■

Corollary, ( i ) I f two complex tori are isomorphic as complex manifolds, then they are also isomorphic as complex Lie groups, (ii) Two complex tori T , = Сп/Г| and T 2 = С п/Г2 are

isomorphic if and only i f the lattices Г , and Г 2 are homothetic. (A homothety is an invertible linear map from Cnto C n.)

Exercise 19. More generally, let T = С п/Г be a complex torus, let G be a complex Lie group, and let tp:T -*■ G be a holomorphic map; prove that

ф:T -+ G

t ^ v K o r V t )

is a homomorphism.[Sketch: Assume first that i¿>(0):

F:T X T - » G

(t,u)>-*4p(t)^u)<p(tur1

e = the identity in G and let

so that F (T , {0 } ) = F ( {0 } , T ) = { e }

I f u is near enough 0, then F (T , {u } ) is in a co-ordinate patch o f G around e and so is constant by the maximum principle. It follows that F (T ,T ) = { e } . C o m m e n t : both the corollary and the statement o f this exercise hold in algebraic geometry (see Shafarevich [13], Ch.Ill, §3, Section 4 .)]

Exercise 20. With the notation o f the proof o f Proposition 7, check that a is well defined by <p,

and that b is well defined modulo the lattice Г 2 •

C o m m e n t . It can be shown that the set o f complex structures on the vector space R 2n is in

bijective correspondence with the homogeneous space Jn = G L2n(R )/GLn(C ) (see e.g.Kobayashi and Nomizu [37], Ch.IX, Prop. 1.3). One can deduce from this that the set o f (isomorphism classes o f) complex tori o f dimension n is in bijective correspondence with the set o f orbits o f Jn under the natural action o f G L2n(Z ) ; in particular, this set is uncountable. The following is striking to us: the classification o f complex n-tori, essentially carried out by Proposition 7 (though one can do better: see Section II.2 for the case n = 1), is easy; but to know which tori are “ algebraic” is hard (see Chapter IV ).

Exercise 21. Let Г be a lattice in С and let со G Г — { 0 } be a point as near as possible from the origin (fo r the usual metric on С ). Show that there exists a basis o f Г over Ж which contains со. [Sketch: Let со ' e Г — Жсо be as near as possible from the origin. Then {со,to’ } is a basis o f Г ; otherwise, there is a point in Г which is either in the open triangle defined by 0, со and со or in the open triangle defined by 0, -со and -со ', and in both cases this contradicts our choices for

со and со '. ]

Exercise 22. I f Г is a lattice in C, denote by Г the lattice { z 6 C\z £ С } and say Г is real i f Г = Г . Show that Г is real i f and only i f it has either a basis consisting o f two complex conjugate numbers or a basis consisting o f a real number and a pure imaginary number; in the first case Г is “rhombic” , and in the second case Г is “rectangular” . [Hint: use Exercise 21.]

COMPLEX TORI 117

Exercise 23. Let Г be a lattice in С which is invariant by a rotation p o f the plane o f angle в.Show that one may take в = 2я/п where n € {1 , 2, 3, 4, 6 }. [S k e t c h : I f P e Г — {0 }, (p i(P ))j e N must be discrete, so that в = 2ïr/q for some rational number q. Let then Q = pk(P ) be as near P as possible, and let о be the rotation pk ; show that о is a rotation o f angle 2тг/п for some n £ 2 - {0 } . Let P ' € Г be obtained by a rotation o f angle 2îr/n round Q, and let Q '6 Г be obtained by a rotation o f angle 27r/n round P I f P = Q ' then n = 6. I f P Ф Q ', then the distance

PQ 'is larger than or equal to PQ; this implies n < 4. You may read this, and much more about plane lattices, in Coxeter [32]. I f there exist such a a with n = 4 (or n = 3 or/hence n = 6), the lattice Г is said to be square, and is then both rectangular and rhombic (or, respectively, triangular, and is then rhombic in many ways).]

Exercise 24. Let r be one o f the numbers>/3i, i, exp(Í7r/3). Then the triangle defined by the origin, the number 1 and the number r has angles (7г/2, тг/З, тг/6), (7г/2,7г/4, тг/4) and (ît/З, тг/3 ,тг/3) respectively. Show that, up to similarity, these are the only planar triangles having angles тг/m, 7r/n, тг/р where m, n and p are integers larger than or equal to 2. [C o m m e n t : These triangles are connected with regular tessellations o f the plane and with the so-called Coxeter groups (see Bourbaki [5 ], Ch.V, §4, Exercises 4 and others; see also Serre [42], Section 1.9).]

Exercise 25. Let Г be a lattice in С spanned by 1 and r S С (r ? R ). Let be a continuous endomorphism o f T /Г; then there exists a e С with а Г С Г and such that is induced by

■ ç - ç

z ►+ az

I f a ^ Z , show that there exist m, n, p, q € Z with n Ф 0 and nr2, + (m -q)r — p = 0. [H in t :

write a = m + nr, ar = p + qr. Given Г, observe that {a £ С|аГ С Г } is a sub-ring o f C ; i f you like field theory, read more about that in Borel et al. [30].]

Exercise 26. A fundamental domain for a lattice Г in С is a “reasonable” subset D o f С (e.g. a connected Borel subset) such that the restriction o f the canonical projection is a bijection from D to C /Г. The favourite fundamental domains in classical textbooks are parallelograms; draw others. [C o m m e n t : Escher can make D look like a beetle, or like two knights on horseback; see [34], or [32] i f the first is not available to you.]

2. CLASSIF ICATIO N OF COMPLEX TO R I OF DIMENSION 1

We follow in this section Serre [7 ], Ch.VII, §2.Let _ ^ b e the set o f pairs (co1; u>2) o f non-zero complex numbers with Im iu j/ iJ ;) > 0,

and let Sf be the set o f all lattices in Ç. There is an onto map from to â? which assigns to (со,, oo2) S -^ th e lattice Г (со,, c j2) = Z oj¡ © .

Let us denote by SL2 (Ж) the group o f (2 X 2 ) matrices with entries in Ж. and with determinant + 1. Then SL2( Z ) acts (from the le ft ) on

SL2( Z ) X

(* )

* k J , (co1; co2)j ►+ (aco1 + b w 2, ccoj + dcj2)

118 DE LA HARPE

Indeed, i f (co,, co2) £ Л and i f r denotes to,/co2, then

/ ato, + bto2\ / ar + b cF + d \Im I --------------- J = Im ( ------------ 2 ----- ) = 1er + d| 2 (ad — b c)Im (r)

\cco, + dco2 / \ cr + d cr + à j

= |cr + d r 2Im (r) > 0

so that the map ( * ) is well defined. We leave it to the reader to check that it is an action.

Lemma. Tw o elements (to ,, to2) and (to,', co2) o f define the same lattice i f and only if they are on the same orbit o f SL2(Z ).

Proof. I f there exists ( 3 | g SL2( Z ) with co,' = ato, + bto2 and co2 = cco, + dco2, then clearly\c à)

Г (со ,, co2) = Г(со,’ , co2). Conversely, i f the two lattices are the same, then there exists a matrix

with entries in TL such that со, = aco, + bco2 and co2 = cco, + dco2. As the matrix must be

invertible over Z , its determinant must be invertible in Z and so is either +1 or -1 . The computation just above the lemma shows that the signs o f

/a b\Im(co,/co2) and o f det Im(co,/co2) are the same; the lemma follows. ■

\c à)

The lemma allows us to identify if and SL2( Ж ) \ Л ? .

Exercise 27. Let Ж2 = Ж ® Z i 6 С and let со, = m + ni e Ж2.Show that the follow ing are equivalent:

(i ) m and n are coprime;(ii) There exists co2 e Z 2 such that { co,, co2} is a basis o f the lattice Z 2 ;

(iii) The open segment from the origin to со, does not meet Ж2;_ /m n\

(iv ) There exist p, q € Z with ( G S L 2(Z ) .\ p q /

Generalize to Z n C K n for n > 2. [H in t : use Bezout’s theorem. C o m m e n t : the main point here is that Z is a principal ring; see Godement [35], Exercise 2 in § 18.]

Now let C * operate on .Jt by

Ç* X

(X, (со,, сог))!-» (XcoI; Xco2)

and on Я * by

Ç * X ^

(X, D ^ -ХГ

We shall identify the quotient set o f ^ b y the action o f C * to the Poincaré half-plane P, = { r £ C|Im (r) > 0 }; the identification is o f course that induced by

^ f ^ P ,

(co,, to2)t->- co,/co2

COMPLEX TORI 119

For all g E SL2(2Z), X e C * and (co1; o j j ) Е Л ' , one has g(X(co,, co2)) = X (g (c ju co2)); it follows that the action ( * ) induces an action o f SL2 (Ж ) on C * \ ^ = P , , which is clearly given by the standard expression

which is the quotient o f SL2(Z ) by the central subgroup with two elements { I , —I } . It is easy to check that the quotient action o f PSL2( Z ) on P, is effective.

We shall call the quotient set o f Sf by the action o f C * the set o f isomorphism classes o f lattices in C, two lattices being isomorphic i f they are homothetic (this terminology is justified by the Corollary to Proposition 7). The onto map from to Я ’ defines an onto map from P, to this set C*\ Sf. And the lemma above says precisely that two elements in P, define the same isomorphism class o f lattices i f and only i f they are on the same orbit o f PSL2(2£).

Proposition 8. For each r in the Poincaré half-plane Px, let Гг be the lattice TL ® 7Lt in C. Then(i ) Any complex torus o f dimension one is isomorphic to С/Гт for some r E P , .

(i i ) Two tori С/Гт and С/Гт»аге isomorphic i f and only i f there exists an element in the modular group PSL2( Z ) which transforms r into r

In other words, complex tori o f dimension one are classified by the quotient set PSL2(2T)\P,.

Proof: follows from the Corollary to Proposition 7 and from the beginning o f the present section. ■

As remarked after Exercise 20, Proposition 8 implies in particular that there are uncountably many isomorphism classes o f complex tori o f dimension one (indeed, PSL2(Z ) is countable and P j is not). Compare with Proposition 3.

It is intentional that the word “ isomorphic” in Proposition 8 is not precise: you can choose “ isomorphic as complex Lie groups” or “ isomorphic as complex manifolds” , again because o f the Corollary to Proposition 7.

Exercise 28. Show that {+1, —1} is exactly the centre o f SL2(Z ) .

Exercise 29. Let SL2(R ) be the group o f (2 X 2) matrices with entries in R and with determinant+ 1. It is a closed subset — indeed a closed submanifold — o f the vector space R4 o f (2 X 2)real matrices, and in fact a linear Lie group. [G L2(R ) is an open submanifold in R 4 and

defines a smooth action by holomorphic transformations o f P, which is almost effective and transitive.

As the matrix —I = q I acts as the identity on P, , one introduces the modular group PSL2(Z ),

det: ( )<-»• ad-bc is o f constant rank on SL2(R ).l\c dj

(i ) Show thatС

ar + b

cr + d

120 DE LA HARPE

(ii ) Let B be the closed subgroup

P q \0 1/pj

D

£ S L 2(R ) p > 0, q £ R o f SL2(R ),

let N =

and let SO(2)

q £ R , let A = C 0, ) a > 0VO l/a/

- I ( Ja b

ba2 + b2 = 1

be the special orthogonal group. Show that the multiplication

SO(2) X В -> SL2(R )

(k, s )«-k s

is a smooth diffeomorphism. [H in t : To show it is onto, apply the Gram-Schmidt orthonormalization process to the columns o f a matrix in SL2(R ):

a b

с dW p 4Y (aHc2)5 * )/ \r s Д 0 (a2 + c2r v

To show it is one-to-one: ks = k's1 => к'4 к = s's-1 £ SO(2) П В = { I } . ]

SO(2) X A X N->-SL2(R )(iii) Show that the multiplication map

(k, a, n)*-> kan

is a smooth diffeomorphism ( “ Iwasawa decomposition” for SL2(R )). State and prove an analogous decomposition for SLn(R ), n > 2.

(iv ) Show that for the action defined in (i), one has Is(i) = SO(2), so that P j can be identified to the homogeneous space SL2(R )/SO(2). Check that SO (2) is a maximal compact subgroup and also a maximal torus in SL2(R ), hence that the same is true o f Is(t) for any т £ P , .

SL2(R ) - * S L 2(R ) , b. , 0 „ , b. . 0 „ . j

, also given by ( c J " 0Д с d j ( - l o j ’(v ) Observe that а :

(g * )is an automorphism o f SL2(R ) with o2 = id; what are the fixed points o f o'! what is the transformation o f P) induced by a?C o m m e n t : It can be shown that SL2(R ) — or more precisely the quotient PSL2(R ) == SL2(R)/ { +1, —I } — is the group o f all holomorphic diffeomorphisms o f P t . In particular, the latter is a real Lie group (see e.g. Cartan [8 ], Section V I.2.6). There are many other complex manifolds (but not all: see Ç 2) for which the group o f holomorphic diffeomorphisms is a Lie group; this is, e.g., the case o f E. Cartan’s “ bounded domains” , by a famous theorem due toH. Cartan; for an expository proof, see Narasimhan [40].

Exercise 30( i ) Let S and T be the two elements o f PSL2( Z ) defined by

Check that S2 = 1 and (S T )3 = 1.

0 -

1 0.and (i!> respectively.

( ii) Let D = {т £ P ] I |r| > 1 and |Re(r)| < . Draw a picture o f the sets D, TD, T ' 1 D,

T ±2D , ..., SD, TSD, T _1SD, T ±2S D ,..., STD, S T - ‘ D.

(iii) For any fixed r in P ,, show that the function

PSL2(Z ) ->■ R+g ~ lm (g r ) has a maximum.

COMPLEX TORI 121

(iv) Let G 'be the subgroup o f PSL2(Z ) generated by S and T and let r be an arbitrary point in P ,. Show that there exists g G G 'such that gr 6 D.

(v) What is the set { т G D| there exists g G PSL2(Z ) with g Ф 1 and gT G D } ?

(vi) For each r G D, what is the subgroup GT = {g G PSL2(Z)|gr = r }?

(vii) Show that PSL2(Z ) = G !

The solution is e.g. in Serre [7], Ch.VII, § 1. It is known that PSL2(Z ) is the free product Ж2*Жз where Z 2 is generated by S and Ж3 by ST; see Serre [43] for a geometric proof, or Appendix В to Kurosh [38] for an; algebraic proof.

We have shown (Propositions 7 and 8, Exercise 29 (iv]) that the double coset space PSL2(Z)\SL2(R)/SO(2) is in natural bijection with the following sets:

1. Lattices in С (identified when they are homothetic).2. Compact complex connected Lie groups o f dimension one, i.e. complex tori o f dimension

one (identified when they are isomorphic as complex Lie groups).3. Riemann surfaces o f genus one, with base point (identified when they are holomorphically

equivalent);We shall make it plausible in the next section that this list can be completed by :

4. Non-singular plane cubics o f equation y2 = x3 + ax + b. For more details, see Ch. I in Robert [6].

5. One could also add to this list an important item on positive definite binary quadratic forms with real coefficients. See A. Weil: exposé 1 in [41 ] and/or Borel [29], No.2.3.

6. A flat 2-torus is a 2-dimensional real torus T together with the Riemannian structure induced by a scalar product on the universal covering R2 o f T. The study o f flat 2-tori, including their classification, has much in common with that o f complex 1-tori. See Berger et al. [27].

3. ONE-DIMENSIONAL TORI AS NON-SINGULAR CUBICS

In this section, we translate into our context some standard material from the theory of elliptic functions. Proofs can be found in numerous classical textbooks, from which we quote:

H. Cartan [8], Sections V.2.5 and V I.5.3 A.I. Markushevich [9], Ch.5 o f Vol. Ill A. Saks and A. Zygmund [10], Ch. V III E.T. Whittaker and G.N. Watson [12], Ch. XX

or in more modern language, two Lecture Notes which are among our main sources:A. Robert [6], Ch.IH.P.F. Swinnerton-Dyer [11], Section 1.2

122 DE LA HARPE

Let r be a point in the Poincaré half-plane P,; le tL T = Ж®ЖТand T r = Ç/LT be the corresponding lattice and torus respectively. A meromorphic function on TT can be considered (or actually defined) as a LT-elliptic function, i.e. as a meromorphic function f on С such that f(z + co) = f(z ) for all z in С and co in LT.

It is shown that the expression

I*-— ' I ( z —co)2co € L r — {0} \

[F o r techn ica l reasons 3& is used instead o f a go th ic p.]

defines a LT-elliptic function known as the Weierstrass function corresponding to LT. (The proof that 3 T is meromorphic is an easy corollary o f Exercise 4, where it is sufficient to consider n = 2 and s = 3; the proof that @>r is LT-invariant is best seen by first looking at the derivative £Э'Т o f 9%.) I f the point r in P t is understood from the context and fixed, we also write L instead o f LT and & instead o f £^т.

Exercise 31. Let Г be a lattice in С with basis {со,, u>2}. A theta-function for Г is a function 0 (holomorphic here, as in Ref.[l 1 ] or [14], but notice it may be meromorphic in Ref.[6]) such that

0(z + co,) = 0(z)exp(a,z + b ,)( * ) fo ra llzG C

0(z + co2) = 0(z)exp(a2z + b2)

where a,, b,, a2, b2 are complex numbers.( i ) Show that a2co, — a,co2 = -i27rn for some n £ l .

(ii) Check that z*-*- exp(az2 + bz + c) is a theta-function for any triple (a, b, c) o f complex numbers. Show that any theta-function without zeros is o f this form.

(iii) I f 0 satisfies (* ), show that

\p(z) = 0(z) expa,z(z—со,) b ,z

2co, co,

satisfies

(* * ) ф(г + со,) = ф(г)for all z 6 С

(* * * ) <f/(z + co2) = i//(z)exp(—\2im-----1- b)CO,

where n is as in (i) and where b is some complex number.(iv) From (* * ), write

L zckexp(i27rk — )

CO,к E ж

and show formally that ( * * * ) implies ck+n = ckexp(i27rk(co2/co,) - b ); i f Im(co2/co, ) > 0

and i f n > 0, observe that the sequence ck, ck+ n, ck+ 2 n> ••• *s *ast decreasing.C o m m e n t : Of course, the quotient o f two theta-functions satisfying (* ) with the same a's and b's is an elliptic function. Starting from the formal identity o f (iv), one can prove the existence

COMPLEX TORI 123

o f non-trivial elliptic functions without using &>. In higher dimensions, there are analogues o f theta-functions, but not o f the Weierstrass function.

Exercise 32. Notations being as in Exercise 31, suppose Im (w2/w, ) > 0, and consider С as R2 with the basis {со 1; co2} . Define

R2 X R2 ---------------------------------------► RA: , x

(x j coj + x2co2, У1 ^ ! + y2cj2)*->-det f 1\У1 У 2

Check that A(coi, co2) = ±1 if and only if {a>i, co2' } is a basis o f Г, and that, moreover, the sign +- holds if and only i f Im(co2/co,') > 0; in other words, A depends on Г in С but not on the choice of any particular basis. Now both A and

Ç X Ç - R

(x, y)*-* Im(x y)

are R-bilinear alternate non-zero forms on R2, so that there exists X £ R * with A (x,y ) = (l/X)Im (xy) for all x,y £ С; give a geometrical meaning to the number X. [Hint: X = ImicS, ■ co2) = I to, I2 Im(co2/co,) = Lebesgue area o f a domain in R2.] Finally, show that (x, y)*-» A (x,iy) defines on С an R-bilinear form which is symmetric positive definite.Comment: To sum up the outcome o f Exercise 32:

(i) A (x ,y ) = -A (y ,x ) for all x,y G С

(ii) А (Г , Г ) С Ж(iii) (х,у)>-+ A (x,iy) is a scalar product on R2.

It is hoped that, after Chapter IV, the reason will be clear for placing this exercise just after asserting the existence o f non-constant elliptic functions, and not with other exercises on lattices.

We recall now some facts about the Weierstrass function.

List o f some properties o f 3 & and o f its derivative

(i) and 3&' are meromorphic functions on C.(ii) ¿2>has a double pole at each point o f L, and no other pole.(iii) S^is an even function: £^ (-z ) = ~P(z) for all z S С (this means in particular: —z is a

pole if and only if z is a pole).(iv) i^has a triple pole at each point o f L, and no other pole.(v) ^ ' i s an odd function: z) = — &>Xz) for all z € C.(vi) ^satisfies the differential equation ( ' )2 = 4 á ^ 3 - g2 - g3 where g2 = 60 S 'co"4

and g3 = 140 Б 'c j"6Xsee Exercise 4).(vii) (g2)3 - 27(g3)2 ^ 0 .(viii) ^ 'h as simple zeros at points o f С which are congruent modulo L to i , a n d i ( l + t),

and no other zero.(ix ) For each A E C U { » } , the equation (z ) = A has exactly two solutions (counting

multiplicities) in any “period parallelogram” , e.g. in

D = {z € C | z = a + bT with a,b real and — i < a,b < i }

(x ) The two solutions o f 3й(z ) = A coincide only if either A = 00 (z G L is a double pole)or A e { ¿ г (4), ,$?(è(i + T) ) } (such z's are zeros o f & ') .

124 DE LA HARPE

Exercise 33. Show that z exp( ^>T(z )) defines a LT-periodic function which is not LT-elliptic, because it is not meromorphic. [Sketch: Near z = 0 one has exp( ¿^r(z )) = exp(z-2 + f(z ))= F(z)exp(z"2 ), where F is holomorphic but where exp(z-2) has an essential singularity.]

Define E0 to be the complex curve

{ (x ,y )G Ç 2|y2 = 4x3 - g2x - g3}

and E to be

{[X , Y , Z ] 6 P 2(C)|Y2Z = 4X3 - g2XZ2 - g3Z3}

Lemma. E is a complex submanifold o f P2(C ) (i.e. an algebraic subvariety o f P2(C ) without singularity).

Proof. Notation is as in Robertson’s lectures (these Proceedings):

Uj = { [ x , y , z ] e p 2(C ) i x ^ o }

<p,: Uj -+C 2 is given by [X, Y, Z]h- jlidem forip2 and <p3.

Let f:Ç 2 ->-C

(x, y)M- y2 4x3 + g2x + g3

Then clearly,

¥>3(U3 П E) = E0 = { (x, y ) E C2|f(x, y ) = 0}

Let us first check that E0 is a submanifold o f C2 by showing that 3f/3x and 3f/3y have no common zero on E0.

Suppose 3f/9x = —12x2 + g2 = 0 and 3f/3y = 2y = 0. Let s be one o f the square roots of g2. Then x = ±s/2^/3 and y = 0. Suppose this point is on E0: then f(±s/2->/3,0) = (—4x2 + g2)x + g3 = 2g2/3 (±s/2i/3) + g3 = 0 so that ±g2s = —3</3 g3,which contradicts property (vii) above. This proves our claim about E0.

Now the “points at °° on E” are points [X, Y, 0] satisfying the equation o f E; the latter gives0 = 4X3, so that E has only one point at °°: [0, 1, 0]. Call it P, so that E = E0 U { P } . The chart

tp2 :U2 -► Ç2

[X , Y , Z ]~ ( x ' = * - z ' = |

maps E П U2 onto {(x ', z ’ ) G C2|f'(x', z ' ) = 0 } where f ' ( x ’, z ' ) = ^-z' + 4 x '3— g2x' z '2— g3z '3.It is easily seen that f'(v>2(P )) = f'(0 , 0) = 0 and that (3 f’/3z ) (0, 0) = 1 Ф 0. By the implicit function theorem (see e.g. Dieudonné [33], Nos. 10.2.1 and 10.2.4), this implies that there is a map

Ç - * - Ç

x'1-»- z' = IKx ')

holomorphic in a neighbourhood o f zero and with f (x ' ф(х' ) ) = 0 in this neighbourhood. It follows that x ' (x', i//(x')) defines a local co-ordinate on у(Е П U2) around v>(P), i.e. a local co-ordinate on E around P.

COMPLEX TORI 125

We leave it to the reader to check compatibility conditions. Up to this, we have proved the lemma. ■

Exercise 34 = Second proof o f the lemma (actually the same, but looked at differently). Let F be the function from C 3 to С given by (X, Y , Z ) «■ Y 2Z — 4X3 + g2XZ2 + g3Z3 and let S = {(X , Y, Z ) € C 3- {0 } |F(X, Y, Z ) = 0 }. Show that 3F/3X, 3F/3Y and 3F/3Z have no common zero on C 3 outside (0, 0, 0). Hence # is a conical complex submanifold o f C 3— {0} .I f it : C 3— { 0 } -*■ P2(C ) is the canonical projection, this implies that E = 7r( S ) is an algebraic subvariety o f P2(C ) without singularity. (This last implication is proved, e.g. in Auslander and Mackenzie [26], §3.4.)

Now let

С — L -* C2 ipo: and let

z » (& > (z ) , & >'(z ))

ip0 :T — id -*■ С 2 be the quotient map (recall it is understood that a point r has been chosen in the Poincaré half-plane P,). Then ip0 has the following properties:

— It is injective, by properties (ix), (iii), (viii) and (x ) above. Indeed: Either z ^ -z(m od.L); then á^ (w ) = Зй(ъ) and w z(mod.L) imply w = —z(mod.L), so that $>0(—z) Ф % (z) by(viii). Or z = —z(mod.L); then ^ ( w ) = implies w = z(mod.L) by (x).

— It maps С — L to E0, because o f (vi) and the equation o f E0.— It is onto E0, by (ix).— It is holomorphic by (i), (ii) and (iv).— Its derivative never vanishes, by (viii).

It follows that <p0 has a holomorphic inverse, so that T — id and E0 are isomorphic as complex manifolds.

Extend <p0 to ip: С -*■ P2(C ) by defining <p(L) = P (that point P defined in the proof o f theabove lemma; this is coherent with ¿p(z) ~ z '2 and .50'(z) ~ z -3 if z — 0). Let ip: T -*■ P2(C )be the quotient map. Then <p is holomorphic at id € T (this is merely to say that ¿5й and are meromorphic), and ip defines a holomorphic bijection from T to E which clearly has a holomorphic inverse.

Proposition 9. Let T be a complex torus o f dimension one. Then there exist an algebraic submanifold E o f P2(C ) of degree 3 and an analytic isomorphism ip: T -*■ E. Moreover, any holomorphic map T -*• T is algebraic. In other words, T is an algebraic group in a natural way.

Proof. The first statement was shown above, and the second follows from Proposition 7. ■

Corollary. Meromorphic functions separate points on one-dimensional complex tori.

Proof. They do on P2(C). ■

Comments

(i) The second statement in Proposition 9 holds for any complex submanifold o f P2(C ); see Shafarevich [13], Ch. V III, §3, Theorem 2.(ii) A holomorphic map T -»■ Pj (C ) cannot be injective because there is no elliptic function o f

order 1. In this sense, Proposition 9 is the best possible.(iii) It is not true that Proposition 9 carries over to higher dimensions; see Chapter IV.(iv) It is known that anymon-singular cubic in P2(C ) is analytically isomorphic to a complex torus;

see e.g. Robert [6].(v ) The terms “abelian curve” and “complex torus o f dimension 1” may be used one for the other.

126 DE LA HARPE

Exercise 35. Notation is as before Proposition 9. Let u, v G C 2 with u ^v (m od .L ). Then ^ (u ) = [u3^ ( u ) , u3^ ' ( u ) , u3] G E and <p(w) = [v3^ ( v ) , v3 ¿ ^ '(v ), v3] G E span a line in P2(C ), say l(u,v). Let w G C2 be the point defined b yu + v + w = 0. A standard property o f the Weierstrass function is that

í3^ ( w ) w 3^ ' ( w ) W"

Show that this is equivalent to the statement that J"(w ) G l(u ,v ). [Conversely, this may be used to define an abelian multiplication on any non-singular cubic; see comment (iv ) above; see also Point 4 at the end o f Section П.2.]

Exercise 36. Let Г be a lattice in С and for each integer n > 3 let Sn = £ 'o j -n (see Exercise 4).( i ) Check that Sn = 0 i f n is odd.

(ii) When Г is a square lattice (Exercise 23), check that Sn = 0 i f n ^ 0 (mod 4); in particular

g3 = 0 in this case.(iii) When Г is a triangular lattice, check that Sn = 0 i f n ^ 0 (mod.6); in particular g2 = 0

in this case.

(iv ) Show that Г is real (Exercise 22) i f and only i f g2 and g3 are both real.(v ) Conversely to (i i ) and (iii), show that Г is square [rectangular] as soon as g3 = 0 [g2 = 0].

[Hint: Sn(a r ) = anS (r ) and Sn( r ) = Sn( r ) for any lattice. The “ i f ” part in (iv ) and question (v ) are the only non-trivial points in this exercise; see e.g. du Val [44], Section 22, and Robert [6],Nos. 3.11 and 3.16 o f Ch.I. Question: can you characterize as in (ii) and (iii) the third isomorphism class o f lattices singled out in Exercise 24?Note: We are grateful to A. Robert, who has indicated to us the good reference for the question

o f Exercise 36. Let j = 26 33 g2(g2 — 27g3)', which is a function o f r (or o f Г ). I f Г is square, j = 26 33. I f Г is triangular j = 0. I f Г is the “ third class” , j = 243353. See J.P. Serre, “Complex multiplication” , pp 292—296 o f “Algebraic Number Theory” , edited by J.W.S. Cassels and A. Frohlich, Academic Press (1967).

Another point is worth mentioning: let Г be a lattice, let E n d (r ) = { а £ | а Г С Г } and let A u t (r ) = {a G E n d fn ia "1 € E n d (r ) } (see Exercise 25). I f Г is square or triangular it is easy to check that A u t (r ) J { + 1, — 1 }. I f r = r r with т = iy/3, then A u t (r ) = { + 1, —1} S E nd (r).]

1. MEROMORPHIC FUNCTIONS ON C "

In this section, by [F ] we refer to Field’ s paper in these Proceedings.Let n be a fixed integer (n > 1); a point in C n will be denoted either by z or by ( z 1 ;..., zn).

We denote by A (C n) the ring o f analytic functions on Cn . It is a commutative ring with identity, written 1 ; it is also an integral domain (see [F ], Prop. 2.2), so that the notion o f a unit in A (C n)

makes sense.

det 3 > (v ) v3^ ' ( v ) v3 = 0

ru3^ > (u ) u3^ ' ( u ) u3

Chapter IIITHE GROUP O F PERIODS OF A M EROMORPHIC FUNCTION

COMPLEX TORI 127

Exercise 37. Let f G A (C n). Show that the following properties are equivalent:( i ) f is a unit (it means: there exists g G A (C n) such that f(z )g (z ) = 1 for all z G C n, i.e.

such that fg = 1 );(ii ) f has no zero on C n ;

(iii) There exists h G A (C n) with f (z ) = exp (h (z)) for all z G Cn.[Hint for (i i ) => (iii): observe that C n is simply connected, so that the lifting problem

has always a solution.]

Given f and g in A (C n), then g is said to divide f i f there exists h G A (C n) with f = gh. Given

f in A (C n), not zero and not a unit, it is clear that any unit u in A (C n) divides f, and so does any product u f with u a unit; i f these are the only elements in A (C n) which divide f, then f is irreducible.

Remark on “irreducible" and "prime”. Let A be an integral domain with unit 1 Ф 0. Let f G A and let (0 be the principal ideal generated by f.

I f ( f ) is prime, then f is irreducible. Indeed, suppose f = gh; then either g G ( f ) , or h G ( f )(or both); choose the notation such that g G ( f ) , i.e. such that g = fk for some k G A. Then f = fkh; it follows thatkh = 1, i.e. that his a unit.

Suppose, moreover, that A is a unique factorization domain. I f f is irreducible, then ( f ) is clearly prime. This is why “ irreducible” and “prime” have the same meaning in [F ] Ch.II, which deals with rings like ©г-

Now the ring A (C n) is not a unique factorization domain. Indeed, let n = 1 and let f be the function z sin(27rz). For any finite subset К o f the integers Z, the polynomial

П (z — a) a £ K

divides f. The assertion follows now from the definition (see [F ], Def. 2.6).It is however a fact that, i f f in A (C n) is irreducible, then ( f ) is prime. But there exists a

connected open subset U o f C 3 for which there exists f G A (U ), f irreducible and ( f ) not prime; see Whitney [46], Section 1.13.

Given f and g in A (C n), it is clear that any unit in A (C n) divides both f and g; i f these are the only elements in A (C n) with this property, then f and g are coprime, written (f,g ) = 1. (Though (f,g ) = 1 is not defined here as after Proposition 2.13 o f [F ], the meaning is the same because o f the “nice” properties o f C n.)

Theorem. Let f , g ' G A (C n). Then there exist f, g, h G A (C n) with f = hf, g' = hg and (f, g) = 1.

The p roof goes beyond our project; see e.g. Gunning and Rossi [15], Section V III.B , Prop.13. ■

Now one could define a meromorphic function on C n to be an element o f the quotient field o f A (C n). The theorem above would then imply that any meromorphic function on C n can be written as a quotient g/h with g, h G A (C n) and (g, h) = 1. But this gives little geometrical idea and, worse, cannot carry over to many other manifolds.

Meromorphic functions on Cn are rather defined by local requirements: as an assignment to each z G C n o f a germ in the quotient field Mz o f the ring © z, the various germs satisfying

h

fC n C - {0 }

128 DE LA HARPE

compatibility conditions discussed in [F ] Ch.II. We shall call set o f regular points o f f and denote by 6 (f) the set o f those points z in C n to which f assigns a germ in © г\ otherwise expressed and with the notation o f [F ], the space C n is the disjoint union o f 8 (f), the pole set P (f) and the indeterminancy set T (f). The geometrical meaning o f 6 (f) is that it is the largest subset o f C n on which f defines a complex-valued continuous function.

A fter the local definition, which could be repeated for any complex manifold, comes the global result, which applies only to some o f them.Theorem. Let f be a meromorphic function on C n; then there exist:

(1) A function q G A (C n) with S (f) = ( z G C n|q(z) Ф 0 }.(ii) A function p G A (C n) with (p ,q ) = 1 and f (z ) = p (z)/q(z) for all z G 6 (f).

Moreover, i f q ' and p ’ have the same properties, then there exists a unit u G A (C n ) with q ' = up

an d p '= u p .

See Ref. [15] again for the proof. ■

We shall denote by M (C n) the field o f meromorphic functions on C n . The above theorem says in particular that M (C n) is indeed the quotient field o f A (C n).

I f f and g are in M (C n), we shall write freely “ f (z ) = g (z ) for all z G C n” instead o f “ f = g” . This can be read as:

(1 ) 6 (f) = 6(g)(2 ) f (z ) = g (z ) for all z G 6 (f)

(3 ) P ( f )= P (g )• (4 ) T ( f ) = T (g )

and there is no trouble coming from expressions such as = oo” or “ 0/0 = 0/0” .

Exercise 38. Write down two meromorphic functions f and g for which 6 (f + g) ф 6 (f) U 6(g); same question with 6 (fg ) ф 6 (f) U 6(g).

Exercise 39. What are the zeros, the poles and the indeterminancy set o f f G M (C 2 ) defined by f (z ) = z ^ z j? i f 7 :R -> С 2 is a continuous map with 7 (0 ) = (0,0), what can you say about limit(s) o f f (7 ( t ) ) when t tends towards zero? Same questions i f f is defined by f (z ) = (z 3 + z2)~l. Draw pictures.

2. GROUPS OF PERIODS

Let f be a meromorphic function on C n. A period o f f is a vector oj G C n such that

f (z + co) = f (z ) for all z G C n. The period group o f f is the set G (f ) o f all periods o f f, which is clearly a subgroup o f the additive group Cn » R 2n.

Lemma. For any f G M (C n), the group G (f) is closed in C n.

Proof. Let (copjgNbe a Cauchy sequence in G (f); suppose it has a limit in the space Cn, say co. Choosez0G C n with z0 + co a regular point for f. By continuity, z0 + coj is also a regular point for f, for large enough j. Then

f (z 0 + co) = f (z 0 + lim C0j)= lim f (z 0 + <Oj)= lim f (z 0) = f (z 0)j -> OO j - * OO j -»• OO

by continuity o f f at z0 + со. Similarly f (z + co) = f (z ) for z near enough z0, which implies that it is true for all z in C n by uniqueness o f analytic continuation. The lemma follows. ■

Notation. Let Q be an invertible С -linear map from C n to C n and let f G M (C n). We denote by

Q f the function defined by Q f(z ) = f(Q ~ ’ z). I f Q ' and Q " are two such maps, one checks that

COMPLEX TORI 129

(Q ' Q" ) f = Q ’(Q ”f). [In other words, (Q ,f) ^ f defines a linear representation o f G Ln(C ) in the

vector space M (C n).]

Proposition 10. Let f £ M (C n). Then the following are equivalent:( 1 ) There exist an integer q < n, a С -linear map jr:Cn -»• C q and a meromorphic function

f e M (Cq ) such that f = f - i t .

(2 ) There exists Q e G Ln(C ) such that ° f depends on z ,, ...,zn.¡ only and not on zn.

(3 ) G (f ) is not discrete.(4 ) f is solution o f a partial differential equation

where the aj's are constants, not all zero.

Proof. (1 ) ** (2 ) is an elementary exercise o f linear algebra. (2 ) =» (3 ): I f (2 ) holds, then

V = {(0 , ..., 0, z ) e C n|z 6 C } is in G (Q f), so that Q~‘ (V ) is a non-trivial subspace o f C n which is in G (f).

(3 ) => (4 ): I f (3 ) holds, it follows from the lemma above and from Proposition 2 that there exists a = ( a , , ..., an) e C n- { 0 } with Ra С G (f). I f z„ is a regular point o f f, there is a neighbourhood U o f the origin in С such that the function

U -*C

z<-+f(z0 + za) - f ( z 0)

is holomorphic and vanishes on U D R, hence on all o f U. It follows that Ca С G (f) and that the derivative o f f in the direction o f a vanishes, so that (4 ) holds.

(4 ) =► (2 ): I f (4 ) holds, there exists a matrix Q € G Ln(C ) with

Definition. A meromorphic function satisfying the equivalent conditions o f Proposition 10 is said to be degenerate. Other functions in M (Cn) are non-degenerate.

Corollary 1. Let f be in M (C n) and suppose f depends indeed on n variables, i.e. satisfies the negation o f (2 ) above. Then:

(i) G (f ) is a discrete subgroup o f C n.(ii) I f c o , , ..., cok are in G (f), then they are linearly independent over R [= reel unabhàngig

in Conforto [16] = independent vectors in Siegel [17 ]] i f and only i f they are linearly independent over Ж [= unabhàngig and strongly independent, respectively].

n

j= l

Write w = Q 1 z, so that ^ f ( z ) = f(w ). Then

n n

j = 1 j = l

and Q f does not depend on zn. ■

130 DE LA HARPE

(iii) G (f ) is isomorphic to Z k for some integer к with 0 < к < 2n. In particular, there exist t o , , ..., cok in G (f) which generate G (f).

Proof. Clear from Propositions 1 and 10. ■

Definition. An abelian function on C n is a non-degenerate meromorphic function on C n which has 2n linearly independent periods. I f f is an abelian function, we write T ( f ) instead o f G (f ) and call it the lattice o f periods o f f; a primitive system o f periods o f f is a basis o f Г (0 over Z .An abelian function f can be thought o f as a non-degenerate meromorphic function on the complex torus C n/r(f).

Example. Let T, , ..., rn be points in the Poincaré half-plane Pj. Define

со, = ( 1, 0, ...,0) w n+1 = ( r „ 0, . . . , 0)

= (0, 1, 0,..., 0) con-f 2 = (0, T2> 0, ..., 0)

o)n = (0, ..., 0, 1) co2 n = (0 ,..., 0, r n)

and let Г be the lattice in C n generated by these. With the notation o f Section II.3, the assignment

(z , , z2, ..., zn) » . ^ ( z , ) &>T2(z2) ... 3¿>Tn(z „)

defines an abelian function on C n with T (f ) = Г. (Check it is not degenerate.)This is a very simple case, because it shows “ separation o f variables” . I f Г is an arbitrary

lattice in C n, we shall see in Chapter IV that there may not exist any non-constant meromorphic function f with Г C G (f).

Corollary 2. Let f be a meromorphic function on C 2; then G (f ) is isomorphic to one o f the following groups:

{0 } , Z , Z 2, Z 3, Z 4 (when f is non-degenerate)С, С © Z , С ® Z 2 (when f is degenerate non-constant)C 2 (when f is a constant).

Proof. Clear from Propositions 2 and 10. ■

Exercise 40. Write down nine meromorphic functions on C 2 so that the nine corresponding period groups are pairwise, not isomorphic.

Caution: some authors do not ask for an abelian function to be non-degenerate.

3. STATEM ENT OF A FU ND AM EN TAL REDUCTION THEOREM

Let Г be a discrete subgroup o f C n and let f € M (C n) be such that Г С G (f) (we allow degenerate functions). From the existence result quoted in the second theorem o f Section III. 1, there exist two coprime functions p,q € A (C n) such that f = p/q. For each ы б Г , write now рш and q^ , the functions defined by

Pcj(z) = P(z + cj)for all 2 £ C n

Qcj( z) = q (z + со)

COMPLEX TORI 131

For all z 6 6 (f), one has p (z)/q(z) = f (z ) = f (z + co) = P^C zV q^ fz ). It is easy to check that рыand q ^ are again coprime; from the unicity result quoted in the same theorem, it follows thatthere exist £ A (C n) such that

p (z + co) = p (z ) e xp (v „ (z ) )for all z £ C n

q (z + co) = q (z ) exp(vw (z ))

O f course, the v j s have to satisfy compatibility conditions:

p (z + co+ co ') p (z + со) ехр(уы + ы - ( z ) ) = ---- + ^ --------— — = exp(v<y (z + co)) exp(vw (z ))

for all со, со' £ Г and for all z £ Cn. I f we had multiplied both p and q by the same unit in

A (C n) to get f = p '/q ', we could have obtained other functions The natural question to ask is: how can one use this freedom to get functions vu as simple as possible? Here is the answer.

Theorem. Let Г be a lattice in C n and let f be a meromorphic function on Cn with Г С G (f). Then there exist p, q £ A (C n) with (p ,q ) = 1 and f = p/q, and there exist linear polynomials vOJ (со £ Г ) such that

p (z + co) = p (z ) exp(vw (z ) )for all z £ C n and со £ Г.

q (z + co) = q (z ) exp(vw (z ) )

Proof. See Siegel [17], § 14—16, or Siegel [18], Ch.5, §7 —8, or Conforto [16], Sections 14—20 for a “ classical” p roof ; see also Swinnerton-Dyer [11], § 4 for a more “modern” proof (which, to our mind, is easier). ■

Definition. Let Г be a lattice in C n. A theta-function for Г is a holomorphic function в £ A (C n) such that

в (z + со) = 0 (z ) ехр(уш(г )) for all z £ C n

and where the vw's are (not necessarily homogeneous) linear polynomials on C n (see Exercise 31 ). The collection o f the уш‘ s is called an exponent system for the function 0.

The theorem above now reads: any Г -invariant meromorphic function on C n is a quotient o f two coprime theta-functions with the same exponent system.

Exercise 41. Open your favourite textbook with a chapter on elliptic functions o f one complex variable, and look for the proof o f the theorem above when n = 1. [ H in t : Look out for “Weierstrass sigma function” .]

Exercise 42. (This is quite a long one, which is very much parallel to Section IV.2 below. It is given here for readers who do not like matrices at the dose o f Chapter IV (see the paper by Cornalba in these Proceedings). The notation is almost the same as in Weil [20], Ch. VI, No.3.)

132 DE LA HARPE

F i r s t p a r t . Let 6 be a theta function for a lattice Г in Cn with exponent system . Foreach со € Г , write

(1 ) vw (z ) = i2n (L ^ C z ) + c(co)}

where L u is a С -linear form o n C " . Check that the compatibility conditions imply

(2 ) + + L<y

(3 ) c(co + co') = + c(co ') + c(co) (m od .z ;

It follows from (2 ) that

Г X C n -> Ç

(co, z ) •+ L w (z )

for all oj, oo'G Г

extends to a R-bilinear form F: Ç n X C n -*■ Ç which is С -linear in the second variable. Define now A (w ,z ) = F (w ,z ) — F (z ,w ) for all w ,z G C n . Deduce from (3 ) that A takes integral values on Г X Г, hence real values on C n X C n, so that A : Ç n X C n -*■ R is R-bilinear, alternate and integral

on Г X Г.

S e c o n d p a r t . Using the C-linearity o f F in the second variable, check that A (w ,z ) - A (iw ,iz )= —i{A (w ,iz ) — A (z ,iw )}, so that A (w ,z ) = A (iw ,iz ) for all w,z G C n. Deduce from this that

(4 ) H:

C n X C n -»• Ç

(z,w ) » A (z ,iw ) + iA (z ,w )

is a hermitian form (C-linear in the second variable) and that ф = 2iF — H is a C-bilinear symmetric

form on C n.

T h i r d p a r t . Define now for each co G Г

(5 ) d(co) = c(co) — iF(co,co)

(6 ) g(co) = Im (d(co))

(7 ) L(co) = g(ico) + ig (c j)

Deduce from (3 ) and from the fact that A is integral on Г X Г that d(co + co') — d(co) - d (co') is an integer for all co, co' G Г, so that g extends to a R-linear form on C n and so that L extends to a С -linear form on C n . Check that d(co) — L(co) is real for all co G Г.

F o u r th p a r t . Define

T -+ S 1

со exp { i2îr(d(co) — L (c o ))}

COMPLEX TORI 133

and

T:Cn - C

z «■ — ф(z,z) + L (z ) 4i

I f вт £ A (C n) is defined by 0r(z ) = exp { —i27rT(z)} в (г), check that

(8 ) 0r(z + co) = 0r(z ) i//(co) exp |ttH(cj,z) + H(co,co)j- for all z £ C n and for all со £ Г

Fifth part. Now let f = p/q be as in the theorem above. Define as above a hermitian form H and two “reduced theta-functions” pr and q r, so that one has again f = pr/qr. Deduce from the analogue o f (8 ) that the function

C n - * R ,

|pr(z ) exp j-^H(z, z)||

is Г -periodic, hence bounded, so that

(9p) |pr(z)| < Ц exp j- H(z, z)j for all z £ C n

for some positive real constant /j,and similarly

(9q) |qr(z)| <//exp|^H(z, z)j for all z £ C n

Suppose that there exist z0 £ С n with H (z0, z0 ) < 0. Deduce from Liouville’s theorem and from(9 ) that the functions from С to С given by X >*■ pr(Xz0) and X *+ q r(Xz0) are constant, so that f is degenerate.

S u m m a r y o f E x e rc is e 4 2 . Let Г be a lattice in C n. Suppose that there exists an abelian function for Г. Then there exists a positive definite С -valued hermitian form H on C n such that H(co, со’ ) — H(co', co) £ 2 iZ for all со, со' £ Г. Equivalently there exists a skew-symmetric R- valued R-bilinear form A on C n such that А (Г , Г) С Ж and such that

Ç n X Ç " -»■ R

(z, w ) ** A (z , iw )

is symmetric and positive definite, i.e. is a scalar product on R^n = (C n) R . [Hint for the equivalence: i f H is given, define A (w , z ) = Im (H (w , z )); i f A is given, define H as in the second part above.]See Exercise 32 for the case n = 1.

134 DE LA HARPE

Chapter IV

PERIOD RELATIONS

1. LATTICES IN Ç n AN D M ATRICES

For any pair (m , n) o f strictly positive integers and for any ring .5? (hereafter one o f TL,Q, R, Ç ), we denote by Mm n(á 2 ) the free -module o f all matrices with m lines, n columns, and entries in á ? . I f X is in M m n(¿% ), its transpose is denoted by 4X S Mn m (0£ ). When

= C, its conjugate is denoted by X £ Mm n(C ). We shall often write matrices “in blocks” ; for example, i f X £ М рп( ^ ) and Y £ M q n( ^ ) , then ( y ) is in Mp + q n(.i% ). The product o f two matrices X € M m¡g ( & ) and Y 6 M j п(!Ж) is just written X Y and lives o f course in Mm n(,5?).

When m = n, we write M n(£% ) instead o f Mn n(.5? ). This is in a natural way a (non-abelian) ring with identity, with group o f invertible elements denoted by G Ln(¿% ).

In this chapter (not until now, for typographical reasons), we think o f vectors in C n as matrices in Mn ] (C ) , i.e. as “ column-vectors” . We introduce the C-bilinear form:

ç n x ç " - * ç

so that <z|w> = *zw = *wz = <w|z> for all z,w 6 C n. We introduce in the same way the R-bilinear

form R n X R n -*R , again denoted by <|).

Let Г be a lattice in C n and let {co,, ..., ш2п} be a basis o f Г over TL. The period matrix associated with this situation is the matrix:

(w i,i - ш1,2п\ j

“ п,1 шп,2п J

Any “period” <o £ Г can be written as co = f lm for some m S M 2n i ( Z ) . Geometrically speaking, the matrix П describes a map y: R 2n ->■ C n which sends X 2n (see just before Proposition 1) onto

Г, and which is clearly R-linear and bijective.As in the lemma o f Section II.2, any other basis cof,..., co|n o f Г would be given by

COMPLEX TORI 135

for some matrix M = (Mjjk) G G L 2n(Z ) . (As we have not introduced oriented basis for lattices in C n, we cannot impose here that M has determinant +1 .) The corresponding period matrix

would then be

i i * = (со* co* ... <o2n) =

As in the Corollary to Proposition 7, an invertible linear map Q :z z * from C n to itself would map Г onto a lattice Г * with basis cof = Q c o ,,..., co2n = Qco2n. Thinking o f Q as a matrix in G Ln(C ), we get the period matrix associated with Г * and with the co*'s as

П * = (со* co* ... co|n) = Q ii

These and the proposition below motivate the following definitions.

Definition. A Riemann matrix is a matrix i i G Mn 2n(C ) the columns o f which are vectors in C n linearly independent over R. Tw o Riemann matrices П and П * in Mn 2n(C ) are equivalent i f there exist Q G G Ln(C ) and M G G L 2n( Z ) such that П * = QÍ2M.

Proposition 11

( i ) The relation defined above is an equivalence relation between Riemann matrices.(ii ) Let i i and i i * be two period matrices associated with two lattices Г and Г * in C n,

furnished respectively with basis [с о , , ..., co2n] and {cof, ..., cofn} . Then i i and i i * are equivalent i f and only i f the complex tori T = С п/Г and T * = Сп/Г* are isomorphic as complex Lie groups.

Proof. Assertion (i ) is clear. For (ii), i f íl* = QÜM for some Q 6 G Ln(C ) and M G G L 2n(Z ) , then the linear map Q: C n -*■ C n factors as an isomorphism Сп/Г -»• Сп/Г*. Conversely, let q :T -»• T * be an isomorphism; then q lifts to a linear map Q :C n -*■ C n which maps Г onto Г *(this is a repeat from the p roof o f Proposition 7). In particular {Q c o ,, ..., Qco2n} and {c o f , ..., со|п} are two bases o f Г*, so that there exists M = (Mj k) G G L 2n(Z) with

2n 2 n

j = l j = l

It follows that П * = QÍ2M. ■

Definition. Let Г be a lattice in C n furnished with a basis {c o ,, ..., co2n} . The big period matrix associated with this situation is the matrix

/ П \

p’ ( ñ j eM¡"E)

Proposition 12. Let P be as in the definition above.( i) P is invertible: P G G L 2n(C ).

(ii ) P_1 can be written as (G G ) for some G G M 2n>n(C ) satisfying Í2G (I denotes the (n X n)-unit matrix.)

(iii) I f П * = QSÎM for some Q G G Ln(C ) and M G G L 2n(Z ) , then P* =

= I and Í2G = 0.

(iv) Let 7 '.R2n -*• Cn be the map described by as earlier in this section, then 7 -1 is given by

I

Cn -»• R2n

z

Proof _ J(i) Consider Q.' = i(f2 + П ) and П "= — ( f i - f i ) , both in Mn 2n(R ). One has

136 DE LA HARPE

' f i \ / I i l \ / f i '

P = -n j \ l - i l / \ f i "

and

(\ i l\ /21 O'Det I J = Det ( ) = D et(2 I) Det ( - i i ) = (~ 2 i)n Ф 0

\ I - i i / \ I - i i ,

/П '\ / П лHence it suffices to check that ( , 16 G L 2n(R ); but the columns o f the matrix I ,

are clearly R-linearly independent, because the vectors cok's are R-linearly independent, so that

/ fi '\( i ) follows. [Notice that Det ^ д -J >s the “volume” o f the parallelotop spanned by the ojk's . ]

(ii) Write P ' 1 = ( G ^ ) with G , and G 2 in M2n n(C ). Then P _1P = G , f i + G 2f i = id = ( 1 ° ),_ _ _ /1 o\ __ Vo I/

so that G , f i + G 2f i = ( 1 (because I = I), which reads (G 2G ,)P = id. Hence_ ' _ /

P 1 = (G 2G j) = (G ,G 2) and G 2 = G b written G from now on. Finally PP-1 = I - ) (G G ) _ / f iG f i G \ / I 0\

“ U g f iG / = ld = V О I Г WhÍCh endS the Pr° 0 f ° f (W '

(iii) Is clear. / »(iv ) For all z e M n l (C ): P“ ' I M = Gz + G z 6 M 2n>l(R ) « R 2n.

I f 5:Cn -» R2n

■ * p" ( l

then one has,for all x e R 2n,6(7 x) = 5 (fix ) = P 1 | ) = P 1 ( i s ) x ~ x> and 6 = 7 '.

Exercise 43. Show that any Riemann matrix in Mn 2n(C ) is equivalent to one o f the form (I T ) for some T 6 Mn(C ) with

GO—

COMPLEX TORI 137

i.e. is equivalent to a so-called reduced Riemann matrix. (Compare with the map inSection II.2.) Check that two distinct reduced Riemann matrices may be equivalent. [H in t : consider

2. FROBENIUS RELATIONS

Suppose now that Г is a lattice in Cn with basis {coj, ..., and let f = p/q be a meromorphic function on Cn with Г С G (f), as in the fundamental theorem o f Section III.3. Write vi> •••. v2n f ° r polynomials vw ,..., v^, . The compatibility conditions first express vu foran arbitrary oj G Г in terms o f the vj's and, second, impose that the Vj‘ s satisfy

(1) Vj(z + cjk) + vk(z ) - vk(z + cjj) - Vj(z) G 12ттЖ for all j,k G 1 1 , 2n}

We shall see that (1 ) impose on Г remarkable conditions, at least if we assume that f is nondegenerate; this was discovered by Frobenius (1884). We shall follow Siegel [17], §20, and use essentially his notation. Many o f the steps below may seem quite artificial; they could be made to look more “natural” via cohomological considerations, but this would take us too far from our project and we shall not comment further.

Step one ( Î may be degenerate): Definition o f the matrices B, A and S.

Write the polynomials v,, ..., v2n as

2n

(2) vj(z)= bjçjZk+Pj, j — 1...2n

k= 1

this reads

(3) Vj(z) = <bj|z> + j3j, j = 1, ..., 2n

and the conditions (1) become

(4) <bj|cok> — <bk|ojj> G i27rZ, j,k = 1,..., 2n

Define the matrix В to be

(5) B = (b, b2 ... b2n)=

so that (4 ) becomes

(6 ) (В П - ‘ ПВ = 2А with — A G M 2n(Z )

138 DE LA HARPE

The matrix A is clearly skew-symmetric, and also hermitian because it has purely imaginary entries. Define also

(7) S = è (*B fi + *ПВ)

so that 4ВЯ = A + S.

Step two ( f may be degenerate): There exist a constant p and a continuous map ф : Cn->- С such that ф(z + cjj) _ ф(z) s vj(z) (mod. imaginary terms) (j = 1 , 2n) and |p(z)exp{-0(z)} I < д fo r all z G C n.

Let Sj be the j * diagonal entry o f S (also o f 1ВЯ), let rj = Re(f3j — is j), and let

6 R 2

Г ” '/ 2 п /

Notation being as in Proposition 12(iv), define the quadratic expression

çn p2n

z ** i <x|Sx) + <r|x> with x = 7 _1(z)(8) ф:

Let ej, ..., e2n be the canonical basis o f R 2n, so that e, = 7_1 (w i), •••> e2n = 7~4<J2n). As both the matrix S and the form <|> are symmetric: ф( z + w k) — ф(z) = i<x + ek|S(x + ek)> ++ <r|x + ek> — i<x|Sx> — <r|x> = <Sek|x> + 4sk + rk for к = 1, ..., 2n. Now Re(<Sek Ix)) = Re(<S—A)ek|x>) and <(S—A)ek|x> = (*ИВек|х> = ® ek|i2x> = <bk|z) on one hand, and Re(isk + rk) = Re(0k) on the other hand, so that the first relation stated in step two holds.

Notice that

lp(z + cok) exp{—0(z + 0Jk)}| = |p(z) exp{vk(z )} exp{-tf>(z) - Re(vk(z )) + i(...)}|

= |p(z) exp{-0(z)}|

It follows that the continuous function

Cn -* R+

z «■ lp(z) exp {-0 (z )} I

is continuous and Г -invariant, hence bounded by some finite real number, say p.

Step three ( f may be degenerate): There exists a holomorphic map oj : C" -*■ С such that

|p(z) exp {- o j (z ) } I < ¡i exp{-l-Re((Hz“|z>)} for all z £ C n, where H = ^BG is a hermitian matrix.

One has (first equality as in step two)

Re«x|Sx» = Re«x|‘ BSÎx)) = Re«G z + Gz llBz>) = Re«Gz|*Bz>) + Re((BGz |z>)

and also Re(<r|x)) = Re(<r|Gz + Gz » = 2Re(<r|Gz>). Define the holomorphic function

(9)

COMPLEX TORI 139

Ç n - > Ç

(10) со:

z *4<Gz|‘ Bz> + 2<r|Gz>

and the matrix

(11) H = iB G G M n(Ç )

One has Re(0(z)) = Re(co(z) + (Hz |z>) so that

|p(z) exp{-co(z)}| = |p(z) exp {^0(z)}| exp{+Re((Hz1z>)} </x exp{+Re((Hz|z>)}

for ail z G Cn by step two.By Proposition 12(ii), H may also be written as

H = +4 ‘ (fiG )BG - i ‘G‘ B (fiG ) = i*G (‘ nB - *Bfi)G = - ‘GAG

As A is hermitian, ‘ H = -*G ‘ÂG = - ‘ GAG = H and H is also hermitian.Observe that

/ ‘G\ _ / ‘GAG ‘ G A G \‘P_1AP_1 = ( J A(GG) = ( _ _ )

\ ‘G/ V G A G ‘G A G /

where 2‘ GAG = ‘G ‘ B (fiG ) - (W fi jB G = 0 by (6) and Proposition 12(ii), so that also ‘GAG = 0, and where ‘ GAG = ‘ ( ‘ g ‘ AG) = - ‘ ( ‘GAG) = - ‘H. This implies

/ 0 -H

(12) *P_1 A P '1 = (\*H 0

so that, in particular, A is invertible if and only if H is invertible.

Step four ( f may be degenerate): effect o f a change o f coordinates in Cn.

Suppose U is a unitary (n X n) matrix: *U = U” 1. I f we change the co-ordinates in Cn/ IT1 0 \

according to U, the matrix Í2 is replaced by Í2* — U f i; similarly P* — I — ■ I P and

P * "1 = (G * G *) = (G G) ^ . As <bj |z> does not change, (bj^lz*) = <bj"|U *z> = <‘ U 1 bj lz)

= <bj|z> for all z € C n, so that bj* = ‘ Ubj(j = 1, ..., 2n) and B* = ‘ UB. (Note that 2A* == ‘ B *fi* - ‘ f i*B * = 2A.) Finally, H* = 4B*G* = |‘ UBGÜ = ‘ UHÜ = ^ ( ‘ U )’ 1.

In particular, modulo an appropriate change o f co-ordinates, one may always assume H to be a diagonal matrix; its entries are then necessarily real.

Step five ( { is non-degenerate/- The matrix H is positive definite.

Recall that a matrix is said to be positive definite if it is hermitian and if its (necessarily real) eigenvalues are all strictly positive.

140 DE LA HARPE

By step four, one may assume that

Suppose, e.g., that hn < 0. Let

Then the function z «• p(z) exp {-co(z )} defines an entire function o f the one complex variable X. By step three, this function is bounded: |p(X) ехр {-ш (Х )} К ц exp{hn |X|2} =/¿exp{-|hn| |X|2} for all X £ C; Liouville’s theorem implies that it is a constant. Similarly z » q(z) exp { - cj( z ) }

does not depend on the last variable, and neither does z » f(z ) = p(z)/q(z). But this is impossible if f is non-degenerate, so that step five is proved.

Proposition 13. Let Г be a lattice in Cn furnished with a basis and let ft be the associated period matrix. Then the following is a necessary condition for the existence o f an abelian function f on Cn with G (f) = Г:

There exists a (2n X 2n)-matrix A with

(1 ) ;— A G М2п(Ж) and A is skew-symmetric, iff

(2 ) A is non-singular.(3) ПА-1 ‘ f t = 0.(4) f t A M ‘ f t < 0 .

Proof. We take A given by (6) above. From (12) and step five,A and H are invertible. From (12) and from the definition o f P:

/О ‘H->\ / ft\ _ / П А '^ П *PA ’ 1 ‘P = = A " 1 ( ‘ f t ‘ ft) = _

\-H -1 0 / \П/ \ f t A '“ ft *

so that (3) and (4) hold by step five. - , q

Caution: we do not claim that (l/iîr)A € GL2n(Z ), e.g. ( J is in М2(Ж) and in GL2(Q) but not in G L j(Z ). \0 2/

Exercise 44. Let со, = ( M u>2 = = f TJ\ and cj4 = ''j be as in the example ofоУ“ 2= W “ 3=W and“ *= U

/1 0 r, 0 \Section I I I .2, so that ft = I „ , „ I . Let (d,, d2) be any pair o f strictly positive mtegers.

\0 1 0 т2/

COMPLEX TORI 141

Check that

Оо

diл

0 0 0 à2

1 CU о 0 0

с»

?О

0

satisfies the four conditions o f Proposition 13.

Exercise 45. The Riemann matrix f t £ Mn2n(C ) being given, any (2n X 2n) matrix A satisfying the conditions o f Proposition 13 is said to be a principal matrix for i i . Check that there exists a principal matrix for f t i f and only i f there exist principal matrices for all Riemann matrices equivalent to ft.

Exercise 46. Let Г be a lattice in C n furnished with a basis and let f t be the associated period matrix. Prove that the follow ing conditions are equivalent:

(1 ) There exists a positive definite hermitian form H on C n with 1 т (Н (Г ,Г )) С Ж.(2 ) There exists a skew-symmetric R-valued R-bilinear form A on C n with А (Г ,Г ) С Ж

and such that

C n X C " - » R

(z, w ) ** A (z , iw )

is a scalar product on the real vector space underlying C n.(3 ) There exists a matrix satisfying the Frobenius relations o f Proposition 13.

Hint: For (1 ) (2 ), see Exercise 42. We shall indicate below (2 ) =>(3) and leave the converseto the reader. We hope that there will be no confusion between the meaning o f A here and in Exercise 42 (as in R ef.[20 ]) on one hand, and the A o f Proposition 13 (which is as in [17 ]) on the other hand.

Suppose (2 ) holds and let M 6 M 2n(C ) be defined by

(1 ) A (ftx , f ty ) = — ‘ xMy, x ,y G R 2n17Г —

i.e. by M = (l/ if f) ‘ f tA f t ; it is clearly a skew-symmetric matrix with (l/i?r)M 6 М 2п(Ж). Choose x,y £ R 2n, let z = f tx and w = fty ; then (1 ) implies

(2 ) A (z ,w ) = — ( l z 4 j + ‘ z^ O M ÍG w + G w )irr

and

(3 ) jrA (z,iw ) = V G M G w + * z ‘ GMGw - V G M G w - V G M G w .

Using ‘ zi'GMGJw = tw t(*GM G)z = - tw tGMGz and similar relations, one obtains

(4 ) 7rA(z,iw) — 7rA(w,iz) = 2tztGMGw - 2tzTtGMGw

142 DE LA HARPE

As (4 ) must vanish for any z,w G C n, one has

(5 ) ‘ GMG = ‘GMG = 0

and (3 ) simplifies to

(6 ) 7rA(z,iw) = * z ‘ GMGw + ‘ w k ïM G z

As A (z ,iw ) defines a positive definite form, one must have

(7 ) ‘ GMG > 0

Relations (5 ) and (7 ) imply that there exists a positive definite matrix К with

We have proved that M satisfies the conditions o f Proposition 13.

3. A COMPLEX TORUS W ITHOUT NON-CONSTANT MEROMORPHIC FUNCTIONS

We follow §31 in Siegel [17].

Suppose not. Then there exists a matrix A satisfying the conditions o f Proposition 13.Write К = Í7rA-1, which is a skew-symmetric matrix in G L 4(Q ). Then condition (3 ) o f Proposition 13 reads:

¡пиогсае1

(9 )

or

Consider the

cj4 = 1 in C 2. Let us first show that any Г -invariant meromorphic function on C 2 is

degenerate.

COMPLEX TORI 143

with

1-1,2 = Ki,2 + iV T K , ;, + iV 7 K lj4 — i\/2" K 2j3 — i-\/5" K 2,4 + (VT3- — \ZT4)K3i4 = 0

As the Kj k's are rational numbers, it follows from the last expression that the skew-symmetric matrix К is zero. This is impossible because К = inA-1 is invertible, and we have proved our first claim.

Let us then show that any Г -invariant meromorphic function f on С 2 is a constant. It followsfrom Corollary 2 to Proposition 10 that G (f ) contains a complex line V С С2. Let

, - , /0 Q = j o ) G G L 2 (Ç ) be an invertible linear map which sends V onto^ I)\ У 5 J z2

GC-

let Г * = Q (T ) and let

'a /3 i(a ,/2 + |V 3 ) ¡ ( «ч Л + (Зл/7) '

7 5 i(jy/2 + 5^3) i (7 \ A +

i(a-v/2 + рл/J) = n ,i3b, + n2>3b2

i(a-v/F + (3 s/7) = n1;4b, + n2j4b2

Then the function Qf, defined by Q f(z ) = f (Q -1 z), is invariant by Г * and does not depend on z2; hence this function defines a meromorphic function o f one complex variable, say g, which has a, /3, i(a.s/2 + /3- /3) and ¡(a^/F + /3^/7) as periods. Either these four periods generate a lattice in C, or g and then also f is a constant. We shall show that the first alternative cannot happen.

Suppose the four periods above generate a lattice in C, and let b b b2 be a basis o f this lattice Then there are integers n M ,..., n2>4 such that

“ = n i,i bj + n2|1b2

/3 — n ii2b, + n2)2b2

By elimination o f a and (3:

+ i-\/3 n1]2 - n j)3) + + i-y/3 n2>2 - nJ 3) = о

bi (i\/5 nM + Ц/7 n , 2 - n lj4) + b jii^/Fn 2jl + is/7 n2j2 - n2>4) = 0

This is a homogeneous system o f linear equations which possess a non-trivial solution: b b b2. Hence

i\/2n2>i + iy/7 n2j2 — n2>3 ^

iy/Tn2>1 + Í\Jl n2>2 — n2)4y

( W 2 n l,l + Ч А n l , 2 - n , , 3

i%/5 n M + iV T n 1;2 - n,_,= 0

This is a linear expression in л/Т4, i\/T, iy/2, i\/7, i\/3, 1, уТ 5 , with coefficients in TL. It follows that all coefficients must vanish, which reads (in the order given: \/\A first, and so on):

n i,in 2i2 - пЬ2п2;1 = 0

П 1,1 П 2,3 — П 1,3П 2,1 = 0

П 1,1 П 2,4 — n l ,4 n 2 ,l = 0

П1,2П2)3 — n lj3n2j2 — 0

n l,2 n 2,4 — n l ,4 n 2,2 = 0

П 1,зП2,4 ~ П 2,3П 1,4 ~ 0

144 DE LA HARPE

Written differently, the rank o f the matrix

(:« 1 , 4 П 2}4

n l,2 n 2,2

n 1,3 n 2,3

\

is at most one, and a, j3, i(a^/2 + py/3), i(a^/F + jS- /7) do not generate a lattice in C.

We have proved the following result.

Proposition 14. There exist complex tori o f complex dimension 2 without non-constant meromorphic functions.

Corollary 1. There exist complex tori which are not algebraic.

Proof. Meromorphic functions separate points in Pn(C ), hence in any submanifold o f Pn(C ), and they certainly do not in the example above. Compare with Exercise 12 and Proposition 9. ■

Corollary 2. There exist Kâhler manifolds which are not algebraic.

Proof. See above, and Exercise 18. Compare with Robertson’s paper, Section 11. ■

Propositions 9 and 14 imply evidently that the Corollary to Proposition 4 does not hold in the complex domain.

4. F IN A L REM ARKS

Let Г again be a lattice in C " with basis {c o , , ..., co2n} , and let f t be the associated period matrix. It turns out that the conditions o f Proposition 13 are not only necessary but also sufficient to ensure the existence o f an abelian function with Г as group o f periods.

Theorem. Let f t be as above. Suppose that there exist a (2n X 2n)-matrix satisfying conditions ( 1) to (4 ) o f Proposition 13. Then mere exists an abelian function f on C n with G (f) = Г.

Proof. See Siegel [17], §21 —26, or Conforto [16], most o f Ch.III, or Swinnerton-Dyer [11], §6.

It is also known that the conditions o f Proposition 13 are necessary and sufficient for a complex torus to be algebraic. We refer to the literature for a proof and for further comments, but we emphasize that escaping this p roof is escaping one o f the main points o f the subject.

Tw o topics are not included in this paper which were covered in the course o f lectures delivered: (a ) A report by Hefez on some open problems and on a recent result (this will appear in his thesis); (b ) Siegel’s generalized half-planes (see Siegel [21 ] for a classical exposition, and Siegel [17] for connection with algebraic complex tori).

ACKNOWLEDGEMENTS

I am grateful to J. Guenot for some discussions, and to the Fonds national suisse de la recherche scientifique for partially supporting the writing-up o f these notes.

IAEA-SMR-18/20

SOME PROBLEMS IN THE THEORY OF SINGLE-VALUED ANALYTICAL FUNCTIONS AND HARMONIC ANALYSIS IN THE COMPLEX DOMAIN

M.M. DJRBASHIAN Institute of Mathematics, Academy of Sciences of the Armenian SSR, Yerevan, USSR

Abstract

T H E O R Y O F S IN G L E -V A L U E D A N A L Y T IC A L F U N C T IO N S A N D H A R M O N IC A N A L Y S IS IN T H E

C O M P L E X D O M A IN .

Part I : 1. T h e in tegral form u lae o f Schwartz and Poisson. 2. T h e D irich let p rob lem fo r the disc. Part II:

Th e class o f harm on ic functions representable b y the Poisson-Stieltjes integral. Part I I I : 1. T h e opera to r L ^

and its p roperties. 2. F orm u lae o f the Cauchy, Schwartz and Poisson typ e . Part IV : 1. General classes o f functions

harm onic in the disc and their representations. 2. A genera lization o f the theorem o f H erg lo tz . Part V : T h eo ry

o f fa c to r iza tion and boundary p roperties o f functions m erom orph ic in the disc. Part V I : H arm on ic analysis in

the com p lex dom ain and its applications in the th eory o f analytical and in fin ite ly d ifferen tiab le functions.

Part V I I : Som e open problem s.

INTRO D UCTIO N

Parts I —IV deal with representation and boundary values o f certain classes o f analytical and harmonic functions in the disc |z| < 1 and contain results obtained by the author as well as those known for a considerable time. Part V presents a review o f the author’s principal results in the theory o f factorization o f the functions meromorphic in the disc |z| < 1 and their boundary values. As a rule, the proofs are omitted for the sake o f brevity. Part V I consists o f a review o f the author’s results in the theory o f harmonic analysis in the strictly complex domain and, in this connection, to the representation o f some other classes o f single-valued analytical functions. Here also, limitations o f space prevent the presentation o f proofs. In Part V II some open problems are formulated.

Part I

1. THE IN TE G R A L FO RM U LAE OF SCHW ARTZ AN D POISSON

Let the function f (z ) be analytical in the disc D (R ) = {z;|zi < R } and continuous in its closure

D (R ).From the Cauchy integral formula we have

¿¥7 j — T; '> : - i s ( i . i )i¿ l= R [0 , ieC -D ( i l )

145

146 DJRBASHIAN

Therefore, at each point z* = R2 /z, symmetrical with the point z G D(R),

¿ s i\ ¿ 6 D fR )ftl'R ¿

In view o f f = Relv>, d f = i Re'^dip, passing over to conjugate values, we get

1 * IM=R ISI-R N =R.

S lyÍR ^ ¿ f ISI-R ^ ?

= 0 , ге DÍR.)

Hence for any point z G D(R)

_L \ Т & Л ' J7-T s s r i ) - = W

Itl-R.

Summing Eqs (1.1) and (1.3), we get

|S[=H ъ

where u(H = R e f(f)-When z = 0, this yields

ь м - г М - ф - л

N=R

Hence, from (1.4), the following identical formulae are obtained:

= ¿j„ 1 ( 0 ) + ¿ i \ t f « a ) 4 rN=R

( 1.2)

(1.3)

(1.4)

(1.5)

SINGLE-VALUED ANALYTICAL FUNCTIONS 147

This formula, expressing the values o f the function f (z ) analytical in the disc \z\ < R through the random values o f its real part on the circumference |f| = R, is called the Schwartz integral formula.

Writing z = re*4’ , from the Schwartz formula (1 .5 ) we get

и а ^ ) = у (ъ е ^ )= R e £ (ге ‘л’ )

, 29 _ 1 ’ ~

iff

n j r nl y.1o Hel +ze _ к ~г__________R e "- е е * •

we have

3S¡R- ^ ,,n «n (1 7)

те1Г] âSï \ ff-áñzbHÍ'f-d)**-3-

Thus we arrived at the Poisson formula (1.7), expressing the values o f a function u (z) continuous on the disc |z| < R and harmonic in its interior, through its values cn the boundary o f the disc.

The function

í í r ' S;^ "R 5 -¿ R rc »Cr-3)+t*~ (L8)

itself harmonic in the disc Ire'll < R for any i> S [0, 2rr] by virtue o f (1.6), is called the Poisson

kernel.

2. THE D IR ICH LET PROBLEM FO R THE DISC

. The classical problem o f Dirichlet reads as follows.To find a function u(x, y ) = u (z) harmonic in the disc |z| < R and continuous on its closure

izl < R such that u (f ) = h ( f ), |f| = R where h (f ) is a previously given function continuous on the circumference If I = R.

I f it is also known that the function f (z ) = u (z) + iv (z ) is continuous on the closed disc |z| < R, then, by virtue o f the Poisson formula (1.7), the required solution o f the Dirichlet problem can be represented in the form:

JUT

lèl < R. d - » )

We shall prove now that the Poisson formula (1 .9 ) solves the Dirichlet problem in the given formulation.

148 DJRBASHIAN

With no loss o f generality we can assume that R = 1 and write the relation (1.9) in the form

noting that according to (1.6)-(when R = 1)

Since P(<p - i?; r) is harmonic for any d 6 [0, 2ir], then

^ S ( f - h t ) = ^ r + ^ r = 0 , H i O , s s ]

and, therefore, we also have

i.e. u (z ) is harmonic in the disc |z| < 1. It is also easy to see that

t m . U(i) — k(ei3°) (1.10)°

U K i

Indeed, for the harmonic function u (z) = 1, according to the Poisson formula (1.7) (when R = 1), we have

1ST. i t i - l i l * |a .1 JLsr ) |е‘Э-г|г ; W < I. le - г

Hence

isr( 1. 11)

i, iA .• I -4 . I • 4 . I 1Л( 1.12)u a ■) -/[ (е1Ц - j£ . j|ei»J 2|4 [ k(e^ .) — h fe ^ " ) ] J 5

Since the function h (f ) is continuous on the circumference If I = 1, it is 27r-periodical and uniformly continuous for all S € ( - “ , + °°). Therefore for any e > 0 there is a 6 = 6 (e) > 0 such that

| U e ‘V U e A )l</- (113)

for any pair â0 for which |i> - & 01 < S.

We write ( 1.12) in the form

ы < г ) - и еЛ ) = 1 + I A

where

, â-T w , 1

I i ' j r C ! - J , J j i ï i p r [ > ‘V U e ‘'S- ) J « f land here, by virtue o f (1.11) and (1.13), we have II, I <e/2. Further, denoting

me* I k C e ñ l

we shall have

SINGLE-VALUED ANALYTICAL FUNCTIONS

where

§ ( i ) = тли Ie‘ - î| , ? £ j )

and hence

Д ^ ( О Ч е Л - 1 \> 0

Consequently, we can take z so close to eilJo as to have also |I2| < e/2.

This completes the proof o f the equality (1.10).

3. THE IN TE G R A L OF POISSON-STIELTJES AN D ITS BO U ND ARY VALUES

Let us remember that the real function iK $ ) defined on the interval [0, 2ir) is called a function o f finite variation, i f

where the upper bound is taken over the set o f all possible divisions o f the interval:

o = r < r < - < & Г < r a * 0

149

150 DJRBASHIAN

Assume that a real function ф(&) o f finite variation is given, defined on the points e11? (0 < d < 2tt) o f the unit circumference. An expression o f the type

. ЛУГ ¿

UCte 1 -1ъик(Э-г) + (1-14)

is usually called a Poisson-Stieltjes integral.

As in the case o f ordinary Poisson integrals, it is easily seen that the function u (z) is harmonic in the disc \z\ < 1.

Theorem 1.1. At any point e1 » where the derivative ф ’(д0) exists and is finite we shall have

u i i e A ) = m > ) (1J5 )

Proof. We put tp = t?0 in ( 1.14) and integrate it by parts:

v, ¿S.)-il (L-t*)f(9) f* i f , . я.

or, assuming that the function iJ/(â) is continued to a 27r-periodical function, defined on the entire axis < t? < + we shall have

Ш геД )=-д1г [ W iS r - O j - f f t O ) ]

(1.16)

v i - l J,/, 2 л (1 - г лН0гЭ I q

I f i?0 = 0, then ф(2п - 0) - ф(+ 0 ) = 0, since by the assumption ф(д) has finite derivative ф'(д0), and i f t?0 Ф 0 then

1 - r 2

l - 2 r c o s # + r 2

tends to zero when r -*• 1 — 0. Hence, when г 1 — 0, from ( 1.16) we have in both cases

Further, since ф \ д 0) exists, it is obvious that, putting

Ш ' 3 ) = t W $ + (1-17)

for any e > 0 we can specify r¡ = т?(е) > 0 (tj < ir) such that

| £ (9 ;& )| < ! • when 151 < 1 U -18)

Divide the integral (1 .16 ') into three terms I j, 12 and I 3, performing the integration over

the intervals [—7r, - 77], [—17,-77] and [77, тг] respectively, i.e.

г / ( г е ‘- ^ ) = о ( ' 1 ) + 1 1 + 1 А + 1Ъ, г — i - 0 ( 1 - 15Г)

where

oId

while I2 and I 3 are defined by similar integrals over [ - 77, 17] and [r¡, 7т] respectively.Let Sup lŸ (S) I ~ ñ ■ Then it is obvious that

[0,1*]

IT I i t 1 < Ci~t*)A1 lie I ^ ( i - b w j + г5)2

i.e. for r -> 1 - 0 ,1, = I 3 = 0(1). Hence Eq.(1.19) takes the following form:

UCce19’) — ! ^ o ( i ) ¡ г _ 1 - 0 (1-19')

Now applying (1.17), rewrite I2 in the form

irá- I [ Ш ) -

j ? j|" i - i i t b d + t * - ^

(1.20)

By virtue o f (1.18) and o f the fact that, as easily seen,

we shall have I ® | < e l 2 and hence, by virtue o f ( 1 .20 ),

г А = [ т к Л ‘ ° d-20')

where |a| < e.Returning to Eq.( 1.19'), by virtue o f ( 1.20'), we shall have

и ( г e¿S°) = [ f '(S ,) + 1 ] ;Ц °+ о ( i ) , г - Í -0

152 D)RBASHIAN

The passage to the limit in the latter relation when r -*• 1 - 0 yields the statement (1.15) o f the theorem i f we show that

fcm C = i t-1-0 1

For this purpose let us note that

LW

ЗГ

S' 3*r (L) Í f { — ?“^ = f M + ° a h t - i r O (1.21)

l - t ¿ . JL [ 1 - fc* /qI х ЗГ J 1 * i *'

d m r J 9 = 0z—L-o J 1 -¿гем^,+гг

Noting finally that according to ( 1.11 ),

57 , ГL -Z -1

'S ' j

we can represent (1.21) in the form = 1 +0(1), r -*• 1 - 0, which completes the proof o f the theorem.

Since the function ф(&) o f finite variation on [0, 2эт] has finite derivative almost everywhere, and ip\&) e L(0, 27t), then from Theorem 1.1 follows:

Theorem 1.2. The harmonic function u(z) representable by Poisson-Stieltjes integral, has radial boundary values

U (ze íS)= - U (e ¿S) = - j ' l( ^ ) G Í ( o , l S ¡ ) (1.22)

almost everywhere on the circumference z = e1 .

In particular, the Poisson-Lebesgue integral

¿Ü

uli)="éf \ (1'23)о

with an arbitrary f (f l) G L(0,27r), can be regarded as a Poisson-Stieltjes integral with the function

which is absolutely continuous on [0 ,2n\.Therefore the preceding theorems yield the theorem o f P. Fatou (1906).

Theorem 1.3. A t every point z = eilJ, where ф '(д) = f($), i.e. almost everywhere on |z| = 1

& т о u (z e LS) = г/(е^)=|б9) (1-24)

0

Part II

THE CLASS OF HARMONIC FUNCTIONS REPRESENTABLE BY THE POISSON-STIELTJES INTEGRAL

Let us first prove the Herglotz theorem (1911):

Theorem 2.1. Let the function u(z) be harmonic in the disc \z\ < 1 and

U ( i )& 0 , l i l < l (2.1)

Then u(z) is representable by the Poisson-Stieltjes integral:

i ¿sr i - *UÍZeCf) ^ ¿ S f Í 1 - W f W T ° ( m (2.2)

where ф (д) is a bounded non-decreasing function on [0, 2тт].

Proof. Let

9U($eL* - )4 ¿ , 0 < $ < Í , 0 & 9 *í ¿5 í (2.3)

In view o f (2.1),{v7p (1 )} (0 < P < 1) is a family o f absolutely continuous functions nondecreasing on [0, 2n], for which

¿a Os rs (5) é \ u (S e u )JJL = 2 ï ïu (o )

and obviously

isrV ( K ) = 15Г U (0 )

154 DJRBASHIAN

According to the first theorem o f Helly, this family is compact, i.e. there is a sequence pn t 1 such that

f . ( d ) = Ÿ Q )n .- .~ 01

for all d € [0, 2tt], except for a countable set o f discontinuity o f the function ф($). Thus, the limit function will be bounded and non-decreasing on [0, 2ir],

According to the second theorem o f Helly,

J rZ ï ^ c i m + o a ) (2.4)

when n -*■ + °°.But in view o f (2.3) and the Poisson formula,

¿5T x

0

(2.4')

Finally, passing to the limit in (2.4) and (2.4') when n -*■ + °°, we get the representation (2.2) o f the theorem.

Denote by U the class o f functions u(z) harmonic in the disc |z| < 1 and subject to condition:

j j j = (-2-5'*

Now prove the following general

Theorem 2.2. The class U coincides with the set o f functions u(z) representable by the Poisson- Stieltjes integral and also with the set o f functions u(z) representable in the form

Vd) = UL( l ) -U l ( i ) , | 2 | 0 (j = 1, 2) are harmonic functions in lz| < 1.

Proof. I f u(z) G U, letting again

bVS)=5 u($eu )J¿, 0 <S< 1

* О

we obtain a family for which

, llrI and y ( f j ) 4 ¿ í r ] [ u ]

Therefore, repeating literally the proof o f Theorem 2.1, we shall obtain the representation

=¿f T , , - <•/

now only with "V ( t ) o w . Further, representingО

where ipj(â) (j = 1, 2) are now non-decreasing and bounded functions on [0, 2ir], for u(z) we obtain the representation (2.6).

Finally, i f u(z) permits a representation o f the form (2.6), then

¿ s r ¿ ir я г

$\ u ( t e ¿ f ) \ j Y « \ и Л г е * и Ч + )

о о о

=Д5т[к1(о) + г/4го)] f ( о < г < 1 )

i.e. u(z) € U.This concludes the proof because, according to the Theorem 2.1, the functions o f the type

(2.6) are representable by the Poisson-Stieltjes integral.Note in conclusion that, while proving the preceding theorems, we saw that

№ ) = еы ] и ( $ я е а ) Л 1in"” 1 о

at any point € [0, 2rr], where the function ф(д) is continuous.It can be proved that, i f u(z) £ U is also representable by means o f another function ф\(д),

then

(d ) = ф( i>) + const

for all points, where the functions are both continuous. In other words, the representation o f a function u(z) E U by Poisson-Stieltjes integral is unique.

Theorem 2.3 (the theorem o f P. Fatou). Below, we shall lean upon a statement derived from the theorem o f Vitaly:

Let the function f(z ) be analytical and bounded in a sector

Д и ) = { г ; I eutj г I , 0 <| í | < r J

and the limit

156 DJRBASHIAN

exists. Then fo r any 0 < a¡ < a we also have

lim f (z ) = A (2.7)2->0

z 6 & (£»[)

From now on i f the limit A in (2.7) exists, let us agree to call it an angular boundary value of f(z ) at z = 0. The same term will be used if the vertex o f the sector is at an arbitrary point of the boundary o f the disc |z| < 1.

We now prove the theorem o f Fatou (1906).

Theorem 2.4. Let the function f(z ) be analytical in the disc |z| < 1 and in addition

Hence the harmonic functions u(z) and v(z) belong to the class U and are representable by Poisson-Stieltjes integrals.

Therefore, by virtue o f Theorem 1.2, we can assert that almost everywhere on If I = 1 the radial limits

exist and f* (e il5) 6 L(0, 2it). But, since any radius Of can be regarded as a bisectrix o f a sector Aj- o f the type (2.9), by virtue o f the corollary o f Vitaly's theorem mentioned, from (2.8') follows also (2.8), i.e. the statement o f the theorem. This theorem o f Fatou allows us to supplement Theorem I . I about the existence o f boundary values o f the Poisson-Stieltjes integrals.

Let u(z) G U and u(z) = U j ( z ) - u2(z), U j(z ) > 0 (j = 1, 2), vj(z) be the harmonic functions, conjugate to U j( z ) . Then the functions

Then, fo r almost all points o f the circumference 'If I = 1, the angular boundary values

f ï « 0 = £ ^ ( è ) (2 .8)

exist, when z -* f along an arbitrary way, belonging to a sector o f the type

(2.9)

Proof I f f(z ) = u(z) + iv(z), then

( 2 .8 ')

are analytical and obviously bounded in the disc |z| < 1, and hence, according to the Theorem 2.4, it has definite angular boundary values almost everywhere on the circumference. Hence the functions Uj(z) (j = 1, 2) must possess the same property, whence, with respect to the assertion

и и , ( и ‘ 8) = ^ ] ( j = L , i )>1-0 J •>

o f Theorem 1.2, it follows that uj(z) has angular boundary values ф '(д) almost everywhere on the circumference. Consequently the angular boundary values o f the Poisson-Stieltjes integral u(z) = u ,(z) - u2(z) exist almost everywhere on the circumference f = e“ , are finite and equal to ф '(.» )= ф № )~ ф 'гт

In particular, the angular boundary values o f the Poisson-Lebcsgue integral (1.23) also exist almost everywhere on the circumference £ = eil5 and are equal to f(i3).

Part III

1. THE OPERATOR i/ " ) AND ITS PROPERTIES

Denote by £2 the class o f functions satisfying the following conditions:

(1 ) w (x ) > 0 and continuous on [0, 1),

(2 ) ы (0) = 1, / w (x )dx < + <»,о

(3 ) / I I — co(x)|x_1 dx < + °°.0

Further, let us agree to denote by Рш the class o f functions p (r) for which

1p(0)=iend р ( г ) = Л Ш * - Ж t e ( 0 , i ] (3.1)

x л

for some co(x) e Í2.In view o f ( 1 ) and (2 ) we shall have p (r) e C[0, 1 ], and it is easily seen that

P(*0)=P(0)= 1 . P ( i ) = 0 (3.2)

For р (т) 6 Ры we introduce the operator

<3-3)

assuming that in proper classes o f acceptable functions < (x) on (0 ,1), the right-hand part in (3.3) exists at least almost everywhere.

Note that in the simplest case, when co(x) = 1 for any function <p(x) e L(0, 1) we shall have

¿ïw)[ïïx)] = f(x)

158 DJRBASHIAN

almost everywhere on (0, 1). In fact, in this case

P fr ) = t - S ^ r = l - f г *

and therefore for almost all x £ (0 , 1)

L<“ ,[ n x ) ] = ¿ ¡ x j m t ) c / r } = ¿ { = f ( i )

In certain classes o f acceptable functions <p(x) and under additional restrictions on the function co(x) S П, the operator L ^ ) also permits other representations simpler than (3.3).

Lemma 3.1. (a) Let co(x) € Í2, oo(x) £ C[0, 1 ] and co'(x) £ L(0, 1). Then fo r the class o f functions (Дх), piecewise continuous on [0, 1], the formula

L(U>[ f ( x ) ] = b > (l)W x )-\W x ? )b J ? T )i> /T , xe (0 ,1 ) „ (3.4)Û

holds.(b) For the class o f functions </?(x), possessing a bounded and piecewise continuous derivative

<p'(x) on [0,1],

l

¿<a)[V ( x ) ] = m + x $ r h T ) c ú ( T ) j T , x e (O f i ) (3.5)О

Proof(a) In the case considered, we get from (3.1) by means o f integration by parts

Р(Г)=-ю(1)Г+ с/х r

and therefore

Hence, from definition (3.3) o f the operator l ( “ )[<p], we find

t b M l = - ¿ 1 * j n x T ) ( - u ( L ) J t j

= - £ { j , xe (0,1)

0 %

П > ) ] = -U(l W x ) - x \ W w V -f Jo/Г, X ф , 1)

from which Eq.(3.4) follows.(b) By virtue o f (3.2), integrating by parts we shall obtain

L i

-x f(0)x + X*S M * ? ) P ^b /г

= f f o )x + x ¡¥ ’ , ) p f ^ ) 0/r> x e ( o , i ]

This yields

^ ’ C f i x ) ] = m + ^ r l x j r f a - p & o f ï j

= PÍO) + j у ? г , { p f г } _ г . p ' ^ / o í r , x e (0 , i )

But from definition (3.1) o f the function p (r) it follows that, in general, p (r) - rp '(r ) = co(r). Therefore

Lfu)[ W x ) ] = Ш + \ f(r)b > (-f )olr, x e (0 ,1 )

which yields the correlation (3.5) o f the lemma.As is known, the Riemann-Liouville operator D_<V (x ) ( - 1 < a < °°) is defined as follows.

I f 0 < a < + ° ° ,

x

D V o = j= ^ y \ (x-if L?(OJit, xe.(o,e)w.

and when <p(x) E L(0, S), the right-hand part exists almost everywhere and again belongs to L (0 ,2). A n d f o r - l< a < 0

D'xn x )= ¿ -D - í t u V x ) , xe(C,e)

under assumption that the right-hand part exists almost everywhere. Finally, it is assumed that

DVx)2=yYx), x&(o,e)

The operator L ^ t ^ x ) ] is an essential generalization o f the Riemann-Liouville operator, as the following lemma shows.

160 DJRBASHIAN

wCx) = ( l - x ) ¿ ( - 1 < / < о ° ) (3.6)

the formula

LM [W k) ] = (3.7)

holds almost everywhere on (0,1). And fo r the values 0 < a < + °° it is valid in the class o ffunctions v?(x) G L (0 ,1), and at least in the class Ci [0, 1 ] o f continuously differentiable functionson [0, 1] i f - I < a < 0.

These assertions follow directly from the assertions (a) and (b) o f Lemma 3.1 i f we note that

u > ' ( x ) = - ¿ a - x f ~ L < £ L (o, d

when 0 < a < + °°

Lemma 3.2. When

2. FORMULAE OF THE CAUCHY, SCHWARTZ AND POISSON TYPE

Let us introduce the function

Í

I , when r = 0

R 8 >

obviously continuous on [0, + о»). Note that

/<“ ) Г t l л , V I ( 0 & 1 « - « ИL [x J=A(Z)X , (û^Xé 1 j (3.9)

Indeed, according to (3.5)

LCw)[x°J =Z.f‘*J’[l] = 1-Д(о)х°and for 0 < r < +°o

[1Ы\ х г]= х \ b ( x t f ' Lc ^ ( r ) J r = X iA f Z )0

Putting Дк = Д(к) (к > 0), from definition (3.8) o f the function Д(г) we get

iA 0 = i , Д К= ( K i i ) (3.10)

0

Since co(x) > 0, x G [0, 1), for any 5 (0 < Ô < 1) we have

l0 < ^ ( S)=|ojO)eJx ü \ u ( x ) J Í K = +

and therefore it is obvious that

( K * l )

This yields

= 1 (3.11)^ _ * + Oo 14

Now, consider the power series

C d - , ^ ) = Y L ^ - Г I г i <£. l ) (3.12)(<-=0

and

ádib>) = ác(i}(j)-¿c(0¡ t)

= i- + i Z L ^ ; C i î k U (3.13)

By virtue o f (3.11) the functions C(z; co) and S(z; со) are analytic in the disc |z| < 1 and have singularity at the point z = 1 and,in addition,

ù С ( l ", co) = f i m S (¿ j u ) = +o° г -1-0 t - i - o

Note the special but important case when

It is easy to see that in this case

л A _ P ( W ) r q * K )a . - i . , a K r f l + ¿ + K )

and therefore


C ( ¿ ; c j ) ~ r ^ i ^ ) C / ¿ ) = (- ~ ^ r

S ( ¿ ; c ) = r L( L + ¿ ) S ¿ t ) = (T £ j n z - l

In particular, when a = 0, hence oo(x) = 1, we get

iC(è;i) = CJi) = jèT

162 DJRBASHIAN

Direct applications o f the operator result in analogues to the integral formulae o f Cauchy, Schwartz and Poissoii.

Theorem 3.1. Let the function

i ( 2 > = Z L a K*K (3.14)K=0

be regular in the disc |z| < 1. Then(a) The function

t ' [ f („ « Л Д Л ( и » ) к (3.15)

is also regular in the disc |z| < 1.(b) For any r (0 < r < 1) the integral formulae

Дзг

(3.16)о

iST

= + (3.17)

hold.

P roof(a) Since the function f(z ) is regular in the disc |z| < 1, according to (3.11) and the Cauchy-

Hadamard theorem,

Hence the function fw (z) is also regular in the disc |z| < 1. But, since according to (3.9)

J fc*i) г¿ “>[ У ] = Д 1 ( 1 = 0 , i ,3 L , . . . )

then, using Eq.(3.5) we get

O o 1

(b ) For any p (0 J fl=0 Ъ

converges uniformly with respect to the parameter ù £ [0 ,2тг]. Therefore, using the expansions (3.15) and (3.18) we obtain


K=0 rL-0

where

This yields Eq.(3.16) o f the theorem.To establish the second formula (3.17) we note that, as is easily seen

T ~

= fío) Д. + S(T-k; n ) A K ciKA ¿ ^ ¿ rLK=i n.=0

(i a < s )

since 'b ( - к ; П ) = 0 ( K = l . , ¿ , . . . ; n — 0, í , í , . . ■)■ Thus

W ) = ¿ r \ C ( e ¿s- ( - j c j ) ( U K Î )0 ¿

Adding this to Eq.(3.16), we get

_____ ЛЯГ

î(2î = -ÇiO) + -5 c ie ‘i8-|-;cj)Re|«Jf S ^ e/5 fa/<$) 0

Or, since

C (é ;< ¿ ) = j [ - & ( i ; u ) + j -

and

164 DJRBASHIAN

¿ Т ^ е ^ ( ? е ^ ) с 1 3 = Я . е £ 5 ( К ; 0 ) Д к а ,

= Цел.ав = ReÇ(O)

we arrive at Eq.(3.17) o f our theorem. In the simplest case, when co(x) = 1, Eqs (3.16) and(3.17) become the known formulae o f Cauchy and Schwartz, described in Part I.

Finally, introduce the function

P c o ^ » J) = R . e S ( « e ‘ ^ c j ) = i + ¿ 2 2 £ * < »5 KS

Note that in the special case when cj(x ) = 1, hence Ak = 1 (k = 0, 1, ...), this formula is identical to the Poisson kernel, since

QQ

l\ (3 ;¡ :)= 1 + i Z L K 'SK=L

П,п ч L-t3- /o<t<Lt ) ~ > 0 \ о & д ^ э ж

From Theorem 3.1 follows immediately

Theorem 3.2. Let u(z) be a harmonic function in the disc |z| < 1. Then the function

Uu (^e iS) = l ! Cj)[ u ( z e i9 ) ]

is also harmonic in the disc |z| < 1. Also, fo r any p (0 < p < 1 ) the integral formula

3SC

ïâ S ï,

3Sî ,

holds.

Part IV

1. GENERAL CLASSES OF FUNCTIONS HARMONIC IN THE DISC AND THEIR REPRESENTATIONS

Denote by иш the set o f functions u(z) harmonic in the disc |z| < 1, subject to condition:

where, as usual, co(x) € Я and

U b , ( t e ¿f) =

Since for co(x) = 1

U ^ ( t e ¿,p) = у с г ё * )

in this case the class Uu coincides with the class U o f functions u(z) harmonic in the disc |z| < 1, satisfying the condition

As established in Part II, the class U coincides with the set o f functions, permitting representation as the Poisson-Stieltjes-type integral. The following theorem gives the representation for the class иш with an arbitrary generator gj( x ) € f i .

Theorem 4.1(a) The class иш coincides with the set o f functions u(z), representable in the form

and pn t 1 is a sequence.(b ) The class UJ, С и ы o f functions u(z) harmonic in the disc |z| < 1 ,such that

u u ( i ) > 0

coincides with the set o f functions o f the type ( 4.2) with the function ф{&) non-decreasing on [0,2тг].

Proof(a) According to Theorem 3.2, we have for any p (0 о

166 DJRBASHIAN

On the other hand, since u(z) G и ш, it follows from (4.1) that

where

Thus, for the family o f functions Ф = (0 < p < 1 ) the conditions o f the first theorem o fHelly are satisfied. Hence there exists a real function o f finite variation on [0 ,2n] and a sequence pn 1 1 such that

Finally, fixing the value z = re'4’ (|z| < 1), we shall write Eq.(4.3) for p = pn > r (n > n0). Then, passing to the limit in (4.3), when n -»■ +°°, we get the representation (4.2) o f our theorem.

There remains to be proved the inverse assertion that any integral o f the type (4.2) represents some function u(z) € и ш. Indeed, the harmonicity o f this integral in the disc \z\ < 1 follows from that o f the kernel

By virtue o f Helly’s second theorem this yields

for all the values o f the parameter â G [0 ,2тт]. Further, since

then

OO

Therefore, i f u(z) is representable in the form (4.2),

= ¿ r \ P ( r - 9 ; t ) o l W ) (4.4)

This means that uw (z) G U; in other words, u(z) G иш.(b) Since иш(г) > 0, it is obvious that for each function u(z) G the function ф(д)

does not decrease. Conversely, i f i//(t?) > 0, then from (4.2) and (4.4) it follows that иш(г ) > 0, i.e. u(z) G U*,,

The following theorem shows that the classes for all possible co(x) G f i exhaust the whole set o f functions harmonic in the disc |z| < 1.

Theorem 4.2. For any function u(z), harmonic in the disc |z| < 1, there exists a function ш0(х ) G f i such that u(z) G

Proof Note that

U ( r e Lf) ] = C j ( i ) u a e cf) - \ Lt(zte¿r) c j ' a ) о

( 4 . 5 )

i f c j ' ( t ) G L ( 0 , 1). Put

U(t) = $ l u < r e ¿r)IJr ( o < t < L )0

and assume that

sup U(t) = +<*> 0 <t < 1

Then there exists at least one function Ь(т) continuous on [0,1] such that

i

0

For example, one can take

Introduce now the function

i

where

168 DJRBASHIAN

It is easy to see that co0(x ) 6 П; hence, using Eq.(4.5), we get

L

— \U(r)h(r)dT + ( 0 < Z < 1)0

i.e. u(z) G UUo and the proof is complete.This brings us at last to the theorem about the comparison between the classes and the

class U.

Theorem 4.3(a) I f the function to(x) G П does not decrease on [0,1), then иш С U .(b) //oo(x) G П does not increase on [0,1), then U С иш.(c) I f the co(x) t + °° or co(x) 4 +0 when x t 1, then the corresponding inclusions are strict.

We shall not dwell on the proof o f this theorem. Note that it is based on the following lemma which is easy to prove.

where oo(x) G f i and foAre1 ) = l (“ ) [fire*45)].Further, denote by R<¿ the class o f functions, analytical in the disc |z| < 1, such that

Lemma(a) / / cj( x ) J. 0 , x t I,then

tivn. = +*—1-0

(b) I f <o(x) t + o<=, x 1 1, then

i < ) = 0

where

2. A GENERALIZATION OF THE THEOREM OF HERGLOTZ

Denote by Сш the class o f functions f(z), analytical in the disc |z| < 1, such that

He ÇuU)»0, |èKl (4.6)

(4.7)

Since the integrals

S3t ^

= ж Я е Ъ ( 0 ) , 0 < í < i0 0

are bounded for f(z ) € Cw , it is obvious that Ccj C Ru .In the case co(x) = 1 the representations for corresponding classes have been obtained by

Herglotz and Riesz.For an arbitrary c j (x ) G Q there holds

Theorem 4.4(a) The class coincides with the set o f functions f(z ) representable in the form

i5rf ( í ) = ¿ C + ¿ $ S f e ^ i < j ) o W S ) (4.8)

О

where Im С = 0, ф(д) is an arbitrary non-decreasing function, bounded on [0 ,2тг].(b ) The class Rw coincides with the set o f functions representable in the form (4.6), where is an arbitrary function o f finite variation on [0, 2ir],

Proof. I f the function f(z ) allows the representation (4.8),'then

1ST

0

and according to Theorem 4.1, u(z) € Ucj. This means that the function

= = lie =

satisfies the condition (4.6), i.e. f(z ) 6 Сш i f ф(д) does not decrease and is bounded on [0, 2ir] and the condition (4.7), i.e. f(z ) G Rw if ф{0) is a function o f bounded variation on [0, 2ir],

Conversely, let f(z) G Ru ; then, denoting

?

0<S<1 о

according to Eq.(3.17) o f Theorem 3.1, we have

ХЧГ

f(«) = ¿ H (°)+¿írí S(e-£9f ; u ) o l t s(S) (U H )o

Now, in the same way as in the proof o f Theorem 4.1, we shall get the required representation (4.8) and with the function

170 DJRBASHIAN

ягfor which У O ') 00 . In particular, i f f(z ) G Сш, i.e. R efw(z) > 0, then ф(0) will benon-decreasing. The theorem is proved.

Part V

THEORY OF FACTORIZATION AND BOUNDARY PROPERTIES OF FUNCTIONS MEROMORPHIC IN THE DISC

Part V consists o f a brief survey o f basic results obtained in the period 1964-1972 in the theory o f factorization o f meromorphic functions. A detailed account o f these results would require too much space. For closer acquaintance with the methods o f these investigations we refer to the Bibliography for Part V, at the end o f this paper.

(A ) The new classical formula o f Jensen-Nevanlinna,and the most important notion o f the characteristic function which is deduced from it, constitute the basis o f the modern theory of meromorphic functions. We shall remind you this formula and the definition o f the characteristic function.

Let F (z) be a function meromorphic in the disc \z{ < 1, {a^} and{bp} be the sequences of zeros and poles o f F(z), respectively, different from z = 0, numbered in the order o f non-decreasing modulus and in accordance with their multiplicity. Then, for any p (0 1 and a pole if À < - 1 ); С is a real constant, defined exactly up to the items o f the type 2irm.

The problem o f complete description o f the class o f functions F (z) meromorphic in the disc |z| < 1 for which it is possible to perform the termwise passage to the limit in the Jensen- Nevanlinna formula,and thus obtain a representation for the function lnF(z) in the entire open disc |zl < 1, was posed and solved originally by R. Nevanlinna in the mid-twenties.

As in the investigations in the value distribution theory o f meromorphic functions, the solution o f this problem is also based on the determination o f the characteristic function:

(5.1)

(5.2)

where n(t; °°) is the number o f poles { b^} in the disc |z| < t (0 < t < 1);

6 гX = m a*{6n X ; 0 } ((K X *-*0" )

Defining the class N as the set o f functions meromorphic in |z| < 1 for which

i S u ç T ( ï î F ) < *o= ,

Nevanlinna showed that the termwise passage to the limit in Eq.(5.1) mentioned is possible only for thé functions o f the class N. Thus the following theorem about the parametric representation and factorization o f the class N was proved.

The class N coincides with the set o f functions F (z) representable in the form

= e*p i (5-3)

where

are convergent Blaschke products, ф($) is a real function o f bounded total variation on [0, 2ir], X 0 is an arbitrary integer, and у is an arbitrary real number.

Essentially, this fundamental result by Nevanlinna became the cornerstone and the base o f the construction o f an elegant theory o f classes A, Hg, etc., and their boundary properties. This result has also had substantial applications in other areas o f function theory, particularly in approximation theory o f functions o f the complex variable, in the theory o f operators.

(B) In connection with this basic theorem o f Nevanlinria and its method o f proof, the following problem arises naturally. Do there exist other, more general formulae o f the Jensen-Nevanlinna type, permitting the establishment o f parametric representation and factorization for wider as well as for more restricted classes than N o f functions meromorphic in the disc?

In the early papers o f the present author (1945), the first attempt in the direction o f solving a problem o f this kind was apparently undertaken, although the result obtained there was far from being completed. A generalization o f the Jensen-Nevanlinna formula was obtained there, being, as a matter o f fact, connected with the integrals o f fractional order. However, we obtained only a canonical but not a parametric representation for the classes N* (0 < ol < + «>) o f functions meromorphic in the disc |z| < 1, whose characteristics satisfy a condition o f the type

So, up to 1964, there had in fact been neither natural inner characteristics nor factorization theorems o f the Nevanlinna type, covering his theorem or making it more exact for wider as well as for more restricted classes than N o f meromorphic functions.

Considerably later (1964), this author returned to the problem o f factorization o f meromorphic functions in a general form, constructing a general theory o f parametric representation o f the classes Na (-1 < a < + °°) o f functions meromorphic in the disc |z| < 1, as complete as the now classical theory o f the class N. These results were presented in detail in the monograph

172 DJRBASHIAN

by the author (1966). The peculiarity o f the method o f this investigation was the systematic application o f the apparatus o f the Riemann-Liouville fractional integro-differential operators D '“ (-1 < a < + ° o ) .

By means o f these operators an important class o f new formulae o f Jensen-Nevanlinna type was discovered, depending on an arbitrary parameter a (-1 < a < + °°), coinciding with the classical formula (5.1 ) for the special value o; = 0. These principally new formulae lead naturally to the considerably more general notion o f the а-characteristic function Ta(r; F) also coinciding with the usual characteristic T(r; F) for a = 0.

These classes Na (-1 < a < + °°), possessing the important property o f inclusion »<*, С Na2 (—1 < о;, < ot2 < + °°),and coinciding with the class N o f Nevanlinna only in the case a = 0, were defined by means o f the condition

sup Ta(r; F) < +0 < r < 1

The basic theorem o f parametric representation o f the classes Na reads:

The class Na ( - 1 < a < + °°) coincides with the set o f functions, representable in the form

F ( î ) = c i P j¿ s [ ( L_ )i.^ - I ] o < (5.4:

where С is a constant, ф(д) a real function o f finite total variation on [0, 2jt], and Ba(z; a^) and Ba(z; b„) are certain convergent products in the disc |zi < 1 with zeros {aM} and {b „ } subject to conditions:

2 1 ( 1 - 1 аД )1+< + ° ° , ¿ l ( i - 4 Ê > l ) W^ o = (5-5)W O ’ )

In the case a = 0 this theorem is merely reduced to the theorem o f Nevanlinna formulated above.

The theory o f factorization o f the classes Na , although being considerably general, could not be regarded as final. The point is that in the case when 0 < c t < + «> (N a D N), they were still not able to include the functions meromorphic in the disc |z| < 1, having characteristics T(r; F) growing at an arbitrarily rapid rate when r -* 1 — 0. And in the case when -1 < a < 0 (Na С N), the constructed theory also needed to he made more precise, since for any a ( - 1 < a < + °°) the classes Na covered only the meromorphic functions whose distribution o f zeros and poles was characterized only by means o f a condition o f the type (5.5). So, e.g., the functions from Na without zeros and poles were being represented only by the formula:

t L * f сi me x p ( « ) ( l - i e ^

Thus, the problem naturally arose o f constructing the theory o f factorization o f functions meromorphic in the disc |zl < 1 in the sense that it could simultaneously include the functions with characteristics T (r) arbitrarily increasing when r -> 1 - 0, as well as the functions with arbitrary rare or dense distribution o f zeros and poles.

(C) Such a perfected and complete theory o f factorization o f functions meromorphic in the disc |z| < 1, including the theory o f the classes Na as a special case, was constructed in the recent investigations by the present author (1969).

This brings us to a cursory survey o f these results. Denote by Í2 the set o f functions co(r), subject to the following conditions:(1) oj(r) > 0 and continuous on [0, 1),

(2) Cj(0) = 1, / G0(r) dr < + oo,0

(3) the integral

K . = î ^ ^ o / xо л

converges absolutely.On the proper classes o f acceptable functions сДг), r €E [0, l),the operator L ^ ^ i^ r ) } will

be defined as follows:

where

U . t - 0p m

Note that if

W(O = i , Z 6 [ 0 , i ) ,

In particular, in the class o f functions yp(r) with bounded and piecewise continuous derivative ip'(r), the operator L^ li/K r)} is representable in the form

JL

lT\4(1)] = m + '(1Т)Ы(Х)Jt , (о,1 ]0

The operator L^H vKr)} is an essential generalization o f the Riemann-Liouville operator D"a (-1 < a < + oo), since, as in the special case, the identity

holds, when to ('x )= (-1 The functions

174 DJRBASHIAN

where

ДГ0) = 1 , Д С Д ) = ^ u ( t ) z * ' Lolz , Х&(0, + <» )О

are associated with the operator L ^ .The evaluation o f the simplest functions Л and log r by the operator l (w) results in the

important correlations:

With the operator are also associated the following three functions, playing an important role in the entire theory:

Vi ( « 4 * ) &3I1- 1^1},о

A J * ■>$) = ( i - \ - ) (0<l±UL)

Theorem 5.1. For an arbitrary function F (z) meromorphic in the disc |z| < 1 having the zeros {a^} and poles {b^.} and fo r arbitrary cj(r) e П and p (0 ~}т)* 0<<ar(s5 i<|f,|iS

¿Sí

+ s é - J ¿T ) ¿V |F(§e¿s)IÍ^ (l¿l<S) (5.6)

holds, where X and are defined from the development F (z) = c^z^ + cx+iz^+1 + ... (с^=£0)in a neighbourhood o f the point z = 0.

Note that in the special case cj(r) = 1, when Eq.(5.4) coincides with that o f Jensen-Nevanlinna, here, as in the theory o f Nevanlinna, the general formula (5.4) leads’naturally to the definition o f the functions:

Г \ t ' { U \ F C c e i f ) \\cl f (5.7)^ 0

and

/!£ ,(*; F ) = j — п(р!^ ( Ш + Ш ; ~ ) \ b p - L h (5.8)0

where l (w) = m ax-IL^, 0} and n(t; °°) is the number o f poles o f the function F(z) in the disc |z| < t, counted in accordance with their multiplicity.

By means o f these functions, passing to fundamental functions m(r; F) and N(r; F) o f the value distribution theory o f Nevanlinna, when co(r) = 1, the function

T w ( 4 F ) = t n . ( t ; F ) + 4 / ^ ; D (5.9)

is at last defined. We shall call this function the co characteristic. It coincides with the function T(r; F) o f Nevanlinna when co(r) = 1 and for it an equilibrium correlation o f the type

X ( T ; F ) = c ^ + T w ( t ; J L ) (5.10)

also holds.Finally, with every function co(r) G f i a class N{co} is associated as the set o f functions

F (z) meromorphic in the disc |z| < 1 subject to the condition

Note that the class N {00} coincides with the class N o f Nevanlinna in the special case when co(r) = 1. And in the case co(r) = ( 1 - r)“ (— 1 < « < + “ ) the class N{co} coincides with the class Na mentioned above. The comparison between the classes N{co}, when co(r) Ф 1, and the class N is given in the following theorem.

Theorem 5.2. Let co(r) G П. Then(a) I f co(r) is non-decreasing on [0, 1 ), then the inclusion N{co} С N holds. Moreover it

is strict i f co(r) t + °° as r t 1-0 .(b) //co(r) is non-increasing on [0,1), then the inclusion N С N{co} holds, and it is strict

i f co(r) 4- 0 as r f 1-0 .

(D ) Passing to the most fundamental theorems o f the theory o f factorization o f the classes N{co} brings us first to one o f the central theorems o f that theory.

Theorem 5.3. Let co(r) G f i and {zk} “ (0 < |zk| < |zk+1 < 1) be an arbitrary sequence o f complex numbers.

(a) In order that the infinite product

А ы (г-,гк) (s.ii)

converges and defines a function Вш(г) Ф 0 analytical in the disc |z| < 1, vanishing on the sequence { zk} “ , it is necessary and sufficient that the condition

Oo 1Z L \ < * o ° (5.12)

14,1

holds.(b) For any convergent product Bw (z) we have

(5.13)

176 DJRBASHIAN

The function Bu (z) is the natural analogue o f the classical Blaschke product

when instead o f the known condition

£ . ( l - u j ) < + o°K=i.

the sequence {zk} “ satisfies the condition (5.12). In addition,

Formulate now the basic theorem about factorization and the parametric representation o f the classes N {cj}.

Theorem 5.4. The class N{o>} coincides with the set o f functions representable in the disc |z| < 1 in the form

2Si' , -/S \ i , ..,1

(5.14)

where Вш(г; ад) and Вш(г; bv) are arbitrary convergent products o f the form (5.11), ф({>) is a

real function on [0,27r] with Ÿ ( t ) ‘ is an arbitrary integer, у is an arbitrary real number

and, finally,

\ 1-cjOc) IK « = ) — r ~ J *

0

In the special case when co(r) = (1 - r ) “ ( - 1 < a < + °°) from the representation (5.14) o f this theorem, the representation classes Na are obtained.

In the case when the function to(r) В П is non-decreasing on [0,1) and thus the inclusion N {c j} С N is true, the following theorem holds.

Theorem 5.5(a) i f the sequence {zk} “ satisfies the condition

¿ I w( x ) o/ x < + O O

I Î . 1

then the representation

b w(2 ;2 K) = ( b ( i ; 2 lt)e « p | ik :5-Sie-£% ;w )fllp )| (5.15)

holds, where m(i?) is a certain non-increasing and bounded function on [0, 2ît].

(b) The class N {c j} С N coincides with the set o f functions representable in the disc |z| < 1 in the form

R i ) = c ‘ ' ' “ " 2 i 5 T i ^ e* p

1ST

(5.16)

where the parameters у, X ^ 0 and кш are the same as in Theorem 5.4, B(z; a^) and B(z; bv) are the Blaschke functions with zeros satisfying the conditions

L IS ы (хЫ х < w i \ UCx)o/x < + oo (5.17)

(J-) ' « j l ( v ) t i l

(c) Any function F (z) £ N{a>} С N is representable in the form

R * ) = - | ^ y ( W < i ) (5.18)

where £ / V {u ¡ , | fR( i ) | ^ i ( |ï| < 1 ) are analytical in the disc |z| < 1.

We now come to the theorem giving the complete solution o f the problem o f factorization o f the entire family o f functions meromorphic in the disc |z| < 1.

Theorem 5.6. For any function F (z) meromorphic in the disc |z| < 1 there exists a function cjp(r) £ П such that F (z) £ Nicjp}.

Note, in conclusion, that the passage from the theory o f the classes Na ( - 1 < a < + °°) to the construction o f the theory o f the classes N {io}, containing the completion o f the factorization theory, required considerable efforts in the case o f the disc. Here, at first, the theory o f operators associated with the given function o;(r) £ Í2 had to be constructed. This theory permitted the family o f Eq.(5.6) o f Theorem 1 to be discovered, which became the basis o f the entire cycle of Our investigations.

Just as an example, we point out the following results, which played a principal role in the theory o f classes N{co} С N and their boundary properties.

Theorem 5.7. Let <o(r) ESI be non-decreasing on [0, 1 ). Then(a) The problem o f the Hausdorff moments

a = z o ô = 5x ,1 w ( * = e . i ,

has a solution a (x ) in the class o f non-decreasing functions bounded on [0, 1 ].(b) In the disc |z| < 1

R.€ C ( í ; ы ) ^0

(E ) As is well known, the class N possesses important boundary properties. For any function F(z) £ N the limit

R e £8K % R z e ¿d) (5.19)

178 DJRBASHIAN

exists for almost all i? S [0, 2ir] and if F (z) # 0 then

2ST

4 ^ l F ( e ¿8 )||d3о

(5.20)

But according to Theorem 5.2 we have

/V/u} Э /J/ i f w (z ) Ю , t f l and /Vfujc/l/if c o ( t ) f + 0 o t t l

The boundary properties o f the classes N {oj} have been investigated for the two following cases:

( I ) The case N {o j} 3 N. Here the following theorems are established.

Theorem 5.8. Let F (z) S N {c j} where cj(r) is non-increasing and satisfying the condition Lip 1 on every interval [0, Д ].1 Then

(a) The limit

almost everywhere on [0, 2т].The following theorem o f uniqueness is an enlargement o f the now classical theorem o f

Szegô for the classes N {oj} D N.

Theorem 5.9. Let f(z ) G N {w } be analytical in the disc |z| < 1. Then(a) I f co(r) e Й is non-increasing on [0, 1 ), then its boundary values

(5.21)

exists almost everywhere on [0 ,2л-].(b ) We also have

(5.22)

are such that

(5.23)

(b) There is no function f(z ) Ф 0 analytical in the disc |z| < 1 fo r which

(5.24)

1 Such functions w ill be further attribu ted to the class i l .

or

T L \ cdOr)efo = + ° ° , f f ( a ; ) = 0 } (5.25)O') laj.1

(2 ) The case N D N{co}. In connection with the known boundary properties o f classes N Э N {c j} noted above, the following questions arise naturally.

Does the exceptional set o f linear measure zero E С [0 ,2тг] get thinner for the classes N {gj} С N where the limit F(eit5) o f a function F (z) G N{co} perhaps does not exist? Can anything else be stated about the boundary values F(eil?) o f a function F (z) G N{co} С N besides the boundedness o f the integral (5.20)? The author and V.S. Zakharian succeeded in obtaining the positive solution o f both these problems in Refs [4] and [5].

To formulate the results obtained here, it is necessary to introduce a definition.Let cj ( t ) Sfibenon-decreasing on (0,1) and

¡U r К

be a function associated with it.We shall assume that the В-measurable set E С [0, 2ir] has positive co-capacity if there exists

a measure д — < E such that the integral

l C ( i e ¿d- u ) I J f ( S ) (5.26)0

satisfies the condition

I f there is no such measure, i.e. i f for any measure ¡i — < E, ju(E) = 1

we shall assume that the co-capacity o f the set E is equal to zero. By this, we shall write Q j (E ) > 0 or Сш (E ) = 0. Note that in the special case when oj(r) = ( 1 - r)a( - 1 < » < 0), Сы (E ) is nothing but the (1 + «)-capacity o f the set E in the Frostman sense.

Theorem 5.10. For any function F (z) G N{co} С N the bounded lim it

exists fo r all ù G [0, 2л] except perhaps an exceptional set E С [0 ,2тг] such that СШ(Е) = 0.

Theorem 5.11. Let F (z )G N | w }C N and E С [0, 2ir] be any set fo r which Сш (E) > 0. Further, let ц — < E, m(E) = I be qny measure with the property

L i > ( f ) = % up

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Then the boundary values F(eil?) o f the function F (z) satisfy the condition

) I h I 11 ф ) = SI &a I F(ea)\\ j j f t ) -t- o

The solution o f the above problem about the boundary properties o f the functions o f classes N{co} С N is contained in Theorems 5.10 and 5.11.

As is well known, if f(z ) is an analytical function from the class N, then |?(г!)=0

if m es E > 0 .

Theorem 5.12. Let f(z ) G N{co} be analytical in the disc |z| < 1 and f(eil?) = 0, & G E where mesE = 0 but СШ(Е) > 0. Then f(z ) = 0 (|z| < 1).

The proofs o f Theorems 5.10 and 5.11 lean essentially upon a series o f fine lemmas and, in particular, upon the following theorems about the functions o f Blaschke and S(z; co)-

Theorem 5.13. Let co(r) G i i fee non-decreasing on [0,1). Then the Blaschke function

belongs to the class N{co} i f and only i f

po\ Uí ( k) clx ^

K=L \Ík\

Theorem 5.14. //co(r) E £2 does not decrease on [0, 1) then the integral formula

l

holds, where oi(t) is a non-decreasing function bounded on [0,1].

Part VI

HARMONIC ANALYSIS IN THE COMPLEX DOMAIN AND ITS APPLICATIONS IN THE THEORY OF

ANALYTICAL AND INFINITELY DIFFERENTIABLE FUNCTIONS

Part VI is devoted to a survey o f a large cycle o f investigations by the author and his students concerning harmonic analysis in the complex domain and the most important applications o f the theory developed here to the solution o f certain fundamental problems o f the classical theory o f functions. The major part o f these investigations has been published in different mathematical journals and also in a monograph by the author [7-10].

1. THEORY OF INTEGRAL TRANSFORMS WITH MITTAG-LEFFLER AND VOLTERRA KERNELS (Plancherel-type theorems in the complex domain)

(A ) An important landmark in the development o f functional analysis in general and o f the theory o f harmonic analysis in particular was the fundamental theorem o f Fourier transforms in the classes L 2 established in 1910 by M. Plancherel.

It is well known that for the complex form o f Fourier transforms this theorem reads:

(a) For any function f(x ) E L2 (_0°, + °°) the following lim it in the mean exists:

F M = Ü W \ $ ( i ) e ' i U Í e ¡ i ¿-в

defining the Fourier transform S^l f ] = F.(b) The reverse lim it relation

r ‘ [Fl-G"

and the equality

+ 00 +Oo

S\U*)\m = \ iFMolu— Oo

hold.

Thus, this theorem has for the first time constructed a Fourier operator in L2 and established a complete equivalence between the function and its Fourier transform.giving a unitary mapping o f the whole space L2 upon itself.

A t a considerably later date G. Watson (1933) constructed the general theory o f the Fourier- type transforms in L2 (0, + °°) performing unitary or quasi-unitary mapping o f L2 upon itself with the aid o f formulae o f the form

i I +oo

о

Notice should also be taken o f the significant generalizations o f Plancherel’s theorems discovered within the theory o f singular boundary problems o f the Sturm-Liouville type on the half-axis (0, + °°).

(B) The theory o f integral transform developed in our investigations is very substantially supported by the remarkable asymptotic properties o f the two families o f functions; the functions o f the Mittag-Leffler type :

Е 5(г ;Л = £ п > ^ ) О < м > о )

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and their continual analogues — the functions o f Volt erra:

^ > = 7 ñ ¡ % j í ( ; > - ! , S>0)

Note that the function Ep(z; p) (with ¡u = 1) was first introduced into analysis by Mittag- Leffler at the beginning o f this century (1903 —1905) in connection with his discovery o f a new rather strong method o f summing the diverging series. In this connection he was also the first to discover asymptotic formulae for this function when |z| -» °° in the domain

A s= { * 5 l < * 2 * l < * f f O f )

and in its complement Др = С Др. As to the function vp(z\p) (for p = 1, д = 0), it was introduced into the analysis by V. Volterra (1916) in connection with the solution o f a special integral equation occurring in the theory o f heredity.

The considerable time that has elapsed since then has not enriched the mathematical literature with any more or less serious investigations on the properties or, even more important, on applications o f these functions to contemporary analysis.

The asymptotic properties o f the functions Ep(z; p.) and vp(z; p) are given by the following theorem.

Theorem 6.1(a) For each p E (-«>, +<») and p > 1/2 the function Ep(z; p.) is entire o f the order p and

type o = l , and, i f a is any number satisfying the condition

¿ y < o¿ <. pi in {5Г, -y - j-

then fo r |zl -*■

(i) ^ ¿ S(L~ S K 0 ( - j~ ) for (6.1)

(ii) E %( ¿ ; f ) = 0 í ^ - ) for (6.2)

(b) For each p E (-<*>, + °°) and p > 0 the function vp(z\p) is analytical on the whole Riemann surface

( we call such functions quasi-entire\ has the order p and type о = 1 and, fo r any a G (тг/2р, ir/p) when |z| -*•

(0 = for (6.3)

(ii) у - о ( ф ) (6.4)

The asymptotic properties o f Ep(z; /j) for 0 < p < 1/2 have a different formulation. For us it was particularly important to reveal these properties for p = 1/2.

Theorem 6.2. I/O < y. < 3, then(i) For 0 < arg z < TT or — it < arg z < 0, respectively, when |z| -* °° we have

(ii) For 0 < x < + °° when x -* + °° we have

E i ( - x ; / ) = x r ( b 4 ^ ( J 7 + f 0 - 7 ) ) + O (t -) (6.5')

(C) It will be assumed below that by a given arbitrary p > 1/2 the parameter p is subjected to the condition

r i f + Finally, i f

3 ( y ) e L¿>j.(0 ,*o°) and the famüy o f functions (0 , too) dependingon the parameter <5(0<<5< +o=) is such that

then we shall write

(/ )

S-

The first basic theorem on the transforms with Mittag-Leffler kernels reads:

Theorem 6.3. Let

(6.7)

and thus

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(a) Putting

Л -J1{ a A i 5 E j í (6.9)

we shall have almost everywhere on the half-axis у E (0, + °°)

(6 . 10)

(b) Also putting

S

— ^ E ^ ( ( ¿X)S^ S f ü l d * (6 .H )-G

with respect to the conditions (6.10), we shall also have

O ' )3 0 j î ï r ) = e.L.m. 4 ( 4 ) f ; 5 ) ( 6 . 1 0 ' )

+oc ‘-I **>

Thus, this theorem contains two substantially new statements o f harmonic analysis:(1) For the usual Fourier transform o f functions from L2>#i(0, + °°) there is not one but

a whole family o f inversion formulae with Ep(z; p)-type kernels, where p > 1/2 is any number. Only the special selection o f parameters p and д (when p = 1 /2, д = 1, 2 or p = д = 1) leads to the inversion formulae well known in the Fourier-Plancherel theory.

(2) When the parameter p > 1, the general inversion formula (6.10) has an important property: representing the function g(y) on the ray у = 0,it simultaneously represents the identical zero in the angular domain тг/р < \<p\ < тт.

This permits the construction o f the apparatus o f integral transforms and their inversions for an arbitrary finite system o f rays proceeding from one point o f the complex plane.

Before formulating the theorem we shall introduce some notation.Denote by L { < ..., ipp] the set o f rays

ек 1вг%1=Ъ (К = 1Л->Р)

OaYl<tl<---< f P< f P<i=

emerging from the origin. The system o f these rays divides the z-plane into p angular domains with a common vertex at z = 0. Denote

CO= m cx* J - — [UK ^p I Yk,l -T k J

noting that ir/w is the value o f the smallest o f these angles.

Theorem 6.4. Let p > co and g(z) be an arbitrary function defined on the set o f rays L { i <¿>p} and such that

\ (6.12)

(a) Putting+o°

U ( 0 , + ~ ) ( K = i , A , . . . , p ) (6.13)

о

we shall have fo r almost any t e l ^€ L { f i ip }

№e¿r)=]¿y Cí«>T t ^ cp'r“ ; (6.14)

(b) The Parseval type equality

(6.15)

L K - M

holds.

It is easy to see that the basic theorem o f Plancherel is a special case o f this theorem when a system o f two rays L{0, it} is considered and д = 1, p = 1.

Finally we shall note that the inverse o f Theorem 6.3 holds.

Theorem 6.5(a) For any function f(x ) G L2 (0, + °°) the transforms

А ) = ^ ^ { ^ ,гТ Е ^ е2£л^ ГхТ^ 1)хл - ^ ^ 1‘ } (6.16)

determine functions ^ ) £ ^ ( 0 , * ° ° ) and the inversion formula

; X£(0’^ <6Л7>

holds.(b) The inequalities

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hold.

(D) The results o f Theorems 6.3, 6.4 and 6.5 remain valid i f the function Ep(z; p.) is replaced by Vp(z; ц ),putting everywhere pi = 1/2 and p > 0.

For this reason we do not present here the reformulations o f these theorems, but only give the analogue o f Theorem 6.4 which is a rather extensive generality. Let ..., vp} denote theset o f parallel straight lines

W = U + i Vk, , - o o < и < + oo ( к,= 1,Д, ,.v p)

- < v,. < \/д Vp <.+ oo

See that the value я/со is equal to the width o f the smallest o f the strips generated on the z-plane by the system o f lines £¿’{v ¡ ,..., vp}.

Theorem 6.6. Let the function G(w) be defined on the system o f straight lines ,..., vp } and satisfy the condition

Denote

Then fo r any p > co the following statements hold:(a) For almost any w ..., vp}

where

(b) The equality

\ | G - ( w ) | c K f l e w ) —-OO

P to o

(6 .22)

holds.

In conclusion, we should like to note that Theorems 6.3, 6.4 and 6.6 may be considered as theorems o f approximation by entire functions on the half-axis (0, + °») or on the system o f rays L{v?,,..., yjp} or on the system o f parallel straight lines , ..., vp}.

2. CLASSES OF ENTIRE, QUASI-ENTIRE AND ANALYTICAL FUNCTIONS AND THEIR INTEGRAL REPRESENTATIONS (Wiener-Paley-type theorems)

(A ) In their famous monograph “ Fourier transforms in the complex domain” (1934), Wiener and Paley established two fundamental theorems: on representations o f exponential-type entire functions from L2 (- ° ° , + °°) and functions from H2 analytical in the half-plane. We shall formulate these famous theorems, since they served as a starting point for our investigations in this area.The first o f these theorems reads:

The class o f entire functions from L2 (-«> , + °») o f exponential type < о coincides with the set o f functions admitting representation o f the form

where </>(t) is an arbitrary function from L2 (— a, a)-

The second theorem reads:

The class H2 o f functions F (z) analytical in the half-plane Re z > 0 and satisfying the condition

where L ^ iO ,* 00).

Our results on representations o f analytical functions are far-reaching substantial generalizations o f these two theorems.

(B) Let us first give the formulation o f the most general theorem on entire functions. To this end we shall first introduce some preliminary notation.

We shall assume that the natural number 36 = Э€(р) > 0 satisfies the condition

- 5

coincides with the set o f functions admitting representation o f the form

Г(г) = \ е * г Г(0<Л , R-ei > 0О

(6.23)

for any p > 1/2. Then, for a given p > 1/2 we shall assume that the set o f numbers

(6.24)

{ 9o, S i,..., 9ХН satisfies the conditions

m**{¡> KfL-áKj = -Oi Ké* J 5

Starting with the set Зэен| we f ° rm a sequence o f pairs

(3 k,9r.O i. (6.25)

and then, preserving the mutual order o f their succession, we shall isolate all pairs for which the equality

\ . r \ = Y ( K = 0 > 36) (6.26)

holds. Here, i f p < эе, let us denote the remaining pairs o f (6.25) by , & V ( 9 = ^ £ -p ) .t v . n i i к / i ' Г I /Denote iurther

188 DJRBASHIAN

(6.27)

and assuming that — l < o j < l , a j c> 0 (k = 0, 1, ..., p), associate with the set o f numbers

{ 90, . - . , 3K4l j the class

\x4î} ( o j ; { Э к } ; { GKf )

o f entire functions o f order p and normal type < о satisfying the conditions

T l ^ e ~ A ) r V ^ < + ° ° ( K = 0 , i , . . . , 2 C ) (6.28)0

We have established the following general theorem, based substantially on our theory o f integral transforms with Mittag-Leffler-type kernels.

Theorem 6.7. The class W s (i*>;{dK\',{<b¡¡} coincides with the set o f functions f(z) admitting the representation

P * . . . A , _M

where

Here, i f

(6 .30)о

then almost everywhere we have

¿ T Í

(0,<Sk ) (6.31)

[ 0 , T r e ( e K , t o o ) С к = о д , . . . ? р ) .

By special selection o f the system o f rays { г = §к \0 along which the conditions o f

the form (6.28) are imposed, some corollaries o f a more special character, including the Wiener- Paley theorem itself, may be obtained. For example, the following statement is true.

Theorem 6.8(a) The class o f entire functions f(z ) o f order p > 1/2 and type < о subject to the condition

coincides with the set o f functions o f the form

£(г)=Д L

where д = (oj + 1 + p)/2p and ip(t) e L2(0, a).(b) The class o f entire functions f(z ) o f order one and type < a fo r which

\ I í ( x ) |¿ lx| cix < + 00


G-

where д = 1 + cj/2 and <р(т) G L2 ( - a, a).

Note that the theorem o f Wiener-Paley is contained in statement (b) o f this theorem as a special case when a) = 0 because then ¡i = 1 and E, (z; 1) = ez .

190 DJRBASHIAN

(С) A function analytical on the whole Riemann surface Gœ with

(1) S u p ( 0 <.Z< + c>o)

(2) 6 « Я ( г ) < + оо Z—**0 V

is called quasi-entire. The order p and type о o f quasi-entire functions are determined in a way analogous to that in the theory o f entire functions. The result similar to Theorem 6.7 is also established for the quasi-entire functions with the function Ep(z; /j) replaced by vp(z; 1 /2) and oj = - 1, p > 0 may be arbitrary.

In order to state the result,some new notations are needed. For the given value o f p

the set o f quasi-entire functions f(z ) o f the order p (0 0 and

Further, forming the successive pairs

we shall isolate, retaining their mutual order o f succession, all pairs

for which

Finally denote by

( 1)о

SINGLE-VALUED ANALYTICAL FUNCTIONS 1 91

(6.32)

where

\ + \ .J

Theorem 6.9. The class

Q s X v m )

coincides with the set o f functions f(z ) representable in the form

? ( * ) = £ . ' \ ' ) М * Ч г * у ± \ т ~ * Г к ( г ) ' 1 тK=0 0

where )€ L¿ (0,<oK) f i á K s w ) .

(D ) As to the results o f the type o f the Wiener-Paley second basic theorem, here we have

Theorem 6.10(a) The class ( j - < / < ♦ « ; -j<uKi) o f functions F (z) analytical in the angle

— l e t , 0 < |г|<оо|

/or which

т ^ { Т | № е ‘”)|Дг"Л Ь ' ~


R ¿ M E § ( £ ^ A / ) vh « H

+ Т Е , ( А Л / ) v (O í 0 / ' V ¿ , ¿ 6 4 ,-s

where

(6.33)

4 i ) ( t ) € L l ( 0 , +<*>), $ á j j é j f

(6-34)

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(b) The class № ¿ [J i] ( 0<- Á < + » ) o f functions analytical in the domain Aa С and subject to the condition

coincides with the set o f functions representable in the form

R * ) = i *о

(6.35)О

where p > 0 is any number,

An important feature o f the integral formulae (6.33) and (6.35) is that they also represent the identical zero in the domain complementary to Да , i.e. for (6.33) this is the case when p > 2a/(2a — 1 ) in the domain

This remarkable fact permits the construction o f the apparatus and the development o f the theory o f Fourier-Plancherel-type integrals for the sets consisting o f a finite number o f rays and angular domains lying on the z plane or on the Riemann surface G „. Precise formulations o f the theorems in question will not be dwelt upon.

3. UNIQUENESS OF CERTAIN GENERAL CLASSES OF INFINITELY DIFFERENTIABLE FUNCTIONS (Denjoi-Carleman-type theorems)

(A ) As early as 1912, J. Hadamard posed the problem o f determining conditions for a sequence o f positive numbers {Mn} “ ensuring the uniqueness o f the class C{Mn} o f functions infinitely differentiable on some interval S ~ = (a, b) and for values {у з^ (х0) } “ , x0 £ satisfying the

and for (6.35) this is the case in the domain o f values

conditions


In 1921, A. Denjoi established for the first time the existence o f such classes substantially wide as compared with the usual classes C {n!} o f analytical functions. In particular, he proved that the class C{Mn} is quasi-analytical i f e.g.

ÏA n _ — ( n- 6 0 3 и. • • • ) , П ^ - Л / р

T. Carleman (1923-1926) gave a comprehensive solution o f Hadamard’s problem by laying down the necessary and sufficient condition for the class C{Mn} to possess the property o f uniqueness; in other words, for its quasi-analyticity.

In the formulation by A. Ostrowsky, Carleman’s result (usually called the theorem o f Denjoi-Carleman) reads:

The condition

= , T M - а и р ( 6.36)

is necessary and sufficient fo r the class C{Mn} to be quasi-analytical.

In the five decades that followed, many original investigations dealing with the theory o f quasi-analytical functions have come into being. Here a special note should be given to the study by S. Mandelbrojt on the generalized quasi-analyticity in the sense o f Denjoi-Carleman where, however, the condition (6.36) is always assumed to be fulfilled.

(B) According to the theorem o f Denjoi-Carleman, by the condition

the class C{Mn} o f functions infinitely differentiable on the half-axis [0, + °°) or on a segment [0, £] will certainly not be quasi-analytical. That is to say, it is well known that by the conditions (6.37), say, in the case o f the half-axis [0, + °»), there exist non-trivial functions ip(x) G C{Mn} satisfying, moreover, the conditions

( v > o , ft5 d ,x 6 | ;o ,*o o )) (6.38)

f ( n ) ( 0 ) — 0 ( n = 0 , L , ¿ , . . . ) (6.38’)

In this connection it would be natural to put the following question:I f the class C{Mn} is not quasi-analytical on [0, + °°) or on [0, £], then which are functionals

{L n(i¿>)}“ that can determine the functions o f this class in a unique way instead o f the values{Y>(n)( 0 ) }p

It turned out that it is possible to introduce a new general notion of a-quasi-analyticity also covering the notion o f classical quasi-analyticity and to obtain a complete solution o f this problem in the spirit o f the classical theorem o f Denjoi-Carleman.

(C) As a preliminary, we introduce some notation and definitions.

194 d j r b a s h i a n

Let us consider a set C^°) (0 < a < 1 ) o f functions ip(x) infinitely differentiable on [0, + °°) and satisfying the conditions:

§ U p | ( l +'X 'P 60|< t »o ( « , 1 4 = 0 ( 2 ^<*• ’ ’ " - I

Assuming that ip(x) G (0 < a < 1 ) and putting 1 /p = 1 - a (p > 1 ), let us consider the operator o f successive differentiation o f the function tp(x) in the sense o f Weyl o f orders n/p (n = 0, 1,...)

DÍwo = Dl DoT W <*Ы)where

i i f(x) =á!r G Í Wx), C L \(1 )J{x

Note that in a special case when a = 0 (p = 1 ) we shall have

Finally, for an arbitrary sequence o f positive numbers {M n}7 we introduce two classes of infinitely differentiable functions:

The class C^{[0, + <*>); Mn} is the set o f functions <p(x) from for which

n

s u p I D l ^ x J l M f t A I r i ( n = 1 , 2 . . . . ) (6.39)

and the class Ca{[0, + °°); Mn} is the set o f functions <p(x) from for which

( i - u , . . . ) ( 6 .4 0 )

For both these classes a question is put similar to Hadamard’s problem and reducible to this problem when the parameter a = 0.

What must the sequence o f numbers {Mn} “ be in order that fo r each function <fi(x) from the corresponding class the equalities

D l ^ ° > = f ( k y \ х 'А" " 1 Л ) о1х = 0 ( k = 0 ,L ,1 ,... ) (6.41)

yield the identity

f ( ¡ í ) ~ 0 ; •?

Classes o f this kind are called “a-quasi-analytical” , and it is easily seen that the 0-quasi- analytical classes C j{[0 , + °°); Mn} or Co{[0 , + <»); Mn} are identical to the classically quasi- analytical class C{Mn}.

The following basic theorem has been established.

Theorem 6.11(a) The class С «{[0, + °°); Mn} is a-quasi-analytical i f and only if

(b) The class Ca{[0 , + °°); Mn} is a-quasi-analytical i f and only i f

(6.42)

(6.43)

In both these statements

l " -T (*)= 7vTn-i 1 / 'n.

is the function o f Carleman-Ostrowsky. Each o f the statements (a) and (b) is reduced to the classical theorem o f Denjoi-Carleman when a = 0.

Elementary estimates show that if

L+A/ \ПЛ 1 ^ = ( и . 1' '1- • •• , п > Ар

where р > 1 is any integer, then the condition (6.43) is fulfilled.This example shows that in the a-quasi-analytical class Ca{[0 , + <»); Mn} (0 < a < 1) the

successive derivatives o f functions may have a substantially faster growth (as,(l +<*)/( 1 - a ) > 1 when 0 < a < 1) than is possible for O-quasi-analytical classes: this can be observed from the original results o f Denjoi.

The notion o f a-quasi-analyticity is also introduced for the classes o f functions infinitely differentiable on a finite segment [0, £]. Here the classes C£{[0, 8]; Mn} and Ca{[0, £]; Mn} are determined as subclasses o f functions <p(x) from the corresponding classes on [0, + °°) satisfying the additional condition

f ( K ) s Q , t é К » »

The corresponding theorem on a-quasi-analyticity o f these classes has exactly the same formulation as statements (a) and (b) o f Theorem 6.11.

(D ) Let us consider briefly the method o f proving these theorems.As is also the case with the original proof o f the Denjoi-Carleman theorem, the problem o f

a-quasi-analyticity is solved by reducing it to the problem o f Watson. In our case, such reduction is possible only by making use o f the apparatus o f integral transforms and representations with Mittag-Leffler kernels Ep(z; д). Here a substantial role is played by the following basic theorem.

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Theorem 6.12(a) Let the function f(z ) be analytical in the interior and continuous on the closed angular

domain

&*l= { è ’ I f c |ал| ¿l á ^ > 0 < \i I < + <*=]■

and in the neighbourhood o f z = °°

h a x I f c j > l ) (6.44)

Then we have an integral representation o f the form

too é --L

£ ( * ) = \ s> t ) ^ (6-45)

where

y ( i ) - i é r \ z f ( 6. 46)

and Lp is the boundary o f the domain Ap advanced in the positive direction.(b) I f f(z ) is an entire function o f the order p > 1/2 and type 2 (0 < £ < + °°) satisfying

the condition (6.44), then in the representation (6.45) we have

m ) = o , e < i < + o o

These statements permit us to establish the necessity o f conditions (6.42) and (6.43) o f Theorem 6.11. Here we rely substantially upon our discovery o f an important property o f the function

s

to be a solution o f the Cauchy-type problem for a differential operator o f fractional order on the half-axis [0, + °°):

D T âjy,^)-}sj)c,}) = 0(6.47)

D° ( * ’^|x=0 = ^

where

- I

and

is the derivative in the sense o f Riemann-Liouville o f the order 1/p. It is also important that the function

J f > $ ^ 0

is also a solution o f the Cauchy problem, but this time o f another kind:

- f

es(o;A)= оAs to the sufficiency o f the conditions in Theorem 6.

by the fact that if

Пх)=с/{[0л°о)Л1, DlW=0then the function

+o° i_ t*-L *

t 5

admits the representation

foo j — ^

S E °»m>M’ ^for any n > 1.

This representation permits us to establish that the fulfilment o f the conditions of Theorem.6.11 results in f(z ) = 0. Then using the theorem on inverse transforms with the Mittag-Leffler kernel already cited, we come to the conclusion that <¿>(t) = 0.

It should be noted in conclusion that during the last two years the reporter and his students have been continuing their investigations in this direction. First, we succeeded in obtaining an analogous theorem o f uniqueness in the case when the function <p(x) is analytical in the domain o f arbitrary angle

A y = | 2 ; | jjt j ¿| , 0 <

of the span тг/у (0 < у < + °°) on the Riemann surface o f the logarithm. Second, we have extended Theorem 6. í 1 to the case when the parameter a lies within the limits o f - 1 < a < 0.This case yields results o f a completely new quality this time for the classes o f Denjoi-Carleman. Formulations o f the theorems in question will not be dwelt upon owing to the lack o f space.

11, here an essential role is played

( n z O )

198 DJRBASHIAN

Part VII

SOME OPEN PROBLEMS

In conclusion let us formulate some open problems. Except for the last one, they are directly connected with the subject o f this paper.

Problem I. Theorem 4.3 only compares the classes Uu , gj G fi.with the class U. A similar situation is found in Theorem 5.2, which compares the classes N{co} with the class N o f Nevanlinna. However, it is important to have theorems o f comparison for the classes иШ| and UW2 or N {w ,} and N {oj2} in the case when the functions Wj(x) G f i (j = 1,2) generating these classes are arbitrary.

Solution o f this problem is easily reduced to solvability o f the Hausdorff moment problem:

Mn = / x - da(x), (n = 0, 1,2,...)

where

i i

J " u ^ M x ^ d x J ' u>2 (x )x 11*1 dx (n = 1,2,...)

in the class o f functions a(x ) with the finite variation V (a ) < + °°.0

We suppose that the problem ( l ) —( 2) must have a solution i f the functions coj(x) G f i 0 = 1 , 2 ) are monotonie on [0,1); also, the function w, (x)/<o2(x) decreases monotonically on [0,1).

It can be seen from Theorem 5.7 that our hypothesis is true at least for the special case when ( j j (x ) = 1 and co2(x ) is a function uniformly increasing on [0, 1). I f this hypothesis is true, it will result in an improvement o f the inclusion theorems noted above, i.e. it will be possible to establish that if CO] ( x )/ cj2 ( x ) is a function decreasing monotonically on [0, 1) then the inclusions 1)Шг С Uco, and N{co2} С N fo jJ will hold.

Problem II. Part V refers to a work [6] which has not yet been described. This work contains an essential, though incomplete, result directed towards constructing a theory o f factorization o f functions meromorphic on the whole plane and which have an arbitrary growth o f characteristic. The very first question arising while constructing the complete theory is formulated in the closing part o f Ref. [6]. It runs as follows.

Let us denote by f i j , the set o f functions co(x) continuous on the half-axis [0, + °°) and satisfying the conditions:( 1 ) cj(x) is positive, non-increasing on the half-axis [0, + °°), and co(0) = 1.(2) The integrals

exist for к = 1,2,.... Putting Д0 = 1, we introduce the functions:

“ “ Ifl

W£>(z ; f ) = J dx - У js "k f to(x)xk~1 dx -

k= 1 0

ш (х)х"к" ^ х

Ifl

zk— , Izj < °°, 0 < Ifl < °° (7.1)A t

A ( z ; f ) = ( \ - ^ e ‘ W“ (z;f) (7.2)

Further, le t {zk} “ (0 < |zk| < lzk+il < + °°) be an arbitrary sequence o f complex numbers for which

OO oo

X J ^ dx<+ca (73)k = l |zkl

The question is, whether any supplementary conditions besides ( 7.3) are needed to provide convergence o f the product

OO

I lA < ~ ) (z ; z k) (7.4)k= 1

on the whole complex plane.We put forward a hypothesis that this must be the case at least when

dlogco(x) , ... ж .------------ i — o o , with X T + OO

d logx

This hypothesis is true in a special case when e.g.

+ OO

ш(х) = п>) / e'otVP'ldt

where p (0 < p < + °°), д (0 < д < + °°) and a (0 < о < + o°) are arbitrary parameters.

Problem III. Let д (t) be, in general, a complex-valued function on [0, + °°) for which

VM( r ) = J tVm COK + oo у г е [ 0 5+оо) (7 5)

0 -

Then it is obvious that the function

OO

fM( z ) = y * zM/xCt)

0

is regular on the whole Riemann surface.

Go» = {z ; |Argz| < ° ° , 0 < | z | < ° ° )

and it is easy to see that

sup If^Cre14’ )! < VM(r); Vr € [0, + °°)¡^|< + oo

This remark brings us naturally to formulation o f the next problem:

Let the function f(z ) be regular on and satisfy the condition

M{ (r) = sup ifOe1 )! < + °°, V r E [0 ,+ ° ° ) (7.6)Iy>|< + “

Is there a function jUf(t) on [0, + °°) having the property ( 7.5) and such that

OO

f ( z )= J ztdMf (t), z e e » ? (7.7)

o

Note that our hypothesis is true in the special case when

fire*^+2,г)) ^Дге*^), V rE [0 , + °°) and E (- ° ° , + °°)

Indeed, in this special case it is obvious that f(z ) is an entire function and therefore can be expanded in the form

200 DJRBASHIAN

f(z ) = 2^ akz , |zi< + °°

k = 0

This means that the representation (7.7) exists with a certain measure /u(t) concentrated only at the points {k }“ o f the half-axis [0, + °°).

Problem IV. In connection with the main Theorem 6.3, we formulate the problem o f finding its discrete analogue for the functions from the classes

£

L2,ju(0, 2)«> J lg(y)l2y2( _1)ciy < + c0

where £ (0 < £ < + °°) is arbitrary.

The problem is to construct the apparatus o f series by the Mittag-Leffler-type functions representing the function g(y) e L2>/J(0, i ) fo r any p > 1/2 and possessing the property similar to statement (b ) o f this theorem in the case when p > 1, in the domain

Complete solution o f this problem will result in a discrete analogue o f our more general Theorem 6.4 for the functions g(z) from the class

P 8

I f |g(reiv>k)|2 r ^ - ^ d r < + °°

k = l 0

where

1 1 1г < /1 < - + - , p > c o = max2 2 P l < k p < <£p+1 = <p¡ + 2я are arbitrary.'Even in the simplest case o f the finite cross type set,

E{i2} = { - e , 8 }U {-i(2 , i i }

the positive solution o f this problem (here it must certainly be p > 2) is o f indisputable interest. Such a solution will provide an original Fourier-type series apparatus for the sets E{2}.

Problem V. Let

o < x 0 < x , < x 2< . . . < x n < „ .

be an arbitrary sequence and sk > 1 designate the multiplicity o f occurrence for the number Xk in the set {X b X2, ..., Xk} Consider the family o f quasipolynomials o f the form

Дп(х ) = ^ a|.n> e - x ^ x sk - l (n = 0, 1, 2, ...)

k = 0

assuming that for the whole family {Д п(х )}

sup {|Дп(х )| }< M (n = 0, 1,2,...)0 < X < + oo

Denote

sup {|Дп(х)|} = /xn0 < X < + o°

(n = 1, 2, ...)

202 DJRBASHIAN

and

sup {|Д^(х)|} = мп(г, R ) (n = 1, 2, ...)r « x < R

for any 0 < r < R < +°°.The problem o f estimating the numbers {/xn} and {мп(г> R )} from above is put forward.

Consider it likely that the estimates o f the form

n

Mn< C Mexpj 2 ^ J _ j (n = 1,2,...)

к = 1n

Mn(r, R ) < CM(r, R ) exp J ¡T

k = l

may be valid in which CM and CM(r, R ) do not depend on n.In the case Xk = к (к = 0, 1,2,...), our assumption holds by virtue o f the well-known Markov-

Bernstein theorem.

REFERENCES

Parts I and II

[1 ] D U R E N , P ., T h eo ry o f H p Spaces, Ch .I, Academ ic Press, N ew Y o rk and L on d on (1 9 7 0 ).

Parts III and IV

[2 ] D J R B A S H IA N , M .M ., A generalized R iem ann-L iouville op era tor and som e o f its applications, M ath. U SSR

Iz v .2 (1 9 6 8 ) 1027—1064 -

Part V

[3 ] D J R B A S H IA N , M .M ., T h eo ry o f factoriza tion o f functions m erom orph ic in the disc, M ath .U SSR Sbornik 8 4

(1 9 6 9 ) 4 9 3 -5 9 1 .

[4 ] D J R B A S H IA N , M .M ., Z A K H A R 1 A N , V .S ., Boundary properties o f m erom orph ic functions o f bounded

fo rm , Math. U S S R Izv . 4 6 (1 9 7 0 ) 1 2 7 3 -1 3 54.

[5 ] Д Ж Р Б А Ш Я Н , M . M . , З А Х А Р Я Н , B . C . , Г р а н и ч н ы е с в о й с т в а п о д к л а с с о в м е р о м о р ф н ы х ф ункций о г р а н и ч е н н о г о в и д а , И з в . А Н А р м . С С Р , с е р и я М а т е м а т и к а , 6 2 - 3 (1 9 7 1 )

[ 6 ] 1 8 2 -1 9 4 .Д Ж Р Б А Ш Я Н , М . М . , Ф а к т о р и за ц и я ф ункц ий , м е р о м о р ф н ы х в к о н еч н о й п л о с к о с т и ,И з в . А Н А р м .С С Р " М а т е м а т и к а " 5^6 (1 9 7 0 ) 4 5 3 -4 8 5 .

Part VI

[7 ] D J R B A S H IA N , М .М ., “ H arm on ic analysis in the com p lex dom ain and its applications in the th eo ry o f

analytical and in fin ite ly d ifferen tiab le functions” , Lecture N otes in M athem atics 399, Springer-Verlag,

B erlin -H eidelberg-New Y o rk (1 9 7 4 ) 94—118.

[ 8 ] Д Ж Р Б А Ш Я Н , M . M . , И н т е г р а л ь н ы е п р е о б р а зо в а н и я и п р е д с т а в л е н и я ф ункций в к о м п

л е к с н о й о б л а с т и , М о с к в а , Н а ук а (1 9 6 6 ) г л а в ы 111-V111.

[9 ] D J R B A S H IA N , М .М ., A n extension o f the Denjoi-Carlem an quasi-analytic classes, A m . Math. Soc. Transi.

107 2 (1 9 7 4 ).

[1 0 ] D J R B A S H IA N , M .M ., K O C A R IA N , G .S ., Uniqueness theorem s fo r som e classes o f analytic functions,

Math. U S S R Izv . 7 1 (1 9 7 3 ) 9 5 -1 2 9 .

IAEA-SMR-18 /37

A GENERALIZATION OF BEURLING’S ESTIMATE OF HARMONIC MEASURE

M. ESSEN Department o f Mathematics, Royal Institute of Technology, Stockholm, Sweden

Abstract

A G E N E R A L IZ A T IO N O F B E U R L IN G ’ S E S T IM A T E O F H A R M O N IC M E A S U R E .

Let D be an open connected subset o f the open unit disc Д. In 1933, A. Beurling gave an estimate o f the harmonic measure o/3D П { |z| = 1} with respect to D; this estimate depends on the circular projection o f the complement o f D onto a ray from the origin. In this note a more precise estimate is given which also depends on the angular size o/D П { |z| = r } , 0 < r < 1. Then are also results o f this type in Rd, d > 3.

Let D be an open connected subset o f the open unit disc Д and set

a = 3D n{|z| = 1}, 0=ЭОПД

Let co(z, a, D) be the harmonic measure o f a with respect to D, constructed by Perron’s method with the boundary function which is 1 on a and 0 on (3. Let

In this n o te a m ore p recise estim ate is g iven o f the harm onic measure o f Э D n{|z| = 1} due to A . Beurlmg.

co(z, a, D), z 6 Du(z) =

0, z S AND

It is clear that u is a subharmonic function in Д. We also introduce

E = r £ (0 , 1) : infu(rei0) = Oв

F(r) = { в G[ -rr, v ] : cj(reifl, a, D) > 0 }

a(r) = mF(r), r E E; a(r) = r ^E

Beurling [2] has proved that

Max cj(reie, a, D) < const expв

(1)

Е П ( г , 1 )

Let A (r) = sup (2îr/a(t)). r < t < l

203

204 ESSÉN

Theorem. Let D and и be as above. Then we have

7Г 1/2 1nJ u(re*®)2 d6 < 2лу/2п'А(г) exp

f dt nJ ta(t)

. 71 Г

Max u(re‘° ) < 6тг,/2А(г) exp r < 1/2 (3)

2r

Remark 1. (1 ) and (3 ) are equivalent (modulo absolute constants) if a(r) = 2n, r G E, i.e. the domain D is the unit disc Д cut along segments along the negative real axis.

Remark 2. I f the value A (r) is assumed on a small set in such a way that the exponential term does not compensate for the large termy^AW, we first replace D by a larger domain so that A (r) decreases and co(z, a, D, ) majorizes co(z, a, D). Now, we can use estimate (2) or (3).

Remark 3

1

J ( ta it )) '1 dt = J ( t a (t )r 1 dt

r E n (r, 1)

is the “ area” o f that part o f D which is in the “ shadow” o f E taken along concentric circles. I f D П { r, < |z| < r2} is narrow, (2) or (3) imply that our estimate decreases considerably when we pass through the gap.

In the proof, we can either use results o f M. Heins [5] or a theorem o f A. Baemstein [ 1 ]. For reasons to be explained later, we prefer to use Baemstein’s result.

With D and F(r) given as above, we define the circular symmetrization D* o f D in the following way.

I f F (r) = [0, 2n], D* n{|z| = r } ={|z| = r }

I f this is not the case,

D* П { |z| = r } = {z = re^ : |i | < a(r)/2}

D* is also an open subset o f Д (cf. Hayman [4], 4.5.3). Starting from D*, we define a*, (3* and

v(z) =cj(z, a*, D*), z E D *

0, z e Д\0*

v is subharmonic in Д and it can be proved that (fi -*■ v(relv>) is decreasing on [0, a(r)/2). Baemstein’s result (cf. Ref. [1], Theorem 7; a proof can also be found in Essén [3], p. 87) is as follows.

HARMONIC MEASURE 205

Theorem A. Let Ф : [0, °°) -*■ R be a convex non-decreasing function. Then

j Ф(и(ге*0)) d0 < J (Ф(и(ге1В)) d0 1, taking p*11 roots and letting p ■+«, we obtain

Max u(reifl) < Max v(reifl), 0 < r < 1 (4)в в

(ii) Choosing Ф (x ) = x2, we obtain

7Г Я

J u(rei0)2 dO < / 1u(rew)2 d0 , 0] -* [0, °°) be a lower semicontinuous and locally bounded function. Assume that there exists a solution o f the differential inequality

z ' ' ( t ) - p (t)2 z(t) > 0 , - ° ° < t < 0

z(0) = 1, lim z(t) = с exists, 0 < с < °°t - > - o o

Let T be given, T < 0. Let p* be the measure-preserving, increasing rearrangement o f p |[T q] on (T, 0] and let p *(t) = inf p(t), t < T.

t<0

Then there exists a non-negative solution w o f the differential equation

w "(t ) - (p * (t ))2 w (t) = 0, w(0) = 1, lim w (t) = c*t- * - “ >

0 < c* < о»

Furthermore

z(T ) < w (T) (5)

Remark. By a solution o f a differential inequality or a differential equation, we mean a function z such that z and z' are absolutely continuous, z " is the a.e. existing derivative o f z' and the inequality or the equation is satisfied a.e.

The point o f Theorem В is that it is fairly easy to find an estimate o f w (T), and thus also ofz(T).

206 ESSÉN

The Baernstein *-function

Let u be subharmonic in Д. Let r be given, 0 < r < 1. We define

u*(re‘° ) = sup |E| = 20

J u(reia;)dco.I, O < 0 <7T

E

where the supremum is taken over all measurable sets E С [—тг, 7г ] o f measure 20. A. Baemstein has proved that u* is subharmonic in the upper half-disc. (Proofs can be found in Baernstein [ 1 ] or in Essén [3], Ch. 9.)

Proof o f the main result

From Theorem A we see that it suffices to consider the symmetrized region D* and the Carleman mean q(r). Without loss o f generality, we can assume that /3* = 3D* П Д is smooth. It is known that q(r) is a convex function o f log r (cf. e.g. Heins [6], Ex. 12, p. 82). In particular, q' exists a.e. and is an increasing function.

We introduce

Q (t)2 = J v(et+^ )2 Ap, a = a(t), t E S

-a l l

Using an argument o f Carleman (cf. e.g. Heins [6], p. 121 — 123), we see that in the interiorS ° o îê ,

Q(t) = qie4), a (t) = a(e4), t < 0

ê = { t : inf v(et+i^) = 0 }

Then

r

(6)

Outside we use the convexity o f q(r) with respect to log r and obtain

(7)

We do not know a priori that q' is absolutely continuous. Therefore, (7) holds only in the distributional sense, i.e. the left-hand member is a non-negative measure. Let

HARMONIC MEASURE 207

The function p is continuous in<f° and in the complement o f S. It is also lower semicontinuous.Thus

Q ''(t) - p (t)2 Q(t) > 0 (8)

in the distributional sense on (—°», 0), i.e.

Q " - P2Q = dM = dMa + d/is

where dд is a non-negative measure and dMa(t) = h(t)dt and dps are the absolutely continuous and singular parts o f dp, respectively. Consider now Q, which is the classical solution o f the problem

QV(t) — P (t)2 Q ,(t) = h(t), Q i(0 ) = Q(0), lim Q ,(t) exists (9)t — CO

I f Q2 = Q - Q ,, we have

Q 2 - P2 Q2^ 0 , Q2(0) = 0, lim Q2(t) exists (10){ —> — OO

in the distributional sense. It is easily seen that we must have Q2 < 0, i.e. Q < Q l4We now apply Theorem B. I f T < 0 is given, we rearrange p to obtain p* and consider

w” (t) - p * (t )2 w (t) = 0 w(0) = Q(0), lim wt(t) > 0t -*■ — OO

Theorem В shows that Q (T) < Q ,(T ) < w(T).

To estimate w(T), we note that p* is increasing on (—°°, 0). It is now fairly easy to show that

0 0

w (T )< 2Q(0)V/A (¡T ) exp - J p*(r)dr < 2 /2л-А(ет ) exp - f p ( r )d T

T • T

Going back to our original notation, we obtain (2). Once we know that (2) holds, it is easy to prove (3).

Remark. I f we want to use Carleman’s original method to study the differential inequality (8), we must have

inf p(t) > 0 for each T < 0T < t < 0

In the problem studied here, we must be able to handle situations when p(t) = 0 on certain intervals. This difficulty is solved by applying Theorem B: we obtain a differential equation w " - (p * )2 w = 0 where the support o f p* is an interval [d, 0]. From this point on, we could also have used Carleman’s original method (cf. e.g. Heins [5]).

Similar results are true in IRd, d > 2. When d > 3, we use a cap symmetrization. I f r is given, F(r) is the subset o f the unit sphere { |x| = 1} where co(rx, a, D) > 0, and a(r) is the (d - 1 )- dimensional Lebesgue measure o f F(r). We define D* П {|x| = r } to be a spherical cap with the

208 ESSÉN

same (d - 1 )-dimensional measure as the set {x = ry, y € F (r ) } . Theorem A, properly interpreted, also holds when d > 3: this is a result o f C. Borell [7] and J. Sarvas. Borell’s paper is contained in Essén [31 where further references are given (cf. Ref. [3], p. 52).

The reason we preferred to refer to a paper o f A. Baemstein rather than to a paper o f M. Heins in the earlier discussion is that the result o f Baernstein is known to hold also in IRd, d > 3.

The second step in our proof is to deduce a differential inequality for the Carleman mean. To do this, we have to investigate a singular boundary value problem.

REFERENCES

[1 ] B A E R N S T E IN , A ., In tegra l means, univalent functions and circular sym m etrization , A c ta M ath. 133 (1 9 7 4 ) 1 3 3 -1 6 9 .

[2 ] B E U R L IN G , A ., E tudes sur un prob lèm e de m ajoration , Thesis, Uppsala (1 9 3 3 ).

[3 ] E SSE N , М ., Th e cos irX th eorem , Springer Lectu re N o tes in M athem atics 467, Berlin-Heidelberg ( 1975).

[4 ] H A Y M A N , W ., M u ltiva len t Functions, Cam bridge Univ. Press ( 1967).

[5 ] H E IN S , М ., On a no tion o f con vex ity connected w ith a m ethod o f Carlem an, J. Analyse Math. 7 (1 9 6 0 ) 5 3 -7 7 .

[ 6 ] H E IN S , М ., Selected Top ics in the Classical T h eo ry o f Functions o f a C om p lex Variable, H o lt , R inehart andW inston , N e w Y o rk (1 9 6 2 ).

[7 ] B O R E L L , C., “ A n in equa lity fo r a class o f harm onic functions in n-space” , Springer L ectu re N o tes in

M athem atics 467 (1 9 7 5 ) 9 9 - 1 1 2 .

IAEA-SMR-18 /6 4

PSEUDOCONVEXITY AND THE PRINCIPLE OF MAXIMUM MODULUS

A. A. FADLALLA Department of Mathematics,* Cairo University, Cairo, Egypt

Abstract

P S E U D O C O N V E X IT Y A N D TH E P R IN C IP L E O F M A X IM U M M O D U LU S .

Pseudoconvex ity o f a dom ain G С <ГП and the existence o f a function f h o lom orph ic in G such that f

assumes its m axim um at a boundary po in t P € 3G are c losely related to each other. Such relationships are discussed.

Definition 1. A domain G C (Гп is said to be pseudoconvex i f (i ) to every point P G 3G, there exist a neighbourhood U o f P and a real-valued function <fi € C2(U ) such that

Q П (J = {z : z 6 U, ip(z) < 0 ) (1 )

(ii ) dip Ф 0 in U ; (iii) the Hermitian form :

1 Szjj 3zy

for all Х д (м = 1, ■ ■ ■ , n) satisfying

( 2)

X„ = 0,■M P L , bH Ip

xu = о (3 )

I f the Hermitian form (2 ) is >0, G is called a strictly pseudoconvex domain.

Definition 2. A hypersurface H C Œn is called pseudoconvex if ( i ) to every point P £ H, there exist a neighbourhood U o f P and real-valued function <p G C2(U ), such that

Н П U = {z :z G U ,< p (z ) = 0} O ')

( ii) dip Ф 0 in U j (iii) the Hermitian form:

IM, v = 1

3 V

3z|¿ 9z„Хд X¡, (2 ')

* Facu lty o f Science.

209

210 FADLALLA

is semidefinite for all \ц (д = 1, . . . , n) satisfying

(3 ')

If the Hermitian form (2 ') under conditions (3 ') is definite, H is called strictly pseudoconvex. Now we give a short proof o f

Theorem 1. Let G be a strictly pseudoconvex domain and P € 3G. Then there exists a function f holomorphic in a neighbourhood o f G such that If(P)| = Max |f(G)| and |f(Q)| < |f(P)| for all Q e G, Q Ф P.

(i) Let P be the origin, and 3G in a neighbourhood o f P be the hypersurface ip = 0. Furthermore let t be a real parameter and

satisfies the relation S П U П G = { P } , [Cf. Ref. [ 1 ]].

(ii) Grauert [2] proved that there exists a neighbourhood N o f 3G and a strictly plurisubharmonic function g defined in N, such that 3G is the hypersurface g = 0 and N П G ={Q : Q € N, g (Q ) < 0 ]. Whence, for sufficiently small e > 0, the hypersurface g — e = 0 is the boundary o f a strictly pseudoconvex domain Gj. Furthermore, 3G] can be made as near to 3G as we please. Let us now choose 9G, so near to 3G and U so small that:

Then if a > 0 is sufficiently large, the analytic hypersurface

n n

itаду ZpZp - (t2 + - ) = 0, -e < t < e, e > 0

have the following property :

H П U П G = {P }

where U is a sufficiently small neighbourhood o f P and

Л n

Д= 1 H,v = 1

n n

(a) aHzlt +

H=1 p.,v = 1

(b) H1 = (H П G,n U } is a closed connected set (hypersurface).

Let H' = H' П G,

PSEUDOCONVEXITY 211

Now H' is a piece o f the hypersurface <p — (t2 + it/a) = 0. Solving this equation in t, we get

i 1t = ----- ± Ы --------

2a V 4a2

According to (a), we can assume that

1n ^ 2 < a lg

,p -------4a2

Зтг< — in U

2

Thus the function s/^p- 1 /4a2 is one-valued in U, and H' is a piece o f the hypersurface

t = - i/2a + \J*p~ l/4a2 . I f Ф = -i/2a + \/<p- 1 /4a2 then Ф is holomorphic in U, and H' is a piece o f the analytic hypersiirface v = 0, where v = Im Ф. Now all points o f {U-P} П G lie on one side o f H'. Let it be the side v > 0.

Now H' is a piece o f the analytic hypersurface v = Q, i.e. through any point Q E H ’ passes a complex analytic (n - l)-dimensional surface Sq, which is a subset o f H' (see (i)). It is obvious that Sp is the surface Ф = 0 and that Sp П G = P.

Now we define the open covering U, and U2 o f G| as follows. Let U* = {Q : Q € U, y(Q) < 0}, U, = U П G ,, U2 = G , — U*. Let f 2 be any function regular in U2. We associate with U[ and U2 the meromorphic function fj = l /Ф and f2 respectively. It is clear that fj — f2 is holomorphic in U, П U2.

Since G, is a strongly pseudoconvex domain, it is a domain o f holomorphy and, according to Oka [3], Cousin’s first problem can be solved in G ,, and a function f3 can be found which is equivalent — with regard to subtraction — to f, in Ui and to f 2 in U2. Thus f 3 has the points o f Sp as poles and otherwise is regular in Gi- Whence f 3 is regular in [G — P] but singular in P.

(iii) We shall consider the function f' = l/f3 as a mapping o f G in the compact complex plane C. Let U'cc Ube a neighbourhood o f P.

Since f 3 is regular in G — U', it is bounded in G — U'. Now let f3 = h + f| in Uj. Hence h is regular in U, and therefore it is bounded in U' П G . Since all points o f {U1 П G — [P ]} lie on the side v > 0, it follows that Im ( 1 /Ф) < 0 in U' П G. Whence there exists a number к such that Im f 3 < к in G. Generally, we can assume that к < 0. Hence w = f* can be considered as a mapping o f G in the half plane: Im w > 0 o f the complex plane o f the variable w.

Let T be a conformai mapping o f the half plane Im w > 0 on the unit circle |z| < 1, in the complex plane o f the variable z. Then f = T0f' possesses the required properties.

For pseudoconvex domains G С <tn, such a theorem does not exist unless 9G contains strictly pseudoconvex boundary points. For example, Rossi [4] proved that if such a function exists, then P should be the limiting point o f strictly pseudoconvex boundary points, from which we draw the following conclusions.

Let G be a pseudoconvex domain and P 6 9G. Suppose that there exists a function f holomorphic in a neighbourhood o f G such that |f(P)| = Max |f(G)|. I f there exists a neighbourhood U o f P such that U Л 9G contains no strictly pseudoconvex boundary points o f G, then:

(1) In every neighbourhood o f P, there exists points Q & 9G such that |f(Q)| = |f(P)|.(2) In every neighbourhood o f P, there exists Q e 9G such that f(Q ) = f(P).

Proof. I f this were not true, we could find a neighbourhood V o f P such that f(Q ) Ф f(P ) for all Q £ V П G,Q Ф P. Let Ф (z ) = f(P ) + f(z ). Thus |Ф(г)| will assume its maximum in V П G only at P, which would be a contradiction to Rossi’s theorem.

212 FADLALLA

The Sommer [55 conditions

In the same direction Sommer proved the following. Let G С (Tn be a pseudoconvex domain,P £ 3G and 3G in a neighbourhood U o f P be the hypersurface 0, where ¡p is a real-valued function 6 C4(U). Let the Hermitian form (2) under conditions (3) be o f constant rank k,0 < к < n in U. Then there exist a neighbourhood U1 o f P and a co-ordinate system (z b . . . , zn) in U' such that

3 c n u ’ = H ' x v 2 , g n u ' = v , x v 2 , h ' c a v ,

where V2 is a domain in the (n — k)-dimensional complex space (En " k (zjc + i ,. . . , zn) o f the variables Zk + i , . . . , zn; V, is a domain in the к-dimensional complex space <Ek(z , ,. . . , Zk) o f the variables z , ,. . . , z^and H' is a strictly pseudoconvex hypersurface in <Ek(z , ,. .. , Zk).

That is to say, the boundary o f G in a neighbourhood o f P is decomposable into (n - k)- dimensional analytic fibres; each fibre is o f the form P' X V 2, where P' £ H'. We denote the analytic fibre through P by^j>.

Now if f is holomorphic in a neighbourhood o f G and if |f| assumes its maximum at P, it is obvious that f(Q ) = f(P ), for all Q £ Which is in agreement with Corollary 2 o f Rossi’s theorem. Now we prove

Theorem 2. Let f be a function holomorphic in G and continuous in G U {P }, i.e.

lim f(P m)ТО-ЮО

exists for all sequences o f points {Pm} C G converging to P. Obviously this limit is unique and will be denoted by f(P). Furthermore let |f(P)| = Max |f(G)|. Then f can be extended continuously to SÇ and f(Q ) = f(P ) for all Q

Proof. Let P = P' X P" where P’ £ H', P” £ V2. Now let Q therefore Q = P' X Q ", where Q” £ V2. Let [Qm ] С U' be a sequence o f points converging to Q and Qm = Qró X Qm, Qme v i>Qm e V2. Thus [Qm] converges to P' and [Qm] converges to Q''. Now let Pm = Qm Xp" ; thus [Pm] converges to P.

Let J = (zk + i ,. • ■ , zn) £ V 2 and put fm ( f ) = f(Qm ,f). Since [fm( f ) ] is a bounded sequence o f holomorphic functions in V2, it is a normal family; it contains a uniformly convergent subsequence ! fm'(J )} to a holomorphic limit function f0(f). Now fm' (P ") = f(Qm' > P ") = f(P m') f(P ) = f0(P"). Thus |f0(P ")l =|f(P)l = Max |f0(V 2)|; thus f0( f ) is constant and equals f(P ) for all f £ V2.

Now f(Qm’) = f(Qm', Qm') = fm'(Qm') f0(Q ” ) = f(P). which proves the theorem.

REFERENCES

[1 ] B E H N K E , H ., S O M M E R , F., D ber d ie Voraussetzungen des Kontinu itâtssatzes, Math. Ann . 121 (1949/1950 ).

[2 ] G R A U E R T , H ., O ber M od ifilcationen und exzep tion e lle analytische Mehgen, M ath. Ann . (1 9 6 2 ).

[3 ] O K A , K ., Dom aines d ’ho lom orph ie , J. H iroshim a Univ. ( 1937).

[4 ] R O SS I, H „ Ann . Math. 74 3 (1 9 6 1 ).

[5 ] S O M M E R , F ., Kom plex-analytische Blatterung reeller H yperflâchen im C ", Math. Ann . 137 (1 9 5 9 ).

IAEA-SMR-18/53

QUASICONFORMAL MAPPINGS

F.W. GEHRING Department of Mathematics,University of Michigan,Ann Arbor, Michigan,United States of America

Abstract

Q U A S IC O N F O R M A L M A P P IN G S .

1. M odulus o f a curve fam ily . 2. C on form ai capacity o f condensers. 3. Inner and ou ter dilatations.

4. D is tortion and convergence. 5. O ne-quasicon form al mappings. 6. M apping problem s. 7. A n ex istence theorem .

1. MODULUS OF A CURVE FAMILY

G iven a f a m i l y , Г , o f n o n c o n sta n t c u rv e s у in ? , we

l e t ad m (r ) d en o te the f a m i ly o f B o r e l m e a su ra b le fu n c t i o n s

p : Rn -*■ [0 , « ] such t h a t

/ pds 1

Y

f o r a l l l o c a l l y r e c t i f i a b l e у € Г . We c a l l

М (Г ) = i n f I pndm , Х (Г ) = М (Г )~ “p e adm ( Г ) Rn

th e modulus and e x t r e m a l l e n g t h o f Г , r e s p e c t i v e l y .

When Г i s a f a m i l y o f a r c s , we may th in k o f М (Г ) as

th e con duc tan ce and А (Г ) a s th e r e s i s t a n c e o f a system o f

homogeneous w i r e s . М (Г ) i s b i g when th e w i r e s a r e p l e n t i f u l

o r s h o r t , s m a l l when th e w i r e s a r e few o r l o n g .

213

Theorem 1 . М (Г ) i s an o u t e r m easure on th e c o l l e c t i o n s o f

c u rv e f a m i l i e s Г in R11 . That i s ,

214 GEHRING

a) М ( ф ) = 0 ,

b ) М( ГХ) < М( Г2) when С Г

c) м(иг .) 0 we can choose f o r each j a

€. a d m (T j ) such t h a t

/ p!?àm < М(Г . ) + 2~3гRn 3 - Э

Now s e t

p = sup p. , Г = U r . j J j

Then p : Rn ■+ [0 , " ] i s B p r e l m e a s u ra b le . M oreove r , i f

Y S Г i s l o c a l l y r e c t i f i a b l e , then у e Г. f o r some j ,

/ pds :> /p .ds 1

Y Y D

and hence p e adm(T) . Thus

М(Г) <_ ( pndm < f J p^dm < J M (Г . ) + eRn R11 j 3 j 3

Remark 1 . I f we a p p ly th e C a ra th é o d o ry c r i t e r i o n to the

o u t e r m easure M to d e f i n e the n o t i o n o f a m easu rab le cu rv e

f a m i l y , then we can show the f o l l o w i n g :

a ) Г i s m e asu ra b le i f М (Г ) = 0 ,

b ) Г i s n o t m easu rab le i f 0 < М ( Г ) < °° ,

c ) Г may o r may n o t be m e a su ra b le i f М (Г) = » .

QUASICONFORMAL MAPPINGS 215

Theorem 2 . I f each cu rv e y^ in a f a m i ly c o n t a in s

a s u b c u rv e y 2 a f a m i ly , then М (Г^ ) £ '

P r o o f , Choose p £f a d i n f ^ ) and suppose y^ G i s

l o c a l l y r e c t i f i a b l e . Then

where y 2 i s th e su b cu rv e in Г2 » and p e ad m (r^ ) .

Thus

М (Г . ) £ / pndm Rn

and t a k in g th e in fimum o v e r a l l such p y i e l d s

M d ^ ) < М (Г2 )

Theorem 3 . М (Г) i s a d d i t i v e on cu rv e f a m i l i e s i n d i s j o i n t

B o r e l s e t s . Th at i s , i f E a r e d i s j o i n t B o r e l s e t s and i f the

c u rv e s o f Г . l i e in E . , then3 3

E x am p le s . a ) R e c t a n g u l a r _ p a r a l l e l o p ip e d _ . I f Г i s th e

f a m i ly o f c u r v e s у j o i n i n g two

p a r a l l e l f a c e s o f a r e a A and

/ pds / pds

* 1 Y 2

M ( U r . ) = I M ( г . ) j J j J

d i s t a n c e h a p a r t , thenh

216 GEHRING

b ) S p h e r i c a l _ r i n g . I f Г i s the f a m i ly o f cu rv e s

j o i n i n g th e sp h e re w i t h c e n t e r x Q and r a d iu s a to the

c o n c e n t r i c s p h e re o f r a d i u s b , then

М( Г ) = W j ^ d o g l ) 1" 11

where

0 < а < Ь < ° ° , ш = щ (s)— — n - 1 n - 1

Here m . (S ) d e n o te s th e s u r f a c e a r e a o f th e u n i t s p h e re in n — 1

P r o o f f o r a ) . Choose p e adm(T) and l e t y^ be the

v e r t i c a l segment from у in the

b a s e E . Then у e Г and У

— ( / P^s) П £ hn ^ / pnds4 > Yy

T h is h o ld s f o r a l l such у and hence

E Y,/n pnas 1 / / /pnas\ атп_ х > - A _

S in c e p i s a r b i t r a r y ,

М (Г ) >- hn - l

N ex t s e t P = h i n s i d e the p a r a l l e l o p i p e d and p = 0 o t h e r w i s e .

Then p e adm( Г ) and

М (Г ) < / pndm = -1— -

Theorem 4 . I f a l l the c u rv e s in a cu rv e f a m i ly Г p a s s

th rough a f i x e d p o i n t x^ , then М (Г ) = 0.


P r o o f . Suppose f i r s t t h a t Xq ^ » and f o r each j l e t

d en o te the s u b f a m i ly o f y G Г w hich i n t e r s e c t x^ and-

S ( x Q, l / j ) . Then each у e IV c o n t a in s a su b c u rv e y ' in

th e f a m i ly o f a l l c u r v e s j o i n i n g x n t o S ( x g , l / j ) i n B ( x n , l / j )

Hence

S ( x Q, l / j )

When Xq = “ , we a r g u e a s abo ve w ith

S (X q » 1/ j ) r e p l a c e d by S ( 0 , j ) .

Theorem 5. I f f : R11 R11 i s a M obius t r a n s fo r m a t io n ,

then

M ( f ( Г ) ) = М(Г

P r o o f . Choose p ' € a d m f ( r ) , s e t

p (x ) = p ' » f (x ) j f ' (x ) I

f o r x € Rn ~ i f ^ ( ° ° ) } » and l e t be th e f a m i ly o f

у e Г w hich p a s s th ro u gh f 1 ( “ ) . Then

and hence

218 GEHRING

M (Г ) < J pndm = I ( p ' o f ) n | f ' | n dm

= f ( p ' о f ) nJ ( f ) d m = j ( p ' ) n dm Rn Rn

T a k in g th e in fimum o v e r every such p ' g i v e s М (Г ) £ M ( f ( T ) ) ,

The r e s u l t f o l l o w s by r e p e a t i n g the p r e c e d in g argument w i th f

r e p l a c e d by f - 1 .

T ^ § 2 £ S ^ ^ - I f f j » f : Rn + [O ,00] a r e B o r e l m ea su ra b le and

i f f j -*■ f in Ln (Rn ) , then th e r e e x i s t s a subsequen ce { j ^ }

and a c u rv e f a m i ly r Q w i t h m(Tq ) = 0 such th a t

l im / I f . - f Ids = 0

к -*■ «° у

f o r a l l l o c a l l y r e c t i f i a b l e c u r v e s у , у ф Г g .

P r o o f . Choose a subsequence so th a t

/ g?dm < 2 - ( n + 1 ) k , g = | f . - f|Rn k ^k

and l e t TQ be th e f a m i ly o f a l l l o c a l l y r e c t i f i a b l e y in

R*1 such t h a t

l im sup / gv ds > 0

к +oo у

We want t o show t h a t M ( Г ). = 0

L e t be the f a m i ly o f a l l l o c a l l y r e c t i f i a b l e c u rv e s

in Rn f o r wh ich

Then p = 2 g^ £ adm( ) and

QUASICONFORMAL MAPPINGS

V

219

М ( Г . ) < 2 ~ l+ 1

U k = * K U ~ k = f c K

and hence м (Гд) = 0 .

Theorem 7 . I f :*-s an i n c r e a s i n g sequence o f cu rve

f a m i l i e s , then

M (U r . ) = l im М <Г .) j 3 j -»■ 00 3

I d e a o f p r o o f . L e t Г = L/Г . . Then by m o n o to n ic i t y o fj 3

th e modulus ,

М (Г ) > l im М (Г .)j ■+ oo 3

P o r th e r e v e r s e i n e q u a l i t y , we may assume t h a t the l i m i t i s

f i n i t e . Then s in c e L (R n ) i s u n i fo rm ly c o n vex , we can choose

pj e ad m (T j ) so t h a t P j ■+• p in Ln (Rn ) and so t h a t

/ pndm = l im ! p^dm = l im М (Г . )В П т + oo P ^ -Î-+-00 3

By Theorem 6 t h e r e e x i s t s a subseq ue n ce an< a f a m i ly

Гр w i th M ( r Q) = 0 such t h a t

/ pds = l im / p . dsY k -*■ °» y 3 k

f o r a l l l o c a l l y r e c t i f i a b l e у £ Г ~ Г

l i e s i n Г . f o r l a r g e к ,3 k

220 GEHRING

0S in ce each such у

/pds > 1

У

Thus p g adm (r ~ Tq ) , and we co n c lu d e th a t

M (Г ) = М(Г ~ r Q) < ^ pndm = l im М(Г .)

Remark 2 . S in c e the a r e i n c r e a s i n g in Theorem 7,

Г = U r . = l im Г.j 3 j + Я ^

in th e s e t t h e o r e t i c s en se and so we s e e t h a t the c o n c lu s i o n o f

Theorem 7 i s a c o n t in u i t y p r o p e r t y f o r th e m odulus. U n fo r t u n a t e l y

no such r e s u l t h o ld s f o r d e c r e a s in g f a m i l i e s I\ w i th

Г = ПГ . = l im Г .•> j -*■ 00 ■>

2. CONFORMAL CAPACITY OF CONDENSERS

We l e t q ( x , y ) d en o te the c h o r d a l d i s t a n c e between p o in t s

x , y c Rn . That i s

<1 ( x , y ) =

I X - у I

/ l + 1 X 1 2 Á

/ l + | X | 2

i f x , y ?

A c o n d en se r i s a domain R C R whose complement i s the

un io n o f two d i s t i n g u i s h e d d i s j o i n t compact s e t s CQ and Сд^

F o r c o n v en ien ce we w r i t e

A r i n g i s a c o n d en se r R = R (C g ,G ^ ) where Cg and a r e

c o n t in u a . We c a l l Cg and the com plem entary components

o f R

G iven a co n d en se r R = R (C g ,C ^ ) w i t h R c Rn; we l e t

adm (R) d en o te th e c l a s s o f f u n c t i o n s u : Rn •* R^ w i th the

f o l l o w i n g p r o p e r t i e s :

a ) u i s c o n t in u o u s in Rn , (77/t jÿ ^0

b ) u has d i s t r i b u t i o n d e r i v a t i v e s in R ,' Л

c ) u = 0 on Cg and u = 1 on Cn . 4-— '

N ote th a t

r q ( x , C 0 )u ( x ) = min {- jz— рг-т , 1J e adm (R) q '^ o

and hence adm (R) ^ ф . W e c a l l

cap (R) = i n f / j Vu IП dmueadm (R) R

th e c o n fo rm a i c a p a c i t y and modulus o f R , r e s p e c t i v e l y .

E xam p le . I f R i s th e r i n g in Rn bounded by

c o n c e n t r i c s p h e re s o f r a d i i a and b , 0 < a 

then

222 GEHRING

mod (R) = l o g —cl

Remark. I f f : Rn + Rn

th en cap ( f ( R ) ) = cap (R) f o r a l l co n d e n se rs R w i th

R , f (R) с R Hence we can use t h i s f a c t t o d e f i n e cap (R)

f o r a l l co n d en se rs in R

p o i n t .

wh ich c o n t a in «° as an i n t e r i o r

G iven E , F , G c R , we l e t A (E ,F ;G ) d eno te th e f a m i ly

o f a l l c u rv e s у w i t h

a ) one e n d p o in t in Ё and th e o th e r in F ,

b ) i n t e r i o r in G

Theorem 1. I f R = R (C g ,C ^ ) i s a r i n g and i f

Г = MCjjfC jy-R ) / then cap (R) = М (Г)

O u t l i n e o f p r o o f . By p e r fo rm in g a p r e l im in a r y Mobius

t r a n s fo r m a t io n we may assume th a t 00 e

L i p s c h i t z i a n f u n c t i o n u e adm (R) and s e t

Choose a l o c a l l y

P (x ) =|Vu(x)

0

x e R

x g CQ и Сх

I f y : [ a , b ] i s a l o c a l l y r e c t i f i a b l e c u rv e in Г , then

Hence p e adm (Г ) and

М (Г ) <

By a sm ooth ing a rgum ent, we

o f th e r i g h t - h a n d s i d e o v e r

Thus

М (Г ) <

Choose a bounded c o n t in u ou s

u ( x ) = min

f o r x e: r , where th e infimum i s taken o v e r a l l l o c a l l y r e c t i

f i a b l e у j o i n i n g Cq to x in R . Then u has d i s t r i b u t i o n

d e r i v a t i v e s and

l im u ( x ) = 0 , l im u ( x j = 1

x - C Q x - C l

Hence we can e x ten d u t o Rn so t h a t u e; adm (R) . Then

s in c e I Vu I = p in R ,

cap (R) < / pn dm < / pn dm- R - Rn

A n o th e r sm ooth ing argument shows the infimum o v e r such p

g i v e s K\T) . Thus cap (R) < М (Г ).

G iven a r a y L from x^ to » and a compact s e t

E с Rn , we d e f i n e th e s p h e r i c a l s y m m etr iza t io n o f E in

L a s th e s e t E* s a t i s f y i n g th e f o l l o w i n g c o n d i t i o n s :

a ) x Q с e * i f f x Q g: E ,

b ) » e E* i f f «> с E ,

Л Vu In dm R

can show t h a t t a k in g th e infimum

a l l such u g i v e s cap (R) .

cap (R)

p e adm (Г ) and s e t

( l , i n f / p d s )

c ) F o r r e: (0 , ° ° ) , E* П S ( x Q, r ) ^ ф i f f E (1 S ( x Q , r ) i- ф,

in w h ich c a se E* П S ( X g , r ) i s a c l o s e d s p h e r i c a l cap c e n te r e d

on L w i t h the same mn_^ m easure a s E f| S ( X g , r ) .

We see t h a t E* i s compact and t h a t E* i s connected ( F i g . l )

i f E i s .

224 GEHRING

F IG . 1. Spherica l sym m etriza tion o f a set.

Theorem 2 . I f E* i s th e s p h e r i c a l sy m m etr iza t io n o f

E in a r a y L , then

a ) mn (E * ) = mn (E) ,

b ) mn _ 1 (3E * ) < m ^ O E ) .

O u t l i n e o f p r o o f . To p ro v e a) we a p p ly F u b i n i ' s theorem

and o b t a in

mn (E * ) = / " mn _ 1 (E* П S ( x Q, r ) ) d r = /°°mn _ 1 (E П S ( x Q, r ) ) d r

= mn (E)

F o r b ) , assume f i r s t t h a t E i s a p o ly h e d ro n . Then f o r

r e : (0 , ° ° ) th e B runn -M in kow sk i i n e q u a l i t y im p l i e s th a t

E * ( r ) = { x : d i s t ( x , E * ) < r } <r { x : d i s t ( x ,E ) < r } * = E ( r ) *

and hence t h a tm ( E * ( r ) ) - m (E * )

m , ( ЭЕ* ) < l im sup ---------------------------------- —n _ 1 - r - о 2 r

m (E ( r ) ) - m (E)< l im sup — --------------------- 2------ = m (ЭЕ)

r * 0 2 r n”

The g e n e r a l r e s u l t f o l l o w s by a l i m i t i n g argum ent.

F IG .2. Spherica l sy m m etr iza tion o f a ring.

Theorem 3 . I f R = R (C g ,C ^ ) i s a cond enser

and i f Cg and CJ a r e th e s p h e r i c a l sy m m etr iza -

t i o n s o f Cq and C^ in o p p o s i t e r a y s Lg and

, then R* = R (C ^ ,C * ) i s a c o n d en se r w i th

cap (R * ) < cap (R) (see Pig. 2 ) .

I d e a o f p r o o f . Choose a l o c a l l y L i p s c h i t z i a n u e: adm (R)

and d e f i n e u * so t h a t { x : u * (x ) < t } = { x : u ( x ) < t } * .

Then u * e: adm (R * ) and Theorem 2 a l l o w s one to show t h a t

cap (R * ) < / I Vu* In dm < / |Vu|n dm.

Rn Rn

T ak in g the infimum o v e r a l l such u y i e l d s the r e s u l t .

L e t e l f e 2 , . . . , e n d en o te th e b a s i s v e c t o r s in Rn . Fo r

t e ( 0 , “ ) l e t RT ( t ) d en o te th e r i n g domain in Rn whose

complement c o n s i s t s o f th e r a y from

t e ^ t o 00 and th e segment from - e ^

to 0 . ^ ( t ) i s c a l l e d th e T e i c h m ü l le r

r i n g . The f o l l o w i n g p r o p e r t i e s f o r i t s

m odulus can be e s t a b l i s h e d :

a ) mod RT ( t ) - l o g ( t + 1 ) i s n o n d e c re a s in g i n ( 0 ,°°) ,

b ) l im mod R_, (t ) = 0 ,t - 0 1

R j C t )

- e ^ 0 t e ^

c ) l im ^mod R^ - lo g ( t + l ) ) = l o g

d ) X- = 16 and l im X1 = e2 .2 n

n+co

Thus

e ) mod R,j,(t) i s s t r i c t l y i n c r e a s i n g in ( 0 , “ ) ,

f ) l o g ( t + 1 ) < mod RT ( t ) < l o g Лп ( t + 1 )

Theorem 4. I f R = R (C q ,C^ ) i s a r i n g w i t h a , b e: Cg

and c ,° ° g , then ^

mod R < mod R ( | £ ^ Ц )

226 GEHRING

T 4 b - a I ■

Cr0, R W ' C1

P r o o f . By p e r fo rm in g a p r e l im i n a r y g z z z z z a --e , 0 Ic - al

s i m i l a r i t y mapping, we may assume t h a t1 |b - a| ~ 1

0 , b = - e , . Then th e s p h e r i c a l Ct C*1 R* —

<zzzzzd„ <2zzzzzz

*T

sy m m e tr iz a t io n s С * , C* o f CQ, — Q jc

in th e n e g a t i v e and p o s i t i v e h a l v e s ^ ^

o f th e x ^ - a x i s c o n t a in th e com ple

m entary components o f R_ (-{-£— -j-1 .- e , 0 |c -a|

Thus |b — a| 1

cap RT t jb' - a - cap R* - cap R

as d e s i r e d .

C o r o l l a r y 1 . I f R = R (C q ,C^) i s a r i n g w i t h a , b e Cq

and c , d E C j , then

( a , c ) q ( b , d ) ■mod R < mod » (Я a_f.° 4 ,d ) T '■q ( a , b ) q ( c , d ) '

P r o o f . By p e r fo r m in g a p r e l im in a r y c h o r d a l is o m et ry we

may assume t h a t d = °° . Then

Ic - a| _ q ( а , с ) /|c| 2 + 1 = q ( a , c ) q ( b , d )

, b _ a l q ( a , b ) / | b | 2 + i q ( a , b ) q ( c , d )

and we can a p p ly Theorem 4.

C o r o l l a r y 2. I f R = R (C q ,C 1 ) i s a r i n g , then

a ) mod R < mod )

4 q (C 0 ,C 1)b ) mod R < mod RTÍq t c ^ g ^ c j J •

P r o o f . F o r a ) , choose a , b e Cg and c , d e C^ so

th a t

q ( a , b ) = q (C g ) , q ( c , d ) = q (C 1 ) c.( V

Then 4 C0ЖV.

g ( a , c ) g ( b , d ) < 4 >

q ( a , b ) q ( c ,d ) - q (CQ) q (C1)

and we can a p p ly C o r o l l a r y 1. F o r b ) choose a e CQ and

с e C^ so t h a t

q ( a , c ) = q C C g , ^ )

N e x t p i c k b e: Cg , and d e C^ so t h a t

q ( a , b ) ^ j q ( C 0) , q ( c , d ) q i C j )

Then

g ( a , c ) g ( b , d ) 4 q ( C 0 ,C l ) q ( a , b ) q ( c , d ) - q (C Q) g ( C ;L)

and vie a g a in a p p ly C o r o l l a r y 1.

C on vergen ce _o_f_ s ets_. We say t h a t a sequence o f s e t s

c o n v e r g e s u n i f o r m ly to a s e t E i f f o r each e > 0

t h e r e e x i s t s a j g such th a t

sup q ( x , E ) < e , sup q (x ,E _ . ) < e

x e E j xeE J < 7 7 ? / 'a

f ° r 3 > j g •

Theorem 5 . I f th e com plem entary components o f a sequence

o f r i n g s Rj co n v e rg e u n i fo rm ly to th e c o r r e s p o n d in g com plem entary

components o f a r i n g R , then

cap (R) = l im cap (R . )■1-+-0O 3

228 GEHRING

3. INNER AND OUTER DILATATIONS

Suppose D , D 1 a r e domains in Rn and th a t f : D ■* D 1

i s a homeomorphism. We c a l l

к m - ciir) M ( f ( Г ) ) к f f i - sun M ( r )Kl ( } pP м (Г ) ' 0 5 s " p _ M ( f ( D )

th e in n e r and o u t e r d i l a t a t i o n s o f f , where the suprema

a r e ta k e n o v e r a l l c u rv e f a m i l i e s in D f o r w h ich М (Г )

and M ( f ( T ) ) a r e n o t s im u lt a n e o u s ly 0 o r “ . S i m i l a r l y

we c a l l

K * ( f ) = S U D CaP ( £ ( R ) ) K * ( f ) = SU D c a p ( R )KI l f ) cap (R) ' K0 l J “ P cap ( f ( R ) )

th e in n e r and o u t e r r i n g dilatations o f f , where the suprema

a r e ta ken o v e r a l l r i n g s R w i t h R с D . O b v io u s ly

K * ( f ) < Kz (f) , K * ( f ) < KQ ( f )

We say th a t f i s K -q u a s ic o n fo r m a l i f

max (Kj. ( f ) ,KQ ( f ) ) < К < ®

Suppose t h a t f : Rn ■* Rn i s a l i n e a r b i j e c t i o n . We

th e in n e r and o u t e r a n a l y t i c dilatations o f f , where

I f I = sup ! f ( x ) I , Z ( f ) = i n f I f ( x ) I , J ( f ) = Id e t f|I x I = 1 I x I = 1

I f E i s th e image o f th e u n i t b a l l В un der f , then

h f f ) = m h ( f ) = m(BQ)HI ( f ) mtBj . ) ' 0 l ' m(E)

w here B j and B^ a r e the l a r g e s t

i n s c r i b e d and s m a l l e s t c i r c u m s c r ib e d

b a l l a bo u t E


Theorem 1. I f f : D -*■ D 1 i s a d i f f e o m o rp h is m , then

К ( f ) < sup H ( f 1 (x ) ) , К ( f ) < sup H ( f ’ Cx))X E D X € D

P r o o f . F o r th e second p a r t choose a cu rve f a m i l y Г

in D . L e t p ' g adm ( f ( T ) ) and s e t

p (x ) = p 1 o f (x ) I f 1 (x ) I xD (x )

I f y g Г i s l o c a l l y r e c t i f i a b l e , then

/ p (x ) ds = / p 1 o f ( x ) ) | f (x ) I ds > / p ’ (x ) ds > 1

Y Y f ( Y )

and hence p с adm (Г ) . Thus

Hence

230 GEHRING

M ( Г ) < sup H _ ( f ' ( x ) ) / p' ( x ) n dmXGD D 1

and t a k in g infimum o v e r a l l such p 1 y i e l d s

М (Г ) < sup H ( f ' ( x ) ) M ( f ( Г ) ) xeD u

a s d e s i r e d .

Theorem 2. I f f : Rn Rn i s a l i n e a r b i j e c t i o n , then

H j i f ) < K * ( f ) , HQ ( f ) < K * ( f )

P r o o f o f f i r s t i n e q u a l i t y . By p e r fo rm in g p r e l im in a r y

o r t h o g o n a l m appings we may assume t h a t

f ( e i ) = Xi e i ' X 1 i > X > 0 n

f o r i = l , * * * , n . Then

t rc\ X, • • • XHT ( f ) = ( * = — ---------

1 * ( f ) n Лп

and we want t o p ro v e t h a t

( * )Xl * ” Xn - l

{X ) n

n -1 - I< K i ( f )

F o r r e ( 0 , “ ) l e t R be th e r i n g R ( F , C ( G ) ) , where

F = ( x : |x.J < 1, x f

Then we see t h a t ( F i g . 3)

F = ( x : |x^| < 1, x^ = 0 } , G = { x : |x^| < 1 + r , |x | <

p = - XR e adm (Г )

w here Г i s th e f a m i ly o f c u rv e s j o i n i n g F and C (G ) in

R . Hence

cap.. n

(R) = М (Г ) < / Pn dm = ( i ) 2 ( l + r ) * * * 2 ( l + r ) • 2r

n - 1

N e x t i f T± and Г2 a r e th e cu rve

f a m i l i e s j o i n i n g th e b a s e s o f th e in d i c a t e d

p a r a l l e ] o p i p e d s and P 2 i n f ( R ) i then

л 2X, • • • 2Хэт , . X, • • • X ,W / T, \ . Ж* / TI \ ^ 1 П 1 _ п П — 1 1 П —1

! - 2 " Л Р Г --------- _ vn - l ---------------2 , , _ , n - l "(Хп г ) ' (Хпг ) '

F in a l ly , s in c e and Г2 l i e in d i s j o i n t B o r e l s e t s

n Xi -,cap f ( R ) = M ( f (Г ) ) > М (Г 1 ) + М (Г2 ) = 2 — ----------

( X n r ) '

and hence

K * ( f ) > > x i ” • ^n —1- cap R

L e t t i n g r 0 y i e l d s ( * )

n - 1U + r )

1 -n

Theorem 3. I f f . : D. ->• D! and f : D D ' a r e- 3 3 3

homeomorphisms and i f

f = l im f .j -VOO J

f " 1 = l im f T 1j ->-00 ^

232 GEHRING

u n i fo rm ly on compact s e t s in D and D' , r e s p e c t i v e l y , then

K * ( f ) < l im i n f K * ( f ) , K * ( f ) < l im i n f K * ( f . )j -► oo U ~ j co 0 3

P r o o f . F o r f i r s t p a r t choose the r i n g R w i th R C D .

Then R с Dj f o r j > j p and we see t h a t th e complementary

components o f f ^ (R ) c o n v e rge u n i fo rm ly to the c o r r e s p o n d in g

com plem entary components o f

f (R) . Thus by Theorem 5 o f

C h ap te r 2, we have

cap f (R) = l im cap f . (R) < l im i n f K ^ C f . ) cap R j-*co 3 j -► 00 3

whence

£a_P_ Í _tR) . < l im i n f K * ( f . ) caP R - 3 -, 00 1 3

T ak in g the suprem um over all such R yields the result.

Theorem 4 . I f f i s a homeomorphism which i s d i f f e r e n t i a b l e

a t Xq g Rn w i t h J ( f ' ( x Q ) ) ? 0 , then

H j i f ' ( x Q)) < K* ( f ) , HQ ( f 1 ( x 0 )) < K*(f)

P r o o f . We may assume Xq = 0 and f ( * g ) = 0 .

F o r each j l e t

f . ( x ) = j f ( | )

Then f (0 ) : Rn -*■ Rn i s a l i n e a r b i j e c t i o n and

f ' ( 0 ) = l im f . , f ' ( 0 ) _ 1 = l im f T 1j -ЮО j-*00

u n i f o r m ly on compact s e t s in Rn . Thus by Theorems 2 and 3 ,

H ( f ’ (0 ) ) < K* ( f ' (0 ) ) < l im i n f К ( f . ) = К ( f )j 00

H ( f • ( 0 ) ) < K * ( f ' ( 0 ) ) < l i m i n f К ( f . ) = К ( f )j ->• oo u 3

as d e s i r e d .

Theorem 5. I f f : D -*■ D 1 i s a d i f f e o m o rp h is m , then

К ( f ) = K * ( f ) = sup H ( f ' ( x ) )xCD •L

К ( f ) = K * ( f ) = sup H ( f ' ( x ) )X C D

P r o o f . By Theorems 1 and 4 ,

К ( f ) < sup H j i f ' i x ) ) < K * ( f ) < K j C f ) xeD

KQ ( f ) ( x ) ) < К , H0 ( f ' ( x ) ) < К

f o r a l l x e: D . One can p ro v e t h a t a homeomorphism f : D -*• D'

i s K -q u a s ic o n fo r m a l i f and o n ly i f

a) f has l o c a l l y Ln- i n t e g r a b l e d i s t r i b u t i o n d e r i v a t i v e s ,

b ) Hx ( f ' ( x ) ) < К and H ( f ' ( x ) ) < К a . e . in D ^ { « , f ” 1 (<») }.

4. DISTORTION AND CONVERGENCE

Theorem 1 . I f f : D ->■ D ' i s K -q u a s ic o n fo r m a l and i f

С (D) ф ф , then

q ( f { x ) , f ( y ) ) q ( C ( D * ) ) 1 « t e c f e ) Ь

1

f o r x , y e D , w here a = 2Xr and b = n .

234 GEHRING

P r o o f . F i x x , y e D w i t h x ? y and suppose t h a t

q ( x , y ) < q ( x , C ( D ) )

By p e r fo r m in g a p r e l im i n a r y c h o r d a l i s o m e t ry we may assume

t h a t x = 0 . N ex t ch oose r and s so th a t

q (0 , y ) < r = j ---- ^---- 5 - < q (0 ,C (D) )V1 + s ¿

and l e t R denote, the s p h e r i c a l r i n g

R = { z : q ( 0 , y ) < q ( 0 , z ) < r } = { z : |y| < |z| < s }

Then R С D and R s e p a r a t e s

th e p o i n t s 0 , у from C(D) .

A l s o , i t i s e a s y t o see th a t

Iy I ÿ q ( o , y )

and hence th a t

mod R = l o g > l o g

L e t Cq and d e n o te the com plem entary components o f f ( R ) ,

l a b e l e d so th a t

f ( 0 ) , f ( y ) e CQ , C ( D ’ ) с с х

Then by th e f i r s t e s t im a te in C o r o l l a r y 2 o f C h ap te r 2,

mod f (R) < mod q ( C l ) )

- l 0 g Xn (q T c ^ T q T c ]T + -1)

2 Xn

- 1 0 9 q ( f ( 0 ) , f ( y ) ) q ( c ( D * ) )

S in c e f i s K -q u a s ic o n fo r m a l ,

c ap f (R) <_ К cap R

1

mod f (R) > K1 -n mod R

and com b in ing the above i n e q u a l i t i e s y i e l d s

q ( f ( 0 ) , f ( y ) ) q ( C ( D ' ) ) < a ( a i | i Z L )

1

w it h a = 2 ^ and b = K^1 n . L e t t i n g r -*■ q ( 0 , C ( D ) ) c o m p le te s

th e p r o o f in c a s e q ( x , y ) < q ( x , C ( D ) ) .

Suppose n e x t th a t

q ( x , y ) >_ q ( x , C (D) )

Then s in c e a = 2An ^ 32 ,

q ( f ( x ) , f ( y ) ) q ( C ( D ' ) ) < 1 < a ^ ¿ У - l -j b

C o r o l l a r y 1. I f f : D D ' i s K -q u a s ic o n fo r m a l , then

f i s К1 - 1 1 - H o ld e r c o n t in u o u s w i t h r e s p e c t t o th e c h o r d a l

m e t r ic i n D and w i t h r e s p e c t t o the e u c l i d e a n m e t r ic in

D ~ { ~ , f 1 ( “ ) } .

P r o o f . Choose domains and D2 such t h a t

D = Dx U D2 , q ( C ( D | ) ) > 0

w here = f ( D ^ ) . Then Theorem 1 im p l i e s th e d e s i r e d r e s u l t

in , i n D2 and hence in D .

Remark 1 . I f f : В В i s K -q u a s ic o n fo r m a l w i t h

f ( 0 ) = 0 , then one can show by a s i m i l a r argum ent th a t

|f (x ) - f (y) I <_ a|x - y | b

1

f o r x , y G. B , where a = 4X and b = K1 -n .n

Remark 2 . I f b e ( 0 , 1 ] , th e mapping

f ( x ) = |x|b - 1 x

i s K -q u a s ic o n fo r m a l , where K = b 1 n . Thus th e exponent f o r

H o ld e r c o n t in u i t y in C o r o l l a r y 1 i s s h a rp .

C o r o l l a r y 2 . I f r > 0 and i f F i s a f a m i ly o f K-

q u a s i c o n fo r m a l m appings f : D -> such t h a t q ( C ( D £ ) ) > r

f o r a l l f G F , then F i s e q u i c o n t in u o u s .

P r o o f . I f f G F , then C(D^ ) ф ф im p l i e s th a t

С (D) И ф and hence by Theorem 1,

q ( f ( x ) , f ( y ) ) < t q ( f ( x ) , f ( y ) ) q ( C ( D ¿ ) ) <

f o r a l l x , y € D .

C o r o l l a r y 3 . I f F i s a f a m i ly o f K -q u a s ic o n fo r m a l

m appings f : D w h ich i s n o t e q u ic o n t in u o u s in D , then

t h e r e e x i s t s an € D and a sequence f £ F such t h a t

i s e q u ic o n t in u o u s in D ~ *

P r o o f . By h y p o t h e s i s , t h e re e x i s t s an x Q e D and an

r G ( 0 , 1 ) such t h a t th e s ta tem en t

x G D, q ( x ,Xq ) < j im p l i e s q ( f ( x ) , f ( x Q) ) < r

f a i l s f o r i n f i n i t e l y many f G F . Thus we can p ic k a sequence

o f х^ G D ~ ( x q ) and d i s t i n c t £ F such th a t

236 GEHRING

q(x_. , x Q) < i- , q ( f j ( х^ ) , f j ( x 0 ) ) > r

f o r a l l j . Fo r each k l e t

Dk D { x 0 ' Xk ' xk + l ' -

I f k <_ j < “ , then

q ( C ( f j ( D k ) ) ) >_ q ( f ( х^ ) , f.. ( x Q) ) > r

w h i l e i f 1 £ j < k , then

q ( C ( f . ( D ) ) ) > min q ( f . ( x ) , f . ( x n ) ) = s > 0 Э K - ! < j < k 3 u

Hence th e mappings f . a r e e q u ic o n t in u o u s in D. by C o r o l l a r y 1D к

and t h e r e f o r e a t each p o i n t o f

D ~ U Q} = U D k

Theorem 2 . Suppose f j : D a r e K -q u a s ic o n fo r m a l

and t h a t f^ f p o in t w i s e in D . Then one o f th e f o l l o w i n g

i s t r u e :

a ) f i s a homeomorphism and co n v e rg en c e i s u n i fo rm on

compact s e t s ,

b ) f assumes o n ly two v a l u e s , one o n ly a t one p o in t ,

c ) f i s c o n s t a n t .

O u t l i n e o f p r o o f . Suppose f assumes t h r e e v a l u e s a t

x ^ , x 2 , x 3 and l e t

= D ~ { x 2 , x 3 J , D2 = D ~ { x 3 , x 1 } , D3 = D ~ { x l f x 2 }

S in c e f ( x 2 ) i f ( x 3) ,

i n f q ( C ( f . ( D . ) ) ) > i n f q ( f . ( x ) , f . ( x ) ) = r > 0 j "’ j D 3 1

and th e f ^ a r e e q u ic o n t in u o u s in . P e rm ut ing the r o l e s

o f shows the f^ a r e a l s o e q u ic o n t in u o u s in D2 ,

in and hence in D . Thus the co n v e rgen ce i s u n i fo rm on

compact s e t s in D and f i s c o n t in u o u s . We now show f i s

u n i v a l e n t i n D .

F i x x e D • By th e e q u i c o n t i n u i t y o f the f^ , we can

choose a c h o r d a l b a l l U a b o u t x such t h a t

B C D , q ( U i ) £ j whence q ( C ( U ! ) ) >_ j

w here UÍ = f ^ ( U ) . We c l a im f i s e i t h e r u n i v a l e n t o r c o n s ta n t

in U . O th e rw is e we can choose a , b , c £11 such th a t

f ( a ) ? f ( b ) = f (c )

L e t R be the i n d i c a t e d r i n g

and l e t R'. = f . ( R ) . Then3 D

mod R > 0 f_. I

On th e o t h e r hand, by th e second

e s t im a te in C o r o l l a r y 2 o f C h ap te r 2,

2 3 8 GEHRING

4 q ( b ! , c ! )mod r ! < mod R r ( q ( a . , b . ) q ( c ( u - ) )

q ( b ! , c ! )

" m° d Rt [ 8 q T i - T b j T ' ] * 0

where a! = f .fa), b'. = f . (b ) , c! = f. (c ) , since3 J 3 3 3 3

q ( a ^ , b ^ ) -> q ( f ( a ) , f (b ) ) > 0 , q ( b . ! , c . ! ) ■+ q ( f ( b ) , f ( c ) ) = 0


and we have a c o n t r a d i c t i o n .

Now l e t G ^ , G2 be the s e t s o f p o i n t s x e D w h ich have

a n e ig h b o rh o o d in w h ich f i s u n iv a l e n t o r c o n s t a n t , r e s p e c t i v e l y .

Then G ^ , G2 a r e open w i th

I f f i s no t u n i v a l e n t we can show t h a t G2 ^ Ф and hence

D = G2 . By c o n t in u i t y f i s then c o n s t a n t in D , y i e l d i n g

a c o n t r a d i c t i o n .

Theorem 3 . Suppose C (D ) c o n t a in s a t l e a s t two p o i n t s ,

p o in t w i s e in D . Then the con ve rgen ce i s u n ifo rm on compact

s e t s and

a ) f : D -»■ D ' i s K -q u a s ic o n fo r m a l ,o r

b ) f i s a c o n s t a n t c G 3D .

Homogeneous d om a in s . We say t h a t D i s К- homogeneous

i f f o r each a , b e D th e r e e x i s t s a K -q u a s ic o n fo r m a l f : D -*• D

such t h a t

a ) f ( a ) = b ,

b ) f i s hom otop ic t o the i d e n t i t y in D .

D = G^ U g 2 , g x 0 g 2 = ф

t h a t f . : D -*■ D 1 :

i s K -q u a s ic o n fo r m a l , and t h a t f^ -*■ f

We say D i s qua fe icon fo rm a l ly -h om ogen eous i f i t i s K-homogeneous

f o r some К .

Theorem 4 . I f D i s q u a s ic o n fo rm a l ly -h o m o g e n e o u s , then

e i t h e r

a ) С (D) i s c o n n e c te d ,o r

b ) С (D) c o n t a in s e x a c t l y two p o i n t s .

P r o o f . Assume o t h e r w i s e . Then t h e r e e x i s t d i s t i n c t

p o in t s a , b , c G. 3D such t h a t a and b do n o t l i e in the

same component o f C (D ) . Choose a compact

s e t E С D s e p a r a t i n g a and b

and l e t U be a n e ig h b o rh o o d o f

с such t h a t a and b can be U

j o i n e d in C (U ) .

Choose Cq , с^ G D so t h a t c^ с . By h y p o t h e s i s we

can choose f o r each j a K -q u a s ic o n fo r m a l f^ : D -»• D such

t h a t f j ( C g ) = Cj , f j hom otop ic t o the i d e n t i t y in D .

S in c e a , b , c ф D ,

q í C í f j í D ) ) ) = q ( С ( D ) ) > q ( a , b ) > 0

and th e f^ a r e e q u ic o n t in u o u s in D by C o r o l l a r y 2. Hence

we can choose a subsequen ce { j. } such t h a t f . ■* f u n i fo rm lyk 3k

on compact s e t s i n D . Then

f ( c ) = l i m f . ( c ) = l im c . = c e 3D к \ к -»■ " 3k

Thus f e с in D by Theorem 3, and by th e u n i fo rm con ve rgen ce

we can choose an i n t e g e r j so t h a t

f j ( E ) С U

240 GEHRING

The homotopy c o n d i t i o n on f^

im p l i e s t h a t f j (E ) s t i l l s e p a r a t e s

a and b . But t h i s c o n t r a d i c t s


th e way t h a t the n e ig h b o rh o o d U

was ch o sen . We c o n c lu d e t h a t e i t h e r C (D ) i s con n e c ted o r t h a t

C (D ) c o n t a in s e x a c t l y two p o i n t s .

C o r o l l a r y 4 . When n = 2 , D i s quasiconformally-homogeneous

i f and o n ly i f i t i s c o n fo r m a l l y e q u i v a l e n t t o one o f the

f o l l o w i n g dom ains :

R2 , R2 , R2 'v* { 0 } , В

5. ONE-QUASICONFORMAL MAPPINGS

Lemma 1. I f f : Rn -*■ Rn i s 1 - q u a s ic o n fo r m a l and i f

f ( 0 ) = 0 and f { “ ) = th en f I R i s l i n e a r .

P r o o f . Suppose t h a t x , y , z a r e f i n i t e , o r d e r e d p o in t s

on a l i n e L . We b e g in by show ing th a t

Fo r t h i s l e t R d en o te th e r i n g whose

complement c o n s i s t s o f the segm ents o f

L j o i n i n g x to у and z to “ .

Then R i s s i m i l a r t o th e T e ic h m ii l l e r

r i n g R„ l z -У|y-x|

and 0 I z - y l

ly_xl 1

mod R = mod R,'T |y -x

N ex t by th e e s t im a te in Theorem 4 o f

C h a p te r 2,

f (x) £(R)

£ (z )f ( y )

242 GEHRING

F i n a l l y , s i n c e f i s 1 -q u a s ic o n fo rm a l , mod f (R) = mod R ,

and hence

mod R_ y - x< mod R_ I f ( z ) - f ( y )

I f ( y ) - f ( X )

S in c e mod RT (t.) i s s t r i c t l y i n c r e a s i n g i n t ,

I z - y I < I f ( z ) - f (y ) I У- x I I f ( y ) - f ( x )

and hence

I f ( v ) - f ( x ) I < I f ( z ) - f ( y ) I У - х I I z - y I

I n t e r c h a n g in g th e r o l e s o f x and z r e v e r s e s t h i s i n e q u a l i t y ,

g i v i n g ( * ) .

We show n e x t t h a t f o r each x , y ç Rn ,

( * * ) f ( x + y ) = f ( x ) + f ( y )

F o r t h i s l e t L and z be a s i n d i c a t e d .

Then a p p l y in g i n e q u a l i t y ( * ) to th e

f o l l o w i n g t r i p l e s o f p o in t s ;

( x , ^ | £ , y ) , ( ^ ^ , y , z ) , ( x , y , z )

y i e l d s

2 lf ( x ) 1 = 2 |f ( y ) - £ ( ~ 2^ ) I = I f ( 2 ) - £ (y ) I = I £ (y ) - f ( x ) 1I x-y.| I x - y I ¡z -y ¡ ¡y- х I

o r s im p ly

| f ( * ± Z ) - f ( x ) | = |f ( y ) - f (£%L) I = i I f ( y ) —f ( x ) I

The t r i a n g l e i n e q u a l i t y im p l i e s th a t

f ( 2 ^ ) = § ( f (x ) + £ ( y ) }

T h is a p p l i e d to th e p o i n t s x+y and 0 g i v e s

f(2£? 1) = 2 ( f ( x + y ) + f ( 0 )) = \ f ( x + y )

and hence ( * * ) f o l l o w s .

F i n a l l y ( * * ) and th e f a c t t h a t f i s c o n t in u o u s im p ly th a t

f ( a x ) = a f ( x )

f o r a l l a € R1 and x £ Rn . Hence f i s l i n e a r .

Theorem 1 . A homeomorphism f : Rn -»• Rn i s 1 -q u a s ic o n fo rm a l

and o n ly i f i t i s a M ôbius t r a n s fo r m a t io n .

P r o o f . The s u f f i c i e n c y i s c l e a r by Theorem 5 o f C h ap te r 1.

For th e n e c e s s i t y sup pose f : Rn -*• Rn i s 1 -q u a s ic o n fo r m a l and

ch o o se a M ób ius t r a n s f o r m a t io n g w h ich maps f ( 0 ) , f ( «

o n to 0, e ^ , " . S e t h = g » ;f . Then h i s 1 - q u a s ic o n fo r m a l and

Lemma 1 im p l i e s t h a t h i s a l i n e a r b i j e c t . i o n . Hence by


244 GEHRING

Theorem 5 o f C h ap te r 3

= H (h ) = К (h ) < 1I (h ) 1 1

|h,n = H . ( h ) = K. (h) < 1

Hence

J (h ) “ O' ' 0

in f| h (x ) I = 5, (h) = Ih I = sup |h (x ) | x|= l |x|= l

and s in c e h ( e 1 ) = 1 ,

| h (x ) I = 1xI

f o r a l l x Ç. Rn . Thus h i s an o r t h o g o n a l mapping and f = g ~ ^ °h

i s a M obius t r a n s fo r m a t io n .

Remark 1 . Theorem 1 h o ld s f o r n > 2 . However when n > 2

we g e t a much s t r o n g e r r e s u l t u s in g th e f o l l o w i n g c l a s s i c a l

r e s u l t due to L i o u v i l l e .

Theorem 2 . Suppose th a t D ,D ’ a r e domains in Rn w here4

n > 2 . I f f : D - * - D ' i s a С homeomorphism and i f

I f ' ( x ) | n = j ( f ' ( x ) )

i n D, th en f = g|D w here g i s a Mobius t r a n s fo r m a t io n .

Theorem 3 . I f f : D •+■ D' i s 1 -q u a s ic o n fo rm a l w here n > 2,

then f = g |d w here g i s a M obius t r a n s fo r m a t io n .

B e f o r e i n d i c a t i n g a p r o o f o f Theorem 3, we need to make

some comments c o n c e rn in g th e e x t r e m a l f u n c t i o n f o r a r i n g domain.

Suppose t h a t R = R (C g ,C ^ ) i s a r i n g in Rn . Then

c a p (R ) = i n f /|vu|ndm adm(R) R

Suppose n e x t t h a t th e com plem entary components Cg and a r e

bo th n o n d e g e n e ra te , i . e . c o n t a in a t l e a s t two p o i n t s . Then u s in g

th e f a c t t h a t Ln (R) i s u n i fo rm ly convex and an o ld argument on

th e D i r i c h l e t p ro b lem due to L e b e s g u e , one can show t h e r e e x i s t s

a u n iq u e e x t re m a l u 6 adm(R) w i t h

c a p (R ) = /|.Vu|ndm R

T h is f u n c t i o n then s a t i s f i e s th e v a r i a t i o n a l c o n d i t i o n

/ ¡ Vu In 2 Vu* V w dm =0 R

f o r a l l w € C ^ (R ) w i t h compact s u p p o r t i n R , i . e . u i s a

weak s o l u t i o n o f th e e q u a t io n

, d i v ( I Vu In 2 Vu) = 0

When n = 2 , t h i s i s th e L a p l a c e e q u a t io n . When n > 2, th e equa

t i o n i s no l o n g e r l i n e a r , b u t r a t h e r q u a s i l i n e a r .

2When n = 2, W e y l ' s Lemma im p l i e s t h a t u i s С and hence

h arm on ic . T h is a l s o f o l l o w s from e x h i b i t i n g an e x p l i c i t

e x p r e s s i o n f o r u by means o f c o n fo rm a i m apping. When n > 2,

th e s i t u a t i o n i s more c o m p l ic a t e d . I f we know t h a t f o r each

compact s e t E C R t h e r e e x i s t s a c o n s t a n t M £ ( 0 , ° ° ) such th a t

( * * * } 1 i i- i - < |Vu(x) I < M


a . e . i n E , then we can a p p ly regularity theorem s due to de G io r g

246 GEHRING

M orrey and Hopf to co n c lu d e t h a t u i s r e a l a n a l y t i c i n R.

A r e c e n t r e s u l t o f U r a l t s e v a im p l i e s u £ C1 (R) w i t h o u t ( * * * )

E x am p le s . a ) Suppose t h a t R i s th e s p h e r i c a l r i n g

R = R (C q , C ^ ) , w here

CQ = В (x Q, a ) , G = С (B ( x Q , b ) )

and x Q £ R , 0 < a < b < ° ° . Then i t

i s e a s y t o v e r i f y t h a t

u ( x ) =l o g

x - x r

, x € R

l o g J

i s th e e x t re m a l f u n c t i o n f o r R . N o te th a t

IV u (x )l o g I

i n R , and hence u s a t i s f i e s ( * * * ) .

b ) Suppose n e x t t h a t n = 3 and t h a t R = R ( C q , C ^ ) i s th e

r i n g w here

CQ = S (a ) O R , Cx = C ( B ( b ) )

and 0 < a < b < oo . Then th e e x t re m a l

f u n c t i o n u f o r R i s by

U r a l t s e v a ' s r e s u l t and sym m etric i n

th e x i x 2 - P l ane an< ^ е х з - a x i s *

Hence v u (0 ) = ( 0 , 0 , 0 ) , |vu(0)| = 0 ,

and ( * * * ) d oe s n o t h o ld . By s l i g h t l y m o d i fy in g R we can f i n d

a r i n g R 1 w i t h e x t r e m a l f u n c t i o n u ' such t h a t R 1 i s homeo-

m orph ic t o a s p h e r i c a l r i n g and such t h a t j y u ' | v a n i s h e s a t

a p o i n t o f R ' .

Ske tch o f p r o o f o f Theorem 3


We may assume t h a t D , D 'C R . Then by Theorem 2 i t i s

s u f f i c i e n t to show t h a t each p o i n t x^ €T D has a n e igh b o rh o od

w h ich f i s r e a l a n a l y t i c . F i x Xq and choose b G (0 ,°° ) so

th e s p h e r i c a l r i n g

R' = í y : a < I y - f ( x Q)| < b }

Then as we saw a b o v e ,

u * ( y ) =l o g

| y - f ( x 0 )

l o g - 3 a

is equ a l in R ' to the e x tr e m a l

fo r R '. Set

u ( x ) = u ' 0 f (x ) , x Ç R

C \

248 GEHRING

u h as d i s t r i b u t i o n . d e r i v a t i v e s in R and boundary v a l u e s 0

on Cq and 1 on C^. B ecause f i s 1 - q u a s ic o n fo rm a l

cap R ’ = cap R < / | V u (x ) | n dmR

< / I Vu' ° f (x ) jn I f ■ (x ) |n dmR

= / I Vu' ° f (x ) I n J ( f 1 (x)|. dmR

= / | V u ' ( x ) ] n dm = cap R'R'

W e conclu de that u is equ a l in R to the unique e x tr e m a l function fo r R .

N e x t w i t h Theorem 1 in C h ap te r 4, i t i s n o t d i f f i c u l t t o show

t h a t u s a t i s f i e s the c o n d i t i o n ( * * * ) on each compact s e t

E С R. Thusi f ( x ) - f ( x n) I

l o g I------- --------5LIu ( x ) = u ' ° f (x ) ------------------ - ----- r----------

l o g - a

i s r e a l a n a l y t i c i n R , whence

| f ( x ) - f ( x 0 ) I = a ( ! ] u ( x ) = b u ( x ) a 1 - u ( x )

i s r e a l a n a l y t i c i n R . We th en l e t a 0 to co n c lu d e

| f ( x ) - f ( x g ) I r e a l a n a l y t i c i n U .

G iv e n a homeomorphism f : D -*• D' we s e t

H ( x , f ) = l im su p L ( x , f , r ) r -*■ 0 I ( x , f , r )

f o r each x £ D <v< { ° ° , f ■*■(“ ) } , w here f o r s m a l l r > 0

L ( x , f r ) = sup i f ( y ) — f (x ) JIx-y]=r

A ( x , f r ) = i n f I f ( y ) —f ( x )|x -y | = r

The f u n c t i o n H ( x , f ) m easures how much f d i s t o r t s th e shape o f

s m a l l s p h e re a t x and thus m easures how f a r f i s from b e in g

c o n fo rm a i a t x . We can ex ten d th e d e f i n i t i o n o f H ( x , f ) t o a l l

X € D by u s in g a u x i l i a r y M obius t r a n s fo r m a t io n s .

Theorem 4 . I f f : D -*-D' i s a homeomorphism, i f H ( x , f ) i s

bounded in D, and i f

_ L

( * ) H ( x , f ) < K n _ 1

а . e . i n D, then f i s a K -q u a s i c o n f o r m a l m apping.

I d e a o f p r o o f . A l e n g t h a r e a argum ent a l l o w s one to p ro v e

t h a t f has l o c a l l y Ln- i n t e g r a b l e d i s t r i b u t i o n d e r i v a t i v e s in

D and t h a t f i s d i f f e r e n t i a b l e a . e . in D. A t each such p o in t

o f d i f f e r e n t i a b i l i t y Xq , i n e q u a l i t y ( * ) im p l i e s th a t

H j i f ' ( x Q) ) < K, H0 ( f ' ( x Q) ) < К

and th e r e s u l t f o l l o w s from th e l a s t Remark in C h ap te r 3.

C o r o l l a r y . Suppose t h a t n > 2. I f f : D-*-D' i s a homeo

morphism and i f H ( x , f ) = 1 f o r a l l x £ D, then f = g| D

w here g i s a Mfibius t r a n s fo r m a t io n .

б. MAPPING PROBLEMS

Suppose t h a t D and D' a r e domains in Rn . How can

we d e c id e i f D and D' a r e q u a s i c o n f o r m a l l y e q u iv a l e n t ?

That i s , when d oe s t h e r e e x i s t a q u a s i c o n fo rm a l mapping

f : D -*■ D' ? The g e n e r a l p ro b lem i s e x t r e m e ly d i f f i c u l t even

when n = 2 . I f we r e s t r i c t o u r s e l v e s t o the c a se where D 1

i s th e u n i t b a l l В , we can g i v e some r e s u l t s .

250 GEHRING

Theorem 1 . Suppose t h a t n = 2 . Then D i s q u a s i c o n -

f o r m a l l y e q u i v a l e n t t o В i f and o n ly i f C (D ) i s a n o n d ege n e ra te

continuum.

P r o o f . I f C (D ) i s a n o n d ege n e ra te continuum , then

D i s a s im p ly conn ected domain in th e ex tend ed complex

p la n e and C (D ) c o n t a in s a t l e a s t two p o i n t s . Hence th e re

e x i s t s a c o n fo rm a i m apping f : D ■+ В by th e Riemann mapping

theorem .

I f t h e r e e x i s t s a K -q u a s ic o n fo r m a l f : D -*■ В , the f a c t

C (B ) i s con nected im p l i e s C (D ) i s co n n ec te d . Suppose C (D )

c o n s i s t s o f a s i n g l e p o i n t x^ , choose a e ( 0 , 1 ) and l e t

R = f ^ ( R ' ) , where

R' = { z : a < IzI < 1 }

N e x t l e t Г = A C C ^ C ^ R ) , l e t Г 1 =

d en o te th e f a m i ly o f a l l c u rv e s w h ich meet x0

Then

Г с r Q and

M (Г ) < M ( r Q) = 0

by Theorem 4 in C h a p te r 1. On the o th e r hand,

М (Г ' ) = 2тт ( l o g i ) - 1 > 0a

w h i l e th e f a c t t h a t f i s K -q u a s ic o n fo r m a l im p l i e s th a t

М (Г ' ) < КМ(Г)

Hence we have a c o n t r a d i c t i o n , and i t f o l l o w s t h a t C (D )

must c o n t a in a t l e a s t two p o i n t s .

Lemma 1. Suppose t h a t n > 3 , t h a t D i s an open b a l l

P r o o f . By p e r fo r m in g a p r e l im i n a r y M obius t r a n s fo r m a t io n

we may assume t h a t D i s a h a l f space and th a t e F .

L e t E* d en o te th e sym m etric ima—

N e x t l e t L be th e l i n e segment j o i n i n g a and b in 3D

Then F (1 L ф and we can choose a p o in t c e F f| L

S in ce

I с “ a I < I b — a I , a , b e cQ , c,°o e c i

Theorem 4 o f C h ap te r 2 im p l i e s th a t

N e x t i f Г ' = A (C q ,C ^ ;R ) , then i t i s n o t d i f f i c u l t t o show

o r h a l f space i n Rn , t h a t E с D and F с 3D a r e d i s j o i n t

c o n t in u a , and t h a t E j o i n s

a , b £ 3D w h i l e F s e p a r a t e s

w here Pn = f cap RT ( 1 )

a , b in 3D . I f Г = A (E , F ; D) ,

then

М (Г ) > u > 0 - Kn

o f E in 3D and R = R (C q ,C^ )

th e r i n g where

С1F

mod R < mod RT <1 )

252 GEHRING

where Г* i s th e sym m etric image o f Г in 3D , and hence t h a t

M (Г) = i М(Г') = i cap R > i cap R (1)

2heo£em_2_. Suppose t h a t n ^ 3 , t h a t Xq € Rn and t h a t

0 < a _ с ( l o g - )

where с i s a p o s i t i v e c o n s t a n t

w hich depends o n ly on n .

P r o o f . By an e le m e n ta ry l i m i t i n g argum ent, we may assume

t h a t f h as a homeomorphic e x t e n s io n w hich maps D on to В .

By h y p o t h e s i s t h e r e e x i s t

p o i n t s x 1 , x 2 S B ( x Q,a ) which

b e lo n g to d i f f e r e n t components

o f

С (D) П B ( x Q ,b )

T h is then im p l i e s t h e r e e x i s t s a

segment E w i t h e n d p o in t s У^/У^ e 3D such t h a t

E ^ { у ^ У з ^ c D ' E c B ( x Q,a )

and such t h a t У ^ 'У 2 n o t b e lo n g to th e same component o f

3D П B ( x Q,b )

S in c e 3D i s homeomorphic to ЭВ , t h e r e e x i s t s a continuum F

s e p a r a t i n g У ^ 'У 2 ■*-n such th a t

F С 3D П Ç ( B ( x Q, b ) ) .

L e t Г = Л (E , F , D) and l e t Гп be th e c u rv e f a m i ly a s s o c i a t e d

w i t h th e s p h e r i c a l r i n g Rg = { x : a < | x - Xg | < b } .

Then each у e. Г c o n t a in s a subcu rve e Гд and hence

м<г) < м (Гд ) = wn i } o g | ) 1_n

N e x t l e t E 1 , F ' , Г 1 , z £ , z 2 d en o te the im ages o f E , F , Г , У ^ »У 2

un der f . Then F ' s e p a r a t e s zi ' z 2 ■'■n and Lemma 1

im p l i e s t h a t М (Г ' ) yn -

F i n a l l y s in c e f i s K -q u a s ic o n fo r m a l ,

М(Г ' ) < КМ(Г)

and the. above i n e q u a l i t i e s y i e l d

К >_ с ( l o g ^ ) n _ 1

Unw here с = > 0 .

n - l

Theorem 3 . Suppose t h a t n > 3 , t h a t x Q e- Rn and th a t

0 < a c ( l o g | ) n _ 1

w here с i s th e

Theorem 1.

P r o o f . L e t g d eno te i n v e r s i o n in S ( X g , l ) and s e t

h = f o g , D ' = g (D ) .

Then h : D' В i s K -q u a s ic o n fo r m a l ,

C ( D ' ) П В ( x Q ,1 / a ) = g (C (D) f) C ( B ( x Q, a ) ) )

h as a t l e a s t two components w hich meet B ( X g , l / b ) = g ( C ( B ( X g , b ) ) ,

and hence Theorem 2 y i e l d s К с ^Log П ^ = с ( l o g n ^

Remark 2 . Though the lo w e r bounds f o r К in Theorems 2

and 3 a r e n o t s h a rp f o r any v a l u e s o f a and b , th ey a r e o f

the r i g h t o r d e r a s b/a •> °° .

_3Exam ples in R .

a ) The i n f i n i t e p l a t e domain

D = { x : x 2 + x^ < °°,|'X3 | < 1 }

i s n o t q u a s i c o n f o r m a l l y e q u i v a l e n t

t o В . F o r suppose t h e r e e x i s t s

a K -q u a s ic o n fo r m a l f : D -*■ В .

Then f o r each b e ( 1 , “ ) ,

С (D) Л B ( b ) h as two components

wh ich meet B ( l ) . Thus

К 21 c ( l o g b ) 2

and l e t t i n g b ■+■ °° y i e l d s a c o n t r a d i c t i o n ,

b ) The i n f i n i t e c y l i n d e r

D = { x I x 3 I < ” }

B (b )

i s q u a s i c o n f o r m a l l y e q u i v a l e n t to В . For

i f we l e t ( r , 0 , x 3) and (р ,9 ,ф ) d enote3

c y l i n d r i c a l and s p h e r i c a l c o o r d in a t e s in R , then th e f u n c t i o n

f ( r , e , x 3) = (р,е,ф) g iv e n by

p = e IT= J r

maps D q u a s i c o n f o r m a l l y on to th e h a l f spa ce D '

which i s i t s e l f c o n fo r m a l l y e q u i v a l e n t t o D .

c ) I f у i s an a r c , then th e s l i t domain

{ x •: x 3 > 0 }

D = C ( y )

i s no t q u a s i c o n f o r m a l l y e q u i v a l e n t t o В . Fo r suppose th e re

e x i s t s a K -q u a s ic o n fo r m a l f : D -*• В L e t Xq be an i n t e r i o r

256 GEHRING

p o i n t o f y f and ch oose b e ( 0 , " )

so t h a t b o th e n d p o in t s o f у l i e in

C ( B ( X g , b ) ) . Then f o r each a e ( 0 , b )

C (D ) П C ( B ( X g , a ) ) has two components

w h ich meet С ( В ( Х д , Ь ) )

К > c ( l o g I ) 2

Hence

by Theorem 3, and l e t t i n g a -*■ 0 y i e l d s a c o n t r a d i c t i o n ,

d) I f a i s a h a l f p l a n e , the s l i t domain

D = С (с )

can be f o l d e d q u a s i c o n f o r m a l l y on to

a h a l f space D' , and hence D i s

q u a s i c o n f o r m a l l y e q u i v a l e n t t o В .

7. AN EXISTENCE THEOREM

We say t h a t a domain D С R11 i s a Jordan domain i f 3D

i s homeomorphic to th e u n i t sp h e re S .

E xam p le . There e x i s t com plem entary Jordan domains and

in .R 3 such th a t

a )D 1

i s n o t q u a s i c o n f o r m a l l y e q u i v a l e n t to

b )D 2

i s q u a s i c o n f o r m a l l y e q u i v a l e n t to В

c ) 3D. = 3D, .

P r o o f . L e t be the s e m i - i n f i n i t e p l a t e domain

and l e t D2 = С ( ) . Then i t i s e a s y

t o s e e t h a t 3D^ = 3D., i s h om eom orph ic

t o S .

D.'2

Now su p p o se t h e r e e x i s t s a K -

q u a s ic o n fo r m a l f : ->• В and f o r

e a c h b e (1 ,° ° ) l e t Xg = b e .

Then С (D) П B (X g ,b ) has tw o

com p on en ts w h ich m ee t B (x Q , l ) ,

and h e n ce

К > с ( lo g b )2

b y Th eorem 2 o f C h a p te r 6 . L e t t i n g b ■* “ y i e l d s a c o n t r a d i c t i o n

and h en ce i s n o t q u a s ic o n fo r m a l ly e q u i v a l e n t t o В .

To s e e t h a t D2 i s q u a s i c o n fo r m a l l y e q u i v a l e n t t o В

l e t H ^, H2 b e th e h a l f s p a c e s

l e t H £, H2 b e th e q u a r t e r spaces

H| = { x : x^ < 0 , x 2 > 1 } , H2 = { x : X ! < 0 , x 2 < - 1 }

and l e t

H j = { x : X j < 0 , | x 2 | £ 1 } » D3 = { x : Xj < 0 }

F o r i = 1 ,2 t h e r e e x i s t s a q u a s ic o n fo r m a l f o l d i n g f ^ : H^ -*■ H^

Нд = { x : x 2 > 1 } , H2 = { x : x 2 < - 1 }

w it h

f ¿ I H± 0 H3 = i d e n t i t y

258 GEHRING

Then D2 = Hx Ü H2 ü H3 , D3 = (j U H3 and

f ¿ (x ) x e H± i = 1 , 2

f ( x ) =

x e H.

d e f i n e s a q u a s i c o n fo r m a l mapping o f D , on to D, . S in ce D¿ 3 3

i s c o n fo r m a l l y e q u i v a l e n t to В , D_ i s q u a s i c o n fo r m a l l y

e q u i v a l e n t to в .

:г

V V W O O O X0 * *

'/////,F IG .4. C on s tru ction o f the m app ing o f D 2 o n to О ъ.

Remark 1. When n = 2 , we see from Theorem 1 o f C h ap te r 6

t h a t a domain D i s q u a s i c o n fo r m a l l y e q u i v a l e n t t o В i f and o n ly

i f 3D i s a n o n d e ge n e ra te continuum . The example in Fig. 4 shows

t h a t when n > 2 , i t i s no l o n g e r p o s s i b l e to c h a r a c t e r i z e the

domains D w h ich a r e q u a s i c o n fo r m a l l y e q u i v a l e n t to В by lo o k in g

o n ly a t t h e i r b o u n d a r ie s 3D . We s h a l l p r o v e , h ow ever , t h a t i f

th e p a r t o r D n e a r 3D i s

q u a s i c o n f o r m a l l y e q u i v a l e n t

t o th e p a r t o f В n e a r ЭВ ,

then D i s q u a s i c o n fo r m a l l y

e q u i v a l e n t t o В . The p r o o f o f

t h i s i s q u i t e l o n g and r a t h e r i n v o l v e d . We b e g in w i t h a r e s u l t

on Jordan domains from which th e g e n e r a l theorem r e a d i l y f o l l o w s .

Theorem 1. Suppose th a t , D2 a r e Jordan domains in

w i th D1 f) D2 = ф and th a t , B2 a r e open b a l l s i n R

w i t h о B2 = Suppose a l s o

t h a t f : С ( 0 Х U D2 ) -► С (В Х U B2>

i s a homeomorphism such th a t

f j C (D ^ U D ) i s q u a s ic o n fo rm a l

and f (3 D ^ ) = 3B^ f o r i = 1 ,2 .

Then t h e r e e x i s t s a homeomorphism

f * : C (D 2 ) -*■ C (B 2 ) such t h a t f * | C(D.,) i s q u a s ic o n fo rm a l and

f '* I 3D2 = f I 3D2

Remark 2. Theorem 1 say s t h a t th e boundary co r re sp o n d en c e

f I 3D2 : 3D2 -> 3B2

h as a q u a s ic o n fo rm a l e x t e n s io n

f* : C (D 2 ) - C (B 2 )

The p r o o f i s somewhat t r i c k y . We c o n s id e r f i r s t two s p e c i a l c a s e s .

L e n m a l . Theorem 1 h o ld s under the f o l l o w i n g a d d i t i o n a l

h y p o th e s e s :

a ) , D2 , B^, B2 l i e i n th e u n i t b a l l В ,

b ) f ( x ) = x in a n e ig h b o rh o o d o f C (B ) .

P r o o f . By p e r fo r m in g a p r e l im i n a r y o r t h o g o n a l m apping, we

may assume t h e r e e x i s t numbers - 1 < a b }l ¿ n

260 GEHRING

H e re we h a ve made e s s e n t i a l u se o f th e

g e o m e t r i c a l c h a r a c t e r o f and

B2 , e . g . t h a t t h e y a r e n o t l i n k e d in

Rn a n d , in f a c t , can b e s e p a r a t e d b y an

(n - 1 ) - d im e n s io n a l h y p e r p la n e .

N e x t l e t

E = U (D . U D . ) + 3 je .j=o z 1

00E ' = U (B .U B , ) + 3j e .

j= 0 1 z 1

CO

F = (J В + 3j e , j=0

Then i t i s c l e a r t h a t E , E ’ C F

and t h a t

Now s e t

g ( x ) =

F % E = и (В л, (D , U D ,) ) + 3 j e . j = 0 ^

F ^ E ’ = U (B ^ (B , U B - ) + 3 j e . j = 0

f ( x - 3 je ^ ) + 3 je ^ i f x G (в °» (D^ U D2 ) ) + 3 je ^

x i f x € С (F )

Then g i s a hom eom orph ism o f C (E ) o n to C (E ') . M o r e o v e r ,

e a c h x G C (E ) h a s a n e ig h b o r h o o d U su ch t h a t g | U i s e i t h e r

th e i d e n t i t y o r th e c o m p o s i t io n o f f w i t h a t m os t tw o

t r a n s l a t i o n s . H en ce g : С (Ё ) -+ С (Ё ' ) i s a l s o q u a s ic o n fo r m a l .

S e t r = Ц р - > 0 and d e f i n e к : R^ ■+ a s f o l l o w s :

k ( t ) =

0

a + r - tr

-1

i f - “ < t £ a + r

i f a + r < t £ b - r

i f b - r < t < °°

N e x t d e f i n e h : Rn -+ Rn by

h (x )

x + 3 k (x ) e , i f x G R t l ' 1

i f x =

Then h i s a p i e c e w i s e l i n e a r , and hence q u a s i c o n fo r m a l , mapping

o f Rn o n to Rn (see F ig . 5).

b - r

F IG .5. M a p p in g h in Lem m a I .

F i n a l l y l e t

g ^ ° h ° g (x ) + Зе^

x + Зе.f * (x )

i f x 6 R ^ E

i f x G Dx + 3 ^ , j > 0

i f x G D2 + 3^ех , j i 1

i f x = 00

We s h a l l show t h a t f * i s the d e s i r e d m apping. I t i s c l e a r

from Fig . 6 that f is a univalent mapping of C fDg ) onto CfB^).

262 GEHRING

F IG . 6. C on s tru ction o f the m app ing f * in L em m a 1.

We a l s o see t h a t f i s c o n t in u o u s e x c e p t p o s s i b l y a t p o i n t s o f

H = [ U Э0 Х + 3je ^ j у I U^3D 2 + 3 j e x J и { » }

Choose a p o in t x Q £. + Sje^^ , where j ^ 0 . Then x Q has

a n e ig h b o rh o o d U such th a t

g (U 'v E) С (В + 3 ) П { x : xr < a + r }

and i t f o l l o w s t h a t g ^ o h о g ( x ) = x , f * ( x ) = x + 3e^

f o r x G 0 ^ E . We co n c lu d e th a t

( * ) f * (x ) = x + 3e^

f o r x G U . S i m i l a r l y i f Xq € 3D2 + 3 je ^ w here j 1 ,

then Xq has a n e ig h b o rh o o d U such that

g (U ^ E) С (В + 3 j e x ) n { x : xn > b - r }

and we have g ^ о h ° g ( x ) = x - 3e^ , f * ( x ) = x

f o r x e U ^ E . Thus

( * * ) f * ( x ) = x

f o r x e U . F i n a l l y i t i s e a s y to v e r i f y th a t

I f * (x ) - x I < 5

f o r a l l x €. Rn . From th e above i t f o l l o w s t h a t f * i s

c o n t in u o u s a t e=<ch p o in t o f H .

Now l e t x С 3D2 • Then

f * ( x ) = g ^ o h о g ( x ) + 3e^

= g - 1 ( f ( x ) - З е ^ + 3e^

= ( f (x ) - 3e1 ) + 3e1 = f (x )

and th us f * I 3D2 = f | 3D2 . F i n a l l y s in c e h is q u a s i c o n f o r m a l ,

i t i s n o t d i f f i c u l t to see from ( * ) and ( * * ) t h a t f * maps

C (D 2 ) q u a s i c o n f o r m a l l y onto C (B 2 ) . T h is com p le te s th e p r o o f

o f Lemma 1.

Lemma 2. Theorem 1 h o ld s un der th e f o l l o w i n g a d d i t i o n a l

h y p o th e s e s :

a ) , D2 , B jy B2 l i e in th e u n i t b a l l В ,

b ) 0 G D2 ' ,

c ) С (В) С f (С (В) ) .

264 GEHRING

P r o o f . By h y p o t h e s is we can c h o o s e 0 < a < b < l s o t h a t

В (a ) С D2 , Dx U D2 С B (b )

N e x t d e f i n e g : R1* -*■ Î?1 by

g ( x ) = jI x I C i f x €. В

i f x e с ( в )

w h e re с = j-¿ g ¿ • Th en q i s q u a s ic o n fo r m a l ,

g ( B (a ) ) = B (b )

and h e n ce

g ( C (D 2 ) ) С C (D 1 U d 2 ) •

We c o n c lu d e t h a t f o g i s a hom eom orph ism o f C (D 2 ) i n t o

U b 2 ) .

Now l e t

Di = f ° д ( 0 х ) , D2 = C ( f о g ( C ( D 2 ) ) )

Then D J, D'2 a r e J o rd a n d om a in s w i t h

Difl °'2 = Ф r

U С c ( f - g ( С (В ) ) ) С С ( f (С (В ) ) ) СГ В .

N e x t l e t U = f ( C ( B ) ) , h = f ° ( f о g ) _1 .


F IG . 7. C on s tru c tion o f the m apping f * in L em m a 2.

Then U i s a n e igh b o rh o od o f C ( B ) , h i s a homeomorphism o f

CCD^ U D^) on to С (B^ U By), h i s q u a s i c o n fo rm a l in

C (D| U D^) and

h ( 8D!_) = f (Эо± ) = ЭВ.

f o r i = 1 , 2 (see Fig. 7).

A l s o i f x € U, th en f - 1 (x ) e C (B ) and

h ( x ) = f o g - 1 o f ” 1 ( x ) = f o f - 1 (x ) = x

266 GEHRING

Hence by Lemma 1 we g e t a homeomorphism h * : C (D 2) ■+■ C (B 2) such

t h a t h* I C (D ^ ) i s q u a s ic o n fo rm a l and

Now s e t

h* j 3D¿ = h I 3D2

f * = h * o ( f o g )

Then h : C (D 2) С (B2) i s a homeomorphism, f * | C (D 2 ) i s q u a s i

c o n fo rm a l , and

f * I 3d2 = h* I Ъи'2 = h I 3D¿ = f I 3D2

T h is co m p le te s th e p r o o f o f Lemma 2.

P r o o f o f Theorem 1 . O b v io u s ly <*> S C ( B ^ ( J B2 ) . By p e r fo rm in g

a p r e l im i n a r y M ob ius t r a n s fo r m a t io n we may assume th a t

œ = f - 1 (oo) g С ( D j ^ U D 2 )

Then t h e r e e x i s t s Xg G D2 and 0 < a < b < such th a t

d 1 U d2 c b ( x q , a ) , вх и в2 С В ( x Q, b)

f ( C ( B ( x Q, a ) ) ) D C ( B ( x Q, b) )

L e t g ^ , g 2 b e th e s i m i l a r i t y m appings wh ich c a r r y B (X g , a ) ,

B ( x Q, b ) o n to В and s e t


Then h s a t i s f i e s th e hyp oth eses

o f Lemma 2 and t h e r e e x i s t s a*

homeomorphism h : C (D 2 ) С (B^

w h ich i s q u a s i c o n fo rm a l in

C iD ^ ) w i t h

w here DJ = and BJ = g 2 ( B ^ ) . Hence

* *f = g 2 o h о g j

i s th e d e s i r e d e x t e n s io n .

Theorem 2. Suppose t h a t D i s a domain in R , t h a t U i s

a n e igh b o rh o o d o f 3D , and t h a t f i s a q u a s i c o n fo rm a l mapping

o f D o U i n t o В such th a t

l im I f ( y ) y-*-x

f o r each x G 3d . Then t h e r e e x i s t s a n e igh b o rh o od U o f 3D

and a q u a s i c o n fo rm a l mapping f : D -*■ В such t h a t

f * j D П U* = f I D H U *

268 GEHRING

I d e a o f p r o o f . An e le m e n ta ry t o p o l o g i c a l argument and two

p r e l im i n a r y M obius t r a n s fo r m a t io n s a l l o w one to deduce t h i s

r e s u l t from Theorem 1.

IAEA -SM R -18/102

GEOMETRIC THEORY OF DIFFERENTIAL EQUATIONS IN THE COMPLEX DOMAIN

R. GERARD Centre d’équations différentielles, Institut de recherche mathématique avancée, Département de mathématique, Université de Strasbourg, Strasbourg, France

Abstract

G E O M E T R IC T H E O R Y O F D IF F E R E N T IA L E Q U A T IO N S IN T H E C O M P L E X D O M A IN .

I t is w e ll know n h ow useful the n o tion o f fo lia tion is in the study o f d ifferen tia l equations in the real case

but it is n o t an indispensable to o l, although fa ir ly useful. Th is paper shows h ow the n o tion o f fo lia tion can be

used in the th eory o f com p lex d ifferen tia l equations and com p lex p fa ffian equations and h ow the analytic behaviour

o f solu tions can be derived from the properties o f the fo lia tions. T h e use o f fo lia tion s allows us to give geom etric

in terpretations o f som e classical results on d iffe ren tia l equations in the com p lex dom ain , to generalize these

results to o th er situations, and to solve n ew prob lem s in the theory . Th e paper is d ivided in to three parts:

I. C om p lex analytic fo lia tion s on a com p lex analytic m an ifo ld : D efin itions , elem entary properties; Exam ples;

App lica tions. I I . Pa in levé ’s fo lia tion s on a fib ra tion : D efin itions, exam ples; L o ca l properties; Th e main theorem

on Pa in levé ’s fo lia t ion ; A pp lica tions . I I I . A lgeb ra ic p fa ffian equations on a p ro jective space; Non-ex istence o f

com pact leaves; Th e structure o f the set o f algebraic solutions.

Part I

COMPLEX ANALYTIC FOLIATIONS

The fo llow ing d e fin ition s can be found in the rea l case in [1 ] and [2 ] .

1. FOLIATED STRUCTURE ON A COMPLEX MANIFOLD

a) The t r iv ia l fo lia te d structure o f codimension p on СП •

Look at the a ffin e space СП as the product X СП_ and le t

x = (x 1,x2, . . . , x p) the coordinates in CP and y = (y 1,y2, . . . ,Уп_р) the

coordinates in СП • Then the simplest fo l ia t io n in codimension p o f Cn is

the fo l ia t io n in which the leaves are the planes x = Cte . Denote by F th isPi

fo l ia t io n . A lo ca l automorphism o f is by d e fin it io n a lo ca l isomorphism o f

СП preserving lo c a lly the leaves; that means that in the neighbourhood o f each

269

point on which the isomorphism is w ell defined i t can be w ritten in the fo llow ing

form :í X = h1(x )

[ Y = h2(x ,y )

where h is a lo ca l isomorphism o f CP •

I f U is an open subset o f СП , a "plaque" in U is a connected component o f

the trace o f a le a f on U .

270 GERARD

b) Foliated structure on a complex analytic manifold

A fo lia te d structure o f codimension p on a complex manifold o f

dimension n with countable basis is a maximal atlas С = (U . , f . ) . _ „ such that4 i l ' i Ç I

fo r a l l ( i i j ) , h „ is a lo ca l automorphism fo r the fo lia t io n F on СП •

A "plaque" in IL fo r th is fo lia te d structure is by d e fin it ion the inverse

image by f^ o f a "plaque" in ^or the t r iv ia l fo lia te d structure o f

codimension p on СП •

An open subset U o f is ca lled distinguished i f i t is isomorphic to a

product o f two disks P * PpC CP such that the inverse images o f the

"plaques" in P X P are the "plaques" in U . n-p p

I t is easy to see that :

1) a l l distinguished open subsets o f containing a point m form

a fundamental system o f neighbourhoods o f the point m ;

2) fo r a l l ( i , j ) , h^j induces an isomorphism from each "plaque" in

f. (U . П и .) onto a "p laqu e"'o f f . (u .n u . ) ; i 1 3 J i 3

3) the in tersection o f a f in i t e number o f "plaques" is a union o f

"plaques".

DIFFERENTIAL EQUATIONS 271

The la s t property enableeus to define a new complex analytic structure on ;

the open subsets fo r -th is new structure are the "plaques". I t is c lea r that this

new structure is fin e r than the old one; with th is structure V is a complexn

analytic manifold o f dimension п-p .

A le a f fo r the fo lia te d structure o f codimension p on V is by d e fin it io n a ------ n J

connected component o f fo r the fin e structure and with th is structure a

le a f is an analytic submanifold o f dimension n-p o f .

A le a f is ca lled proper i f the topology on the le a f defined by the "plaques" is

the same as the topology induced by the natural topology o f the manifold .

Let 3" be an analytic fo l ia t io n o f codimension p on the complex

analytic manifold and G = » f--} j g j the atlas defin ing th is fo l ia t io n .

A map g : UCV^-C13 ( U open subset) is ca lled distinguished i f i t is lo c a lly

n Po f th e form g = rro f^ where tt i s th e c a n o n i c a l p r o j e c t i o n o f С on to С .

The "plaques" in ^ ^ ( fo r a l l i € I )

are the inverse images o f the points

o f Cp .

I f . f and g are two distinguished maps having a common part in the ir range,

there ex ists a lo ca l isomorphism h o f CP such that f = ho g .

Now i t is easy to see that a fo l ia t io n 3 o f codimension p on is also

given by a fam ily ç j that :

1) ( v . } . , _ is a covering o f V ;i i t J n

2) f . : V. -C P are submersions;i i

3) the fam ily is complete in the fo llow ing sense: i f

f : U -C P ( U open) is a submersion then fo r a l l i such that

UflU. i IÍ and fo r a l l x eU flU . there ex ists a loca l isomorphism hi i

o f CP such that f = ho f^ in the neighbourhood o f x .

This d e fin it io n o f fo lia t io n s is the natural one aris ing in the study o f

d if fe r e n t ia l equations in the complex domain.

Now le t us re ca ll some elementary properties o f the leaves which w il l

be used la te r :

1) Continuity theorem:

I f m and m' are two points

belonging to ;the same le a f then fo r each

distinguished neighbourhood U' o f m'

there ex is ts a distinguished neighbourhooc

U o f m such that each le a f meeting U goes through U' .

2) The equivalence re la tion defined by the leaves is open.

3) The closure in V o f a union o f leaves is a union o f leaves.' n

The leaves' space is the quotient space o f by the equivalence

re la tion associated to the fo l ia t io n . In :general th is space is very

complicated and exceptionally Hausdorff.

In the study o f th is space, the fo llow ing resu lt is known :

THEOREM. I f F is an open equivalence re la tion on a reduced analytic space X

then Х/R is an analytic ( reduced) space i f and only i f the graph o f R in

XXX is an analytic s e t .

For the proof and other resu lts , see [3 ] and [4 ] .

2. EXAMPLES

2.1.F o lia tion defined by a holomorphic d if fe r e n t ia l system

Let ^ = F (x ,y ) be a holomorphic d if fe r e n t ia l system on С X СП; that

n nmeans that F is a holomorphic map from С x С into С

I f (x ° ,y ° ) is a point o f С X Cn then the Cauchy theorem states that there

ex is ts a unique solution у = .cp(x,y°,x°) which takes the value y ° at the point

x ° . And we have also y ° = cp(x°,y,x) ; in other words there exists a disk U

272 GERARD

centered at y ° such that the loca l solution are the inverse images o f the points

o f СП by the mapU X V - €

(x ,y ) - cp(x°,y,x)

The c o lle c t io n (UXV.cp) gives us the distinguished maps fo r the fo lia t io n

defined by the given d if fe r e n t ia l system.

We can also look at th is fo lia t io n in the fo llow ing way : take a point (x ° ,y ° )

in С X Cn and the lo ca l solution at th is point, у = i|/(x,y°,x°) the analytic

continuation o f ф, is a solution o f our

system and the graph o f th is analytic

continuation in С X c” gives us a le a f

o f the associated fo l ia t io n .

This introduction to fo l ia t io n can be found in Pa in levé 's lectures at Stockholm

( 1894) ( see [5 ])and in th is sense Painlevé was the f i r s t fo l ia te r (G. Reeb d ix i t ! ) !

?.2.T ra jectories o f a holomorphic vector f ie ld on IPg(c)

I t is easy to see that a holomorphic vector f ie ld in the complex

projectile plane IPg(c) is induced by a lin ea r vector f ie ld in € . A generic

vector f ie ld on JPg(c) has three singular points a ,b ,c . By the choice o f the

coordinates we may assume that a = ( l ,0 ,0 ) ,b = (0 ,1 ,0 ) , с = (0 ,0 , l ) . This

vector f ie ld defines in IP (c) - fa ,b ,c } a fo l ia t io n . The leaves o f th is fo lia t io n

are tra je c to r ie s o f the group С and the only one parameters connected groups

are : С , cy linder, torus, poin t. In our case i t is easy to give the e x p lic it

parametrization o f the tra je c to r ie s :

tx = x eо

\ tУ = У0 e

in some coordinate neighbourhood.


Now i t is c lea r that no le a f is a torus. This admits the fo llow ing generalization

as we shall see la te r :

I£ ш = 0 is a completely integrable P fa ffia n equation on IP (c ) ,

S the set o f s in gu la rities o f щ then the fo lia t io n defined by ш = 0 in

Pn(c ) - S has no compact l e a f .

Now le t us g ive the description o f the fo lia t io n given by a generic holomorphic

vector f ie ld on 3Pg(c) .

•|) I f X is a ra tion a l number then a ll

leaves are cylinders whose end B(y) = Y - Y

(y is a le a f ) is the union o f two singular

points.

274 GERARD

2) I f X is rea l but not ra tiona l there

are three particu lar leaves which are

cylinders whose end is composed o f two singu

la r points. The other leaves are complex

planes; the end o f such a le a f contains a

three-dimensional manifold (T^ x3R) and

two singular points ( fo r d eta ils see [6 ] ) .

3) I f X is complex but not rea l there are

three particu lar leaves which are cylinders

ending at two singular points; the other

leaves are proper complex planes.

2.3.Foliation defined by a completely integrable p fa ffia n system on a complex

analytic manifold

For more de ta ils look at [7 ] (Frobenius theorem).

A fo lia t io n 3> o f dimension p on a complex analytic manifold

(n a p ) gives us an integrable subbundle o f rank p o f the holomorphic tangent

bundle o f Vn and conversely an integrable holomorphic subbundle o f the holo

morphic tangent bundle gives an analytic fo lia t io n on the complex analytic mani

fo ld V . n

4 ) Other examples are given in the fo llow ing paragraph.

3. APPLICATIONS

З-1-The holomorphic R icca ti equation

A holomorphic R icca ti equation is an equation o f the fo llow ing

type:

( 1 ) y ' = a (x )y2 + b (x )y + c (x )

where a,b ,c are en tire functions.

This equation can also be written in the fo llow ing form :

( 2 ) dy - (a (x )y 2 + b (x )y + c (x ) ) dx = 0

2Equations ( 1 ) and ( 2 ) define the same fo lia t io n in С

Now we extend equation ( 2 ) to an equation on ]P^(c) x С , i f (y ,z ) are homo

genous coordinates on ÍP^(c) this extended equation is

( 3 ) z dy - у dz - (a (x )y 2 + b (x )y z ) + c (x ) z2) dx = 0

The manifold ÜP^C) xC is covered by two coordinate neighbourhoods :

U1 : (y ,x ) (z Ф O)

U2 : (z ,x ) (y 1 O)

Equation ( 3 ) is also given by two equations :

d y - (a (x )y 2 + b (x )y + c (x ))d x = О in U1

-dz - (a (x ) + b (x )z + c (x )z 2)dx= 0 in


Now i t is c lea r that each le a f o f the fo lia t io n defined by the equation (з ) is

transversal to the fib e rs x = C.'te . This means that our fo lia t io n is transversal

to the fib ra tion ]P (с) X С -* С which has compact f ib e rs , then by a theorem o f

Ehresmann [8 ] each le a f is a.covering o f the basis. I t fo llows that each le a f is

isomorphic to the basis С by the canonical p ro jection . This resu lt implies

that every solution o f a holomorphic R iccati equation is a uniform function.

This is a well Toiown c la ss ica l resu lt but everywhere proved by analytica l methods.

P f x y)3.2.D iffe ren tia l equations o f the form y ' =

Let us suppose that P and 0 are polynomials in у with analytic

c o e ffic ien ts and moreover that P and Q are r e la t iv e ly prime.

We re ca ll that fo r such an equation there are two types o f s in gu la rities fo r a

solution (a pole is not considered as a s in gu la r ity ).

1) Fixed s in gu la rities ; These s in gu la r ities can be determined ex p li

c i t l y on the equation. They are :

a) the point Ç£C such that there exists an 1] sa tis fy in g

p (f,T l) = 0 and 0(?,Ц ) = 0 .

The points (§,T)) are s in gu la r ities o f the vector f ie ld defined

2by the equation in С .

b) the points I such that Q (f ,y ) = 0 id en tica lly in у .

2) Movable s in gu la r ities : These s in gu la rities depend on the choice

o f the solution and cannot be given exp lic it| ly ; before knowing exp lic itly .' the

considered solu tion .

Now write the given equation in the fo llow ing form :

W = 0 (x ,y ) dy - P (x ,y ) dx = 0

and extend th is equation to an equation ш = 0 on IP^(CxC) ,

276 GERARD

Denote by E the set o f the s in gu la rities o f щ . Then ш = 0

defines a fo l ia t io n on IP (c) x C -E • This fo lia t io n may have v e r t ic a l leaves (fo r

the pro jection onto C). Theseleaves are parts o f the fib ers x = | fo r which

Q(§ i у ) = 0 id en tic a lly in у . On such a f ib e r there is always a singular

point o f щ .

Let S be the subset o f С made up by the points | which are the projection

o f the points o f £ . The set H has only iso lated points. Denote by 3 the

subset o f С which is the union o f the points o f С such that the f ib e r over

such a point contains a le a f o f the fo l ia t io n associated to the equation ш=0 •

The point o f H U H are the fix ed s in gu la r ities o f our equation in the sense

given before. We shall see la te r that:

a) a point o f H is at most a transcendental singular point fo r

the solution and never an essen tia l singular point ;

b) a point o f H can be an essen tia l singular point fo r a solution.

A movable s in gu la rity is a point at which the le a f has a v e r t ic a l tangent. As

we shall see la ter, such a s in gu la rity is at most a lgebra ic.

Now le t us determine the equations o f the above form without movable singu lari

t ie s . This means that fo r such an equation the associated fo lia t io n in

3P^(C) xC has no le a f with v e r t ic a l tangent plane ; and th is implies that Q

has to be independent o f у and the equation has the fo llow ing form :

ÉZ = p(x ,y ) dx Q(x)

The same thing has to be true at in f in it y . Near in f in it y in у the equation

has the fo llow ing form (change o f variables z = ^ )

dz P (x ,z ) . . _ 4- dx = ~n-2 , s n = de3ree o f P (in y)

dX ZR Q ( x )


To have no v e r t ic a l tangent fo r the leaves i t is necessary that n -2 s 0

A ll this means that our equation has to be a R iccati equation o f the fo llow ing

form :

2v . _ a (x )y + b (x )y + c (x ) y d (x )

where a ,b ,c ,d are holomorphic functions.

I t is easy to see that such aR icca ti equation has no movable singular points.

&3.Uniformity o f the e l l ip t i c functions

The c la ss ica l e l l ip t ic functions are solutions o f the d if fe r e n t ia l equation :

У’ 2 = (1 - y2) ( l - k 2y2)

We are going to prove in a geometric way the uniform ity o f the solutions o f the

e l l ip t i c equation. F irs t we are try ing to do the same as fo r the R iccati

equation. Look at the equation dy - zdx on the surface with equation

2 2 \ 2 2 3z = ( " l - у ) . ( l - k y ) in С , now extend the s ituation to 3P2(C ) (y , z , t ) xC (x)

intotdy - ydt - ztdx = 0

on

2 2 / 2 2 w 2 ? 2 чt z = ( t - у ) ( t - k y )

The f i r s t d i f f ic u lt y aris ing now is the existence o f s in gu la rities fo r our

p fa ffia n equation on the surface. To c la r i fy the situation we make another

com pactification.

Look at the fo llow ing diagram :

F tO p ^C )) X C(x) - tO p^C )) x C ( x ) -1 > 1(с ) X C(x) - C(x)

where :

IP^C ) x C(x) is obtained from C'(y) x C(x) by com pactification o f

the y -p lane;

T&P^C)) is the holomorphic tangent bundle to IP^ (c );

ИРТфР^с)) is the projective tangent bundle to IP^(c) .

278 GERARD

The la s t bundle is covered by the fo llow ing coordination neighbourhoods :

0-, : (X>y ’s) ’ S = to

02 : (x ,Y ,s ) , yY = 1

03 : ( x ,Y ,s ) , S =

04 : (x ,Y ,T ) , TS = 1

Our surface is given in IPT IP (С )) X С by the equations

s2 = (1 - y2)(1 - k2y2) in 0

and

S2 = (Y2 - l ) ( Y 2 - k 2) in 03

and is contained in 0 U0 . The equation on th is surface is now given by

dy - sdx = 0 in 0 and dY - Sdx = 0 in 0

Now we see that there are no s in gu la rities fo r the equation on our surface but

the fo lia t io n defined by our equation is not transversal to the fib ra tion . Put

f ( y . s ) = - s2 + (1 - y2) (1 - k2 y2)

F(Y,S) = - S 2 + (Y2 - i H y2 - k2)

and look at(dy - s d x )Л d f = s щ

(d Y - Sdx) A dF = S ш2

^ 2 2 \ 2 2(U1 = - 2dy A ds + 2sdx A ds + 2 y [( 1 - к у ) + k ( 1 - y )]d xA d y

Û) = - 2dY A dS + 2S dx A dS - 2Y(2^Z - k2 - l)d xA dY

Now i t is easy to see that

a) ui = 0 , ш2 = 0 define our fo lia t io n (we have lo s t , s = 0

and S = o);

b) iu1 , ¡Ï have no s in gu la r ities on the surface.


Further more

Ul1 Л dx = - 2dy ds ф 0

Ш2 Л dx ф 0

So our fo lia t io n is now transverse to a fib ra tion with compact fib er, and Ehresmann

theorem implies the uniform ity of the e l le p t ic functions.

3-4- Equations F (x ,y ,y ' ) = 0 without movable singular points

For an analytic study see [5 ] .

3.5.Uniformity o f the h yp ere llip tic functions

Exercise.

3.6. Geometric study o f B r io t 's and Bouquet's equations (see _[9JI

One o f the problems is the fo llow ing : determine a l l the algebraic d if fe re n t ia l

equations o f the form F (y ,y ') = 0 with only meromorphic solutions. I t is also

possible to g ive in a geometric way the c la ss ica l resu lts o f B riot and Bouquet

concerning the p e r iod ic ity o f the solutions o f F (y ,y ') = 0 .

3.7. On abelian functions

Problem : Determine by using geometry the polynomials F i >F2,F3 ’ F4 such that

the p fa ffian system :

F ^ z . t .p .q .p ^ q ^ = 0 , F (z , t ,p , q, P i , ) = 0

F3( z , t ,p ,q lp1,q l ) = 0 ( F'4 ( z , t ,p , q,p ) = 0

dz = pdx+qdy , dt = p^dx+q^dy

is completely integrable and has only uniform solutions.

3.8.The f i r s t Pain levé equation

I t would be very in teresting to have a geometric proof o f the uniform ity o f the

transcendental function o f Paul Painlevé which is solution o f the d if fe r e n t ia l

280 GERARD

equation:


2" = бу + x ( f i r s t Painlevé equation)У'

This equation can be replaced by the fo llow ing p fa ffian system :

2dz = (6y + x)dx 2

on C (y ,z ) x C(x)

dy - zdx = 0

2Now the c la ss ica l com pactifications o f С into ]Pg(c) or ЗР^С) xIP^C)

or an analogous com pactification as those used fo r the e l l ip t i c functions does

not lead to a nice geometric s ituation allowing the use o f Ehresmann's theorem.

I t would be very in teresting to g ive a geometric in terpreta tion o f the trans

formation used by P. Painlevé in the proof o f the uniform ity o f the solutions 2

o f the equation y" = 6y + x (see [1 0 ]).

Part II

PAINLEVE’S FOLIATIONS ON A COMPLEX MANIFOLD

1. DEFINITIONS AND EXAMPLES

Let E and В betwo connected topo log ica l manifolds o f f in i t e dimension and

rr : E -• В a continous p ro jection .

D efin ition 1. A fo lia t io n 3 in E is c a l l ed simple fo r the projection i f fo r

each point m E there exists a distinguished neighbourhood o f m such that

the "plaque" o f m and the f ib e r o f m (тт ”' ( т г ( ш ) ) cut themself only in the

point m .

Examples

1 ) I f т г : Е - * В is a fib ra tion and 3 a fo l ia t io n which is transversal to

the fib ra tion then 'З is simple fo r the p ro jection тт .

2) I f тт : E - В is a covering then any fo l ia t io n in E is simple fo r тг .

3) Let

Ш = P (x ,y )d x + Q (x ,y )d y = 0 a p fa f f ia n equation in C2

where P (x ,y ) and o (x ,y ) are polynom ials r e la t i v e ly prim e. Let S be the set

o f s in gu la r po in ts o f щ and suppose that щ = 0 has no in teg ra l curves in

2every f ib e r x = C te . Then щ = 0 de fin es in С - S a fo l ia t io n which is

simple f o r the p ro je c t io n on the x - p la n e i fw e a re r e s t r ic t e d to th e f ib r a t io n

тг : С2 - it 1( tt( s ) ) - C (x ) = tt( s )

4 ) Other examples can be g iven by com p letely in teg rab le p fa f f ia n systems.

D e fin it io n 2. A f o l ia t io n S' in E is a P a in levé f o l ia t io n o f the f i r s t

282 GERARD

type fo r the p ro je c t io n тт i f f o r each path {¿ . [ 0 , 1 ] ] in В and each poin t

m Ç тт 1(1 (0 ) ) th is path can be l i f t e d in to the le a f o f m in to a path s ta r t in g

at m . An important consequence o f th is d e f in it io n is that i f 3 is a

Pa in levé f o l ia t io n o f the f i r s t type fo r тт : E -• В then the r e s t r ic t io n o f

тг to each l e a f is s u r je c t iv e .

Examples

1 ) Each f o l ia t io n tran sversa l to a f ib r a t io n w ith compact f ib e r and o f the same

dimension as the basis is a Pa in levé f o l ia t io n o f the f i r s t type.

2 ) In R2 : ^ = p ( l - p 2) p ,9 , p o la r coord inates in R2 d e fin e a Pa in levéU0

2f o l ia t io n o f the f i r s t type f o r the ra d ia l p ro je c t io n o f R - {0 }

1onto S .

З) In R the f o l ia t io n represented in the p ic tu re is not a P a in levé fo l ia t io n

je c t io n on the x - plane and a lso not fo r

» «

o f the f i r s t type f o r the canonical pro-

the canonical p ro je c t io n on the y -p la n e .

4) As we sha ll see la ter, each fo l ia t io n defined by a d if fe r e n t ia l equation in

the complex domain is nearly a Pain levé fo lia t io n fo r a good p ro jection .

Now we are going to define P a in levé 's fo lia t io n s o f the second type.

Let тг : E -* В a continous pro jection

S фф a subset o f E such that fo r a l l x in В , тг 1(x ) OS has only

iso la ted points in тт \ x ) .

3 is a fo l ia t io n in E - S o f the same dimension as В .

D efin ition 3. 3 is a Painlevé fo l ia t io n o f the second type i f fo r each path

{• í i [ 0 , l [ } in В and each point т€ (тт 1 (£ (o ) ) - S П тт 1( ¿ ( o ) ) ) the path

{ JÎ, [0,1 [ } can be l i f t e d in the le a f o f m into a path starting at m and the

path № , 1 ] } (c losed ) in the closure o f th is l e a f .

As a consequence we have fo r such a fo l ia t io n : the re s tr ic t io n o f тг to the

closure o f a le a f is su rjec tive .

Examples

1 ) xdx + ydy = 0 in C2 defines in C2 - (0 ,0 ) a Pain levé fo lia t io n o f the

second type fo r the canonical p ro jection on the x -p la n e .

2) zdz + ydy+ (z 2 + y2)dx = 0 defines in (Г* - fy = z = 0 } a Painlevé fo lia t io n

o f the second type fo r the pro jection on the x -p la n e .

3 ) We shall see la te r that to many d if fe r e n t ia l equations there is associated

a Pain levé fo l ia t io n o f the second type.

2. LOCAL PROPERTIES OF SIMPLE FOLIATIONS

Let (Е,тт,в)Ьеа complex analytic lo c a lly t r iv ia l fib ra tion and denote

by n the dimension o f the complex analytic manifold В and by n + p the

dimension o f the complex analytic manifold E. Consider now a complex analytic

fo l ia t io n o f dimension n in E which is simple fo r the pro jection тт .

Then we have :

Lemma 1. For each point a€E , there ex is ts a distinguished open subset

О Э a such that each "plaque" in 0 meets each f ib e r going through О . a --------------------- — a — —■ a

DIFFERENTIAL EQUATIONS 2 83

I d e a o f th e p r o o f

(E ,tt,b) is a locally trivial fibration and the result of the lemma

is a local result, so it is sufficient to proof the lemma in the following

situation •.

284 GERARD

U X V U a p o l y d i s k i n С

pV a p o l y d i s k i n С

5 a complex a n a l y t i c f o l i a t i o n in

И XV w h ich i s s im p le f o r th e p r o

j e c t i o n p

a = (0 ,0 )

L e t be d i s t i n g u i s h e d open neighb ou rhood o f ( o , o ) such t h a t th e f i b e r

o f (0 ,0 ) meets th e " p la q u e " o f (0 ,0 ) o n l y i n th e p o i n t (0 ,0 ) .

Now Qq i s is om o rp h ic t o a p ro d u ct xV^ o f two p o l y d i s k s U ^ c C *1 and

V c í such t h a t1

Denote b y ho : U X V - U X V 1

( x , y ) - ( x ^ y . , )

th e a p p l i c a t i o n a s s o c i a t e d t o th e isomorphism,

-1Then h h a s e q u a t i o n s :о n

x = h(xi ’yi ) , - v ï t \hQ (0 ,0 ) = (0 ,0 )

у = k ( x 1 . y ^

A " p la q u e " i n 0 i s g i v e n b y

x = h ( x , y ° ) у = k ( x , y ° )

As 3 is simple fo r the pro jection on U the system


о = h (x1, o)

у = k(x ,o )

has the unique solution x = o ,y = о .

Now look at the fo llow ing product :

U XV xu1 XV

(x .y .,) ( x ^ y )

P

U XV

(x .y ^

and the analytic set G defined in th is product by

x = h(x , y )

У = k (x1, y^)

The system

о = h(x ,o )

у = k(x ,o )

having the unique solution x = o ,y = о ; p \ o , o ) in tersects G only in

(o ,o ) and then the theorem o f Remmert-Stein [11] implies that there ex is t

two polydisks II' e l l , VJcV^ such that Gflp ”*(u ' xV]j) is a f in i t e ram ified

covering o f U' XV . Then h^^U ' xV^) has the property announced in the lemma.

Lemma 2. I f U is a polydisk in СП , V a polydisk in CP , 3f a simple fo -

l ia t io n in U X V -• U o f dimension n , then every germ o f patios at 0 Ç lT Can

be l i f t e d into the le a f o f , (o ,o ) in_ UxV .

I f the le a f o f (0 ,o ) is transversal to the f ib e r o f (0 ,o ) then the resu lt

is t r i v ia l . In the other case the assertion fo llow s immediately from the fo llow ing

fac ts :

1) in a distinguished neighbourhood o f (0 ,0 ) the "plaque" o f

( 0 ,0 ) is an analytic subset;

2) the p ro jection p : U x V - * U is a "good" p ro jection to apply

the theorem o f Remmert-Stein to the "plaque" o f (0 ,0 ) which is over a

neighbourhood o f 0£U a f in i t e ram ified covering.

3. THE MAIN THEOREMS ON PAINLEVE’S FOLIATIONS

The lemmas o f §2 imply immediately the fo llow ing theorem.

Theorem 1. Let (Е,тт,в)'Ье a complex analytic lo c a lly t r iv ia l f ib ra t io n , 3 a_

simple fo l ia t io n in E sa tis fy in g dim. 3 = dim. В . Then fo r each germ o f paths

in В at a point aSB and each point m € тт ”*(a) , th is germ is l i f t a b le into

the le a f o f m and the number o f p o s s ib il it ie s is f in i t e .

Now we are able to state a generalization o f a theorem due to Paul Painlevé in

the situation associated to a d if fe r e n t ia l equation.

Theorem 2. I f (E ,tt,b ) is a lo c a lly t r iv ia l complex analytic fib ra tion and

3 a fo lia t io n in E which is simple and having the same dimension as B; and i f

moreover the fib ra tion has compact fibers,then 3 is a Pa in levé 's fo lia t io n o f

the f i r s t type.

Remark. This resu lt remains true i f тт is only a proper submersion.

P roo f. The cornpacticity o f the f ib e r and Lemma 1 g ive us immediately the fo llow ing

fa c t : i f you can l i f t the path f¿2>[0,e[} then you can l i f t the closed path

f-í, [0 , e] } in any le a f going through a point o f тт 1 ( ¿ (o ) ) . Then with th is pro

perty and the lo ca l l i f t in g theorem i t is easy to conclude ( fo r d e ta ils see

[1 2 ]) .

As a consequence o f th is resu lt we have :

" " " Theorem (p. Pa in levé ). Une in tégra le y (x ) de F (y ',y ,x ) = 0 polynôme

en y ’ ,y à c o e ffic ien ts analytiques en x ne peut admettre comme points singu

l ie r s non algébriques que certains points f ix e s x = | qui se mettent en

■ M i l l

évidence sur 1 'equation même.

286 GERARD

Let us give the proof o f this resu lt fo r an equation o f the fo llow ing form :

у ' = р (х >у )- where P and 0 are polynomial r e la t iv e ly0 (x ,y )

prime.

The general case is given as an exercise.

We w rite the associated p fa ffian equation

Q (x,y)dy - P (x ,y )dx = 0

and extend th is equation to an equation U) = 0

on ^ .,(0 ) X C(x) - C(x)

Denote by :

the set o f singular points o f ш ;

S the subset o f 3P (c) XC which is the union o f v e r t ic a l leaves2 1

fo r the pro jection on C(x) (leaves o f the fo lia t io n defined in ЗР^С) x C -S 1

by ш = o) .


= т г ^ ) U tt(S2) (th is is a d iscrete subset o f C (x )).

Let 3 be the fo l ia t io n defined by i = 0 in

^ ( C ) X (C -H ) -* C(x) - S

This fo l ia t io n is simple fo r th is t r iv ia l fib ra tion (lo c a l existence and unique

ness theorem fo r a d if fe r e n t ia l equation). So by theorem 2, 3 is a Painlevé

fo l ia t io n o f the f i r s t type and theorem 1 says that the movable s in gu la rities

are at most algebraic ; each "plaque" is lo c a lly a f in i t e analytic ramified

covering o f an open subset o f the basis.

As a consequence o f the theorems 1 and 2, the resu lt o f P. Painlevé is also true

fo r certa in p fa ffia n systems in which th is general geometric s ituation arises.

In the general s ituation ( e ,tt,b ) with compact f ib e r and 3 a simple fo lia t io n

in E , it is also possible to give a generalization o f:

"""Theorems (P. Pa in levé ). Soit yQ la valeur de y (x ) pour x = xq e t so it

y = св(х,уо ,хо) l 'in té g ra le générale de F (y ',y ,x ) = 0 1 Si x e t xq sont deux

valeurs numériques quelconques d istin ctes des valeurs | la fonction y= C0(x ,y o lx^)

ne présente dans le plan des y^ que des points singu liers a lgébriques."""

Let (Е,тг,в)|>е a lo c a lly t r iv ia l complex analytic ■ fib ra tion with

compact f ib e r , S an analytic subset o f E which meets each f ib e r only in

iso la ted points , 3Í a complex analytic fo lia t io n in E -S which is simple fo r

the pro jection rr and whose dimension is the same as the dimension o f В .

Theorem 3. ÍF is a Painlevé fo lia t io n o f second type.

The only thing to prove is the fo llow in g ; i f f j í , [0 , e [ } ( e < l ) is a path in

В which is such that {^ , [0 , e [ } can be l i f t e d in the le a f o f the point

тётт 1 U (o ) ) into a path f£ , [0 , e [ } then f i , [ 0 , e ] } can be l i f t e d in the c lo

sure o f the le a f o f m ; that means i (e )€ F (m ) .

Using the compacticity o f the fib er, there is a point b Ç тт "*(jí( e) ) and belonging

to i ( [ 0 , e [ ) .

288 GERARD

Now three cases are possible :

l ) b € E -S ; then the lemma 2 gives the conclusion


2) b€S and l ( [ 0 , e [ ) Птт 1( i ( e ) ) = fb }

then the resu lt is also true and

¿ ( e ) ÇF(m) - P(m) , where F(m) denotes the le a f of

____ ï» in S

З) b Ç S and Т ([0 ,е [П т г 1( jî( e: ) ) c S has at least two points b and

b' . But th is hypothesis im plies,as i t is easy to see,that

S Л rr " '(jî( e) ) has non-isolated points (see the p ictu re ) .

There are now in fin ite points o f S in

t h e f i b e r o v e r ¿ ( e ) b e c a u s e i f a p o i n t

“ 1с б Ж О . е С П т т ( 4 0 ) i s n o t i n S we c a n

a p p ly l ) , a n d b i s n o t i n t h e c l o s u r e o f

^£[0, e [ w h ic h in c o n t r a d i c t i o n w it h o u r

h y p o t h e s i s .

4. APPLICATIONS

4-1.Singularities o f the solution o f a d if fe r e n t ia l equation

290 GERARD

For s im p lic ity we look at a d if fe r e n t ia l equation o f the fo llow ing form :

â z = p ( x - y )dx Q (x,y) where P and 0 are polynomials r e la t iv e ly

prime

Let ш = 0 be the extended associated p fa ffian equation to IP ( C) xC .

Denote by :

S the set o f singular points o f щ;

the union o f the pro jection o f

v e r t ic a l solutions;

Sg the set o f p ro jection o f the singular

points o f щ which are not in S1 •

the p fa ffian equation ш = 0

gives us a fo l ia t io n which is simple fo r the pro jection on C(x) . hs S has

the property given in theorem 4, the fo l ia t io n is o f the second type.

This means an a ly tica lly that i f у = cp(x) is a solution o f our equation, this

solution is in general a multivalued function which has a lim it along each path

going from a regular point xq to a point x € Sg . This lim it is the same fo r

a l l homotopic paths in C(x) - S US^ . Such a point is ca lled a transcendental

point by P. Pa in levé. Now le t us see what happens when we try to l i f t a path

in C(x) -S^ LIS with extrem ity in There are two p o s s ib il it ie s :

l ) In 2 [0 ,1 [ there is a point o f S then

the point x is at most a transcendental

singular point fo r the considered solution

o f our equation.


2) In 2 [0 ,1 [ there is a point of

it V x _ ) -S : then i f we look at the fo lia t io n' 1

in (С) X C(x) - S ,

тт 1(x ) - S D tt 1 ( x -] is a le a f o f this fo

l ia t io n and then the whole le a f is in the

closure o f the le a f o f m .

This means that our considered solution has

at the point x an essen tia l singular point.

As a co ro lla ry o f the preceding considerations we have an analogy to P icard 's

b ig theorem on en tire functions fo r the solutions o f an algebraic d if fe r e n t ia l

equation : I f a solution o f the d if fe r e n t ia l equation

y Q (x,y)

has an essen tia l singular point at x then i t takes near x a l l values except

perhaps the values y which are solutions o f P (x ,y ) = 0 .

For an analytic study o f th is question see [13] and [14 ].

4 .2 .D ifferentia l system

Consider a d if fe r e n t ia l system o f the fo llow ing form

to = F1 ^ - y ’ z ) to = P2( x ’ y ' z )

where F„ and F„ are rational functions.1 2

As i t is done in the work o f P. Painlevé, we consider the associated system

in Л>2(с)хс(х):

fv ) dx _ tdy - ydt _ tdz - zdt X t A - yC “ tB - zC

where X,A,B,C are homogenous polynomials in y , z , t . Denote by q the degree

o f X; then A,B,C are o f degree q + 1 . ( in a l i t t l e more general situation

the co e ffic ien ts o f these polynomials are holomorphic in x) . In his lectures

at Stockholm,Painlevé proves the fo llow ing resu lt :

U II II

i ) Si les polynômes X,A,B,C sont les plus généraux de leur degré,

l 'in té g ra le générale

y = cp(x,yo ,z o ,xo) , z = i,(x ,y0,z 0,x0)

ne peut admettre de singu larités mobiles non algébriques.

I I ) Pour que l 'in té g ra le générale y = cp , z = ф

admette des singu larités transcendantes mobiles, i l faut (mais i l

ne s u f f i t pas) que les éga lités

X = 0 , t A - y C = 0 , y B - z C = 0 , z C - t B = 0

soient compatibles (quelque so it x ) pour les valeurs y ,z , t qui

ne soient pas toutes nu lles.

I I I ) Pour que l 'in té g ra le générale y = cp , z = ф admette des

singu larités essen tie lles mobiles, i l faut (mais i l ne s u f f i t pas)

que le polynôme x (y , z , t ,x ) (ou l'u n de ses diviseurs X ^ (y ,z ,t ,x ))

défin isse une in tégra le première pa rticu la risée .о o n

Now we are giving a geometric in terpreta tion o f the se resu lts .

Denote by

тт : 1P2(C) X C(x) - C(x)

51 : the set o f singular points in ]P2(C) xC fo r the f ie ld o f

d irections given by our system ;

S : the set o f points (y , z , t , x ) S1 such that тт 1(x ) f lS 1 has

no iso la ted points ;

52 : the union o f S j and o f the in tegra l manifolds which are in

a f ib e r x = Cte .

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The projections ttCs ^) and tt(S2) are analytic subsets o f С . So we have

three cases :

l ) tt(S ) Ф С and ttCSj) Ф С (generic case)

ï l ) tt( s )= C and tt(S2)H C

I I I ) tt(S2) = С which implies n(S ) = С .

Geometrically these three cases g ive us :

i ) The fo lia t io n associated to (s ) in

( C - tt(S1 XS2) ) XP2(C) - C -n (S 1 US2)

is a Painlevé fo lia t io n o f the f i r s t type and as a consequence

the movable s in gu la r ities are at most a lgebra ic.

I i ) The fo lia t io n associated to the system in

(C - tt(S2) ) XT=2(C) - S1 - С - tt(S 2 )

is a Painlevé fo lia t io n o f the second type. This means that

the movable s in gu la r ities are at most transcendental singular points

I I I ) Movable essen tia l singular points can occur.

4 .3 . As an exercise prove geom etrically the fo llow ing resu lt o f P. Pain levé on

second-order d if fe r e n t ia l equations o f the form :

( 1 ) y " =K 0 (y , y , x )

where P and Q are polynomials in у ' , у with holomorphic c o e ffic ien ts in

x and which are r e la t iv e ly prime. Denote by p the degree o f P and q the

degree o f Q . Assume that p è q + 2 (th is is in the case o f the general

s itu a tio n )•


The resu lt o f P. Pain levé:

f.MI ) Si Q (y ',y ,x ) n'admet pas de zéro y = G(x) indépendant de y '

et s i p > q + 2 , l 'in té g ra le générale y (x ) de ( 1 ) et sa dérivée

y ' ( x ) ne peuvent présenter de s ingu larités essen tie lles mobiles.

I I ) Si Q (y ',y ,x ) admet au moins un zéro y = G(x) indépendant de

y ' et s i p > q + 2 , l 'in té g r a le générale y (x ) de ( l ) ne peut pré

senter de singu larités essen tie lles mobiles. Mais y ' ( x ) peut en

présenter.

I I I ) Si Q (y ',y ,x ) n'admet pas de zéro indépendant de y ' mais s i

p = q+ 2 , la fonction y ' ( x ) peut présenter des singu larités

essen tie lles mobiles mais y (x ) n'en présente pas.

iv ) Si Q (y ',y ,x ) admet au moins un zéro indépendant de y ' et s i

p = q+2 , les fonctions y (x ) et y ' ( x ) peuvent présenter des

singu larités essen tie lles mobiles.

H int: Look at the associated system in

ЗР^С) XP^C ) X C(x) - C(x)

I t is easy to see how to generalize many resu lts o f Painlevé to p fa ffian

systems.

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Part III

ALGEBRAIC PFAFFIAN EQUATIONS ON A PROJECTIVE SPACE

1. DEFINITIONS AND EXAMPLES

Consider in СП+ (n s 2 ) a d if fe r e n t ia l form

sa tis fy in g the fo llow ing conditions :

1) fo r a l l i , «J1 is. a homogenous polynomial o f degree m ;

n + 1 ^2) T tu = 0

i = 1

The second condition means that the in tegra l manifolds o f ш = 0 in СП+1[0 }

n+1are cones with top at the o rig in o f С . I f we denote by X the vector f ie ld

n + 1 ,£ x i »

i = 1 l S x i


thenn + 1

S x^U)1= шJX (in te r io r product)i= 1

An algebraic p fa ffian equation on 3P ( C) o f degree m is a class*

modulo m u ltip lication by elements o f С o f d if fe r e n t ia l forms sa tis fy in g the

two conditions 1 ) and 2 ) .

The equation is ca lled irreducib le i f the greatest common d iv iso r o f the

U^'s is 1 ; th is means that the singular locus o f w in pn(c ) contains no

hypersurfaces. I t is easy to see that i f S is the singular set o f щ (the set

o f common zeros o f the yj1) then S ф ф unless n = 2p + 1 and m = 1 .

The p fa ffia n equation is ca lled completely integrable i f the equation ш = 0

n+1is completely integrable in С

Example.

Ш = xdy - ydx + ydz - zdy + zdt - tdz = 0

is a completely integrable p fa ffia n equation on IP (c) and S = 0 .

Proposition 1. I f ш = 0 is a comple t e ly integrable algebraic p fa ffia n equation

on IP (c) then the singular set S contains always an irreducib le component

o f dimension n - 2 .

For the complete proof see [15 ]. In the general case the resu lt is a consequence

o f a theorem o f Bott [16 ].

Proof in a particu lar case

I t is su ffic ien t to prove the resu lt fo r the one-form щ in Cn+'* .

Denote by S the singular locus o f ш . I t is known that at each point Ç€S

at which duj(ç) фО an irreducib le component o f S o f codimension smaller or

equal to two goes through th is point. So i t is su ffic ien t to prove that dxjj

is not id en tica lly zero on S . I f dui is id en tica lly zero on S we must have b}

the "N u lls te llen sa tz" o f H ilbert a re la tion o f the form

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fo r a l l i , j ( g - - | ï~ ) 1J € (tu1,0I2 ........<»n+1)

where, n „ is fo r a l l i , j an integer greater or equal to 1 and

(ш1 it«21 . •. i » n+1 ) is the ideal generated by the u/'s .

But in the case where S is reduced we must have

Й: = tr £or a11 iijj i

So there exists a homogenous polynomial g o f degree m+ 1 such that

W = dg

But th is is wrong because of

0 = ujJX = dgJX = (m+l)g (E u ler's formula)

An algebraic solution o f codimension one o f the algebraic equation ш = 0

on 3Pn(c ) is by d e fin it ion an irreducib le homogenous polynomial f such

that

d f Л ш = 0 on f = 0

Theorem 1. Assume na 2 . Then


a) The equation ш = 0 in 3P (C) -S has no compact in tegra l manifold

o f codimension one.

b) I f codim. S г 2 each algebraic solution Z o f codimension one o f

U) = 0 whose singular points are in S contains an irreducib le

component o f codimension 2 o f S . In particu lar, i f codim. S г 3

the equation w = 0 has no a lg e b ra ic solution o f codimension one

which is regular outside o f S .

c ) I f codim. S 52 , each normal algebraic solution Z o f ш = 0

is o f degree smaller or equal to the degree o f w .

C oro llary . I f щ is completely integrable then the fo lia t io n defined by

id = 0 in ! > (c) - S has no compact leaves.

Proof o f the theorem

a) Let V be a compact in tegra l manifold o f uj = 0 ; th is is a connected regular

algebraic submanifold o f HP (c) and so is defined by a homogenous polynomial

f . Denote by у the cone defined by f = 0 in СП+ and y = y - f О} ; у is

a complete in tersection and is regular outside the top o f the cone, which is o f

codimension a 2 . This means that у is normal.

As

d f фо at each point o f V

Ш Ф 0 at each point o f V

and V being an in tegra l manifold, we have

d f Л ш = 0

where d f . Q _ ^ 0Ш * * л •Y V 0 vY Y çïl+1 Y

are the induced morphisms*

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The cone у being normal, the section a can be extended to an in ve rtib le element

where h is a polynomial one-form ; but the degree o f f is greater then the

degree o f d f and th is implies h = 0 .

Now we have ш = a d f and the Euler formula gives us the fo llow ing

contradiction :

and h is a polynomial one - form.

I f we have deg. f >m then fh has no homogenous components o f degree m ; th is

implies that ш = a d f , which contradicts Eu ler's formula.

The fo llow ing more general resu lt is also true.

Theorem 2. Let Y be a Stein manifold and E a complex analytic vector bundle

on Y and тт : P (e ) -* Y the associated projective bundle. Then every complex

also denoted by a o f the С - algebra r (y ,0 ) ; that means to an element aÇC .Y

The ir re d u c ib il ity o f f implies that

*ID = a d f + h f , a € С

o = w JX = a (d e g . f ) f

b) The only remark we have is that

codim. (S n z ,z ) й 2

which implies that the cone associated to Z in С,n+1is also normal and the

proof fo llow s as in case a ).

c ) The hypothesis on S and Z implies e a s ily that

m = a d f + fh

where

a€ C[x ,x2 » • • *

analytic fo lia t io n 3* o f p os it ive dimension in an open subset U o f P (e ) has

no compact l e a f .

The proof o f this theorem fo llow s from the fo llow ing lemma.

Lemma 1 [17 ]. I f X is a smooth compact submanifold o f dimension m>0 o f

]Pn(C) then there does not ex is t on X a complex analytic vector bundle which is

both ample and f l a t .

Idea o f the proof o f theorem 2

I f X is a compact le a f , X is in a f ib e r because Y ^is a Stein manifold.

Now we can r e s t r ic t ourselves to the fo llow ing situation :

тг :ЗРП(С) XV - V

Ywhere V is an open subset o f С containing 0, and moreover we can assume

that ХСЛ=П(С) X {0 ] . I f N is the normal bundle o f X in JPn(C) , since X

is smooth and p ro jec tive , N is ample (see [ 18] p. 105, example 2 ); then det.N is

also ample (see [19] p. 68, co ro lla ry 2 .б ).

But a theorem o f Ehresmann (see [ 1 ] p. 384, theorem 2 .Я) implies that the normal

bundle N to X in IP (C) xV is f l a t .n

But the exact sequence

0 - > N - > Ñ - > X x C r - 0

implies that det. N is isomorphic to det. N and so det. N is also f la t ,

which is a contradiction to the resu lt o f the lemma.

D e fin it ion . Let U be an open subset o f a complex analytic manifold Z and

'S a complex analytic fo lia t io n in U . A compact solution o f 3 is by d e fin i

tion a compact analytic subset A o f Z such that А П U is a le a f o f the f o l i a

t io n .

Lemma 2. ( fo r the proof see [1 7 ]) . Let S be a closed analytic subset o f 3Pn(C)

and U = P n(c ) - S. Then there does not ex is t a closed algebraic submanifold X


o f 3Pn(c ) o f p os it ive dimension such that V = UflX is smooth and satisfies

the fo llow ing two properties :

1 ) the normal bundle N to V in IP (*-) i s f l a t and the res tr ic tion

to V o f an ample bundle over 3P (C) ;n

2) Codim. (S n x ,x ) â 2 .

From th is lemma we deduce the fo llow ing.

Theorem 3. In the s ituation o f theorem 2 assume that S = P (e ) -U is a closed

analytic subset and that 3 is o f codimension one. Then each compact solution

X is in a f ib e r and

codim.(S П X , x ) s 1

In particu lar, a f ib e r P o f a point y 6 Y does not contain a compact solution

i f cod im .(snP ,P ) & 3 .— У У

Proof. We know that X is in a f ib e r P . Let N be the normal bundle o f X

in P; then from the exact sequence

0 - N - Ñ - X X C r - 0

we deduce that N is the re s tr ic t ion o f an ample bundle on IP (C) and .moreover,n

th is bundle is f la t , which contradicts lemma 2.

The theorems 2 and 3 can be generalized to p fa ffia n systems by using the same

lemmas 1 and 2 ( fo r d e ta ils see [1 7 ]) .

2. ALGEBRAIC SOLUTIONS OF A PFAFFIAN EQUATION ONPn«R

The structure o f the set o f algebraic solutions o f a p fa ffia n equation on a

p ro jec tive space was f i r s t given by Darboux [20] in the case n =2 .

Let us consider a p fa ffia n equation on ]P (c), i.e. an equation o f the fo llow ing

form :

300 GERARD

sa tis fy in g the conditions;

1 ) the («'"''s are homogenous polynomials a l l o f the same degree


n + 12 ) £ x.u)1 = u ) j x = 0

i = 1

n + 1 ,where X is the vector f i e ld £ x. т—i dx.

i = 1 1

We have :

Theorem [15 ]. For щ = 0 there are only two p o s s ib il it ie s :

1) A ll the solutions are algebraic ;

2 ) The equation has only a f in i t e number o f algebraic solutions

and th is number is smaller than

1 / n_1 + n N = -? m(m-l) , ’ I + 22

1 n-2

A proof o f th is resu lt was given by Darboux in [20] fo r the case

n = 2 .

C oro lla ry . I f we have N algebraic solutions F ,F g i...|F jj o f щ = 0 then

there ex is t N complex ntanbers or , . . . not a l l zero such that :

N1 ) £ cr deg. (F^) = 0

i = 1

N . dF.2 ) £ у - i л «j = 0

i = 1 i

and in particu lar щ is completely in tegrab le.

Notation

A = C[x^, x ^ ,. • . >xn+^]

К is the f ie ld o f fra c tion o f A ;

302 GERARD

Па(СП+ ) the A-module o f algebraic 1 - d if fe r e n t ia l forms on Cn+1;

Л(01(С +1)) the ex te r io r algebra associated to Q1 . a a

I r e ca ll that the fo llow ing complex o f Koszul is acyc lic :

n+1 JX n JX 2 JX 1 „ JX e0 - A (O') - a (q ) Л(П') - Л(П') - A -* С -> 0

a a a a

n+1 gwhere JX is the in te r io r product by X = E x . —

i=1 1 i

e(P) = P(0 )

1 1 1 [ A (n ) ] is the set o f elements o f A (o ) whose c o e ffic ien ts are a l l o f a * a

degree p •

A rational integrant fac to r o f w = 0 is by d e fin it ion a non-zero rational

Pfunction R = — homogenous such that

dR = h щ where h is an other rational function.

But the la s t re la tion implies that

d R jX = 0 since o)JX = 0 so (deg.R ).R = 0

and R is homogenous o f degree 0 .

I re ca ll that an algebraic solution o f щ = 0 is by d e fin it ion an irreducib le

homogenous polynomial F such that

2 .dP = hF h 6 А (СГ)

Denote by G the set o f a l l a lgebraic solutions o f щ = 0 and consider the

С - vector space with basis G : .

Proposition 2. The С - lin ea r map

ф : CG ---- 0^®A K

dF.u = E a i ф(и) = Í » . у 1

is in jec tive.

We make use o f the fo llow ing lemma

Lemma 2 . Let = {F_ = 0 } l £ i á p be p d iffe ren t algebraic hypersurfaces

in lPn(c ) ; then there ex istsa p ro jec tive lin e D which cuts the surface

in an iso la ted point and which is not contained in the other surfaces.

Proof o f the proposition

Assume thatdF.

s » i I T = 01

fo r some a^ 's and that, fo r example, a ф 0 .

Let D be a lin e given by the lemma fo r С' which has equation F = 0 , and

8 a parametric representation o f the lin e D . I f we denote by = F^= 0

and z the coordinate on D, we have


fo r a l l s b . Ф i

F- = M>1(z - a ) t П (z-b? ).1 1 b.^a 11

F = ii, П (z-b1?) p p i

Then _dF. dF.

E a . -jr~o 6 = E a . ~ = t » , — + (e — !— ) dzl F. 1 p 1 z-a 4 z-c,i f к

dF.and E a . -=— = 0 implies a „ = 0, which is in contradiction with ouri F. r 1

ihypothesis.

The proposition 2 allowed us to consider as a subvector space o f •

Denoteby e (F ) the vector o f the basis o f associated to the algebraic solu-

Gtion F and consider now the subvector space W o f С defined by

W = {E 0 'i e (F i )| EQ'i (deg.F i ) = 0}

The mapCG— С

304 GERARD

£ a i Fi S ° 'i d e g . (F i )

is a lin ea r form on CG and W is the kernel o f th is map and as a consequence

codim. W = 1 .

Consider now the С - lin ear map :

: W — ■ [ Л (Q1) ] m-1

dF. dF.Б Q'.Fi — шЛ£ Q'i

i i

Proposition 3

cp(w )c[Z2] m 1 (space o f 2 - cycles o f degree m-1 o f the Koszul complex).

I t is easy to v e r i fy that

dF.(So^ Ш Л ^ ) JX = 0

i

Proposition 4 . For the 1 - form ш there are only two p o s s ib il it ie s :

1) i t has a rational integrant fa c to r ;

or

2 ) Dim. (Ker. cp) = 1 .

Let and £ be two elements o f Ker. cp. By adding a f in i t e number

o f terms we can assume that these elements are o f the fo llow ing form : s s2 Q\F. and £ B .F. (with some zero c o e ffic ie n ts )

« 1 1 1 1 1i= 1 i= 1

By assumption we have

dF.шЛ £ a . = 0

1 Г .

dF.* Л Е p. - j 1 = 0

i


Denote G = П F. a n d G, = F . F . . . F . . . F l к 1 2 k si = 1

Then we have :

шЛ E a i G. dF. = 0 i = 1

W Л . E g . G. dF. = 0 i = 1 i i i

As щ / о outside S, which is o f codimension g r e a t e r than 0r equal to two, there

ex is t two polynomials kff and k^ such that

S d . G. d F . = к * k ^ Oi i i a a

E p. G dF = к ш к ¿ °1 1 1 P P

And then we have

and b y d i f f e r e n t i a t i o n

d F . к1 O'

s a i T 7 = T ш

dF. к

^ 1 ^ = G *1

d A n t - ? d i| i= 0

kRx kRd(-pf) Лш + d(u = 0

By taking the d ifferen ce o f this two expressions we obtain

кd (T r ) л U) = 0

And two cases appear :

к кi ) t— ф Cte; then — is a rational integrant fac to r o f ш and a ll

9

solutions o f да = 0 are a lgebra ic;

306 GERARD

к2) — = с (constant)

which implies eas ily :

dF.

S ^ i " c 7 ^ = °

That means

a . = с 8. fo r a l l i1 1

and the dimension o f ker. cp is 1 .

Proof o f Darboux's theorem. We have only to study in more d e ta il the case 2.

We have the exact sequence :

Ф0 -* Ker.cp — W -• V -• 0

where V = cp(w) с [Z ] jn_ 1

Now D im .[Z.] < + “ , Dim.W <+ “> and since2 m-1

Codim.W in CG is 1 , this implies that Dim.CC<+«> .

Moreover, Dim.C^ = Dim.W+ 1 and Dim.W = Dim.V+ 1 ; that means Dim.C^ =

Dim.V + 2 .

But ï c [ Z „ ] „ so Dim.C^ S Dim [Z „] . + 2L 2 m-1 L 2 m-1

and an easy ca lcu lation gives us

/ m-1 + n '/ m-1 + n \Dim. [Z g ] m_ i = j ™ 0 "- l)^ ^ J

Which ends the proof o f Darboux's theorem.

You can find in Darboux's o r ig in a l paper an example o f p fa ffian equation on

P g (c ) which has exactly N algebraic solutions and no more.

Now le t us g ive an other important resu lt on the fo lia t io n defined by a p fa ffian

equation on P (c) . Denote by ш = 0 the p fa ffian equation and S i t s singular

s e t .

Theorem [21 ]. I f a l l the leaves o f the fo l ia t io n 3> defined in ]P ( c ) - S by

Ш = 0 are proper then да = 0 has an algebraic so lu tion .

The proof o f th is resu lt fo llows from the fo llow ing lemma.

Lemma. The set

E = (F I F is a le a f o f 3 }

is an inductive s e t .

The proof o f th is lemma is very long and make use o f P a in levé 's fo lia t io n s .

Idea o f the proof o f the theorem

Let F be a le a f o f the fo lia t io n 3 and consider the set

E = fF j F is a le a f o f 3 }

This set is inductive by the lemma. This implies that F fo r the considered le a f

contains a minimal set, say К . Consider now the end В(к) = К - К o f К .

As К is proper we do not have в(к)зк .

As К is minimal we do not have

В ( К ) П К = 0 and В(К) ГПР (C) -S фф

So в(к) CS and codim.SS2 implies that К is an algebraic submanifold of

3Pn(c ) and gives us an algebraic solution o f the p fa ffian equation щ = 0 .

In particular,we have also the fo llow ing resu lt :

Each le a f o f the fo lia t io n 3> is algebraic or contains in i t s

closure an algebraic le a f .

BIBLIOGRAPHY

[1 ] H A E F L IG E R , A ., Variétés feu illetées, Ann . Sc. N o rm . Super. Pisa, Sci. F is. M at. 16 (1 9 6 4 ) 36 7 —397.

[2 ] R E E B , G ., Sur certaines p ropriétés topo log iqu es des variétés feu illetées, A ctua lités sc ientifiques et industrielles,

Herm ann, Paris (1 9 5 2 ).


308 GERARD

[3] SU ZU K I, М., Sur les re la tions d ’équivalence ouvertes dans les espaces analy tiques, A nn. Sc. Ec. N orm . Sup.Paris, 4° série , t . 7 , fasc. 4 (1974 ).

[4 ] KAUPP, B., Ü ber offene analy tische À quivalenzrelationen au f kom plexen R aum en , M ath. A nn. 183 (1969).[5 ] PA IN LE V E , P ., Oeuvres, E d itio n du CN RS, T om e 1 (1972 ).[6 ] M A RTIN ET, J ., D oc to ra t de 3° cycle, G renoble (1962).[7 ] N ARA SIM H A N, R ., A nalysis on Real and Com plex M anifolds, Masson (Paris) 1968.[8] EH RESM AN N , С., “Les connex ions infinitésim ales dans un espace fib ré d iffé ren tiab le”, Colloque de topologie:

espaces fibres, L ibrairie universita ire , B ruxelles (1 9 5 0 ) 31.[9 ] D IE N E R , F ., D o c to ra t de 3° cycle, IRM A, S trasbourg (1974 ).

[10] PA IN LE V E , P ., Oeuvres, to m e III, E d itions du cen tre na tionale de la recherche scien tifique, Paris (1975 ).[11] REM M ERT, R ., STEIN , K ., Ü ber die w esentlichen S ingularitâ ten analy tischer M engen, M ath. A nn. 126

( 1953 ) 26 3 - 3 0 6 .[12 ] G E R A R D , R ., SEC, A ., Feuilletages de Painlevé, Bull. Soc. M ath. F rance 100 (1 9 7 2 ) 4 7 - 7 2 .[13 ] K IM U RA , T ., Sur les po in ts singuliers des équations d ifférentie lles ord inaires du prem ier o rd re , C om m ent.

M ath. Univ. Sancti Pauli 2 (1 9 5 3 ) 4 7 —53.[14 ] K IM U RA , T ., SIBUY A , Y ., Essential singular po in ts of so lu tions o f an algebraic d iffe ren tia l eq ua tion ,

Jou rnal fur M ath. 2 7 2 (1 9 7 3 ) 1 2 7 -1 4 9 .[15 ] JO U A N O LO U , J .P ., E qu a tio n s de P faff algébriques sur u n espace p ro jec tif, IRM A p rep rin t,S trasbourg (1975 ).[16 ] BO TT, R ., L ectu res on characte ristic classes and fo liations, L ectu res N otes in M athem atics 279 , Springer (1972).[17 ] G E R A R D , R ., JO U A N O LO U , J.P ., E tu d e de l’ex istence de feuilles com pactes p o u r certains feuilletages

ana ly tiques com plexes, CR A cad. Sc. Paris 277 (1973 ).[18 ] H A R T SH O R N E , R ., L ectu re N o tes in M athem atics 156. Springer, Berlin (1970 ).[19] H A R T SH O R N E , R ., Pub lication N o. 29 , In s titu t des hau tes é tudes scien tifiques, B ures-sur-Y vette (1966 ).[2 0 ] D A RBO U X , G ., M ém oire sur les équa tions d ifférentielles algébriques du p rem ier o rd re e t du prem ier degré,

Bull. Sc. M ath. (1 8 7 8 ) 6 0 - 9 6 ,1 2 3 - 1 4 4 ,1 5 1 - 2 0 0 .[21] TR A N HUY H O A N G , D o c to ra t de 3° cycle IRM A, S trasbourg (1975).

SECRETARIAT OF THE COURSE

DIRECTORS

A. Andreotti

J. Eells

F. Gherardelli

EDITOR

Department o f Mathematics,

School o f Science,Oregon State University, Corwallis, Oregon 97331, United States o f America

andIstituto Matematico,Università di Pisa,Via Derna 1,Pisa,Italy

Mathematics Institute, University o f Warwick, Coventry CV4 7AL, Warwickshire,United Kingdom

Istituto Matematico “U. Dini” , Università di Firenze,Viale Morgagni 67/A,Firenze,Italy

Miriam Lewis Division o f Publications, IAEA , Vienna, Austria

The fo llow ing conversion table is provided fo r the convenience o f readers and to encourage the use o f SI units.

FACTORS FOR C O N VERTIN G UNITS TO SI SYSTEM EQ UIVA LEN TS*

SI base units are the metre (m), kilogram (kg), second (s), ampere (A ), kelvin (К ), candela (cd) and mole (mol).[For fu rther inform ation, see International Standards ISO 1000 (1973), and ISO 31/0 (1974) and its several parts]

M u ltip ly fry fQ obtain

Mass

pound mass (avoirdupois) 1 Ibm 4.536 X 10“ ‘ kgounce mass (avoirdupois) 1 ozm = 2.835 X 101 9ton (long) (= 2240 Ibm) 1 ton = 1.016 X 103 kgton (short) (= 2000 Ibm) 1 short ton = 9.072 X 102 kgtonne (= metric ton) 1 t 1.00 X 103 kg

Length

statute mile 1 mile = 1.609 X 10° kmyard 1 yd = 9.144 X 1 0 '1 mfoo t 1 f t 3.048 X 10"1 minch 1 in - 2.54 X 10’ 2 mmil (= 1 0 '3 in) 1 mil 2.54 X 10’ 2 mm

Area

hectare 1 ha = 1.00 X 10“ m2(statute m ile)2 1 mile2 = 2.590 X 10° km 2acre 1 acre = 4.047 X 103 m 2yard2 1 yd2 = 8.361 X 10’ 1 m2

o' о ы 1 f t 2 = 9.290 X 10‘ 2 m2inch2 1 in2 = 6.452 X 102 mm2

Volume

yard3 1 yd3 = 7.646 X 10’ 1 m3fo o t3 1 f t 3 = 2.832 X 10~2 m3inch3 1 in3 = 1.639 X 104 mm-gallon (B rit, or Imp.) 1 gal (Brit) = 4.546 X 10-3 m3gallon (US liquid) 1 gal (US) = 3.785 X 10 '3 m3litre 1 I = 1.00 X 1 0 '3 m3

Force

dyne 1 dyn = 1.00 * 10 5 Nkilogram force 1 kgf = 9.807 X 10° Npoundal 1 pdl = 1.383 X 1 0 '1 Npound force (avoirdupois) 1 Ibf = 4.448 X 10° Nounce force (avoirdupois) 1 ozf = 2.780 X 10‘ ‘ N

Power

British thermal unit/second 1 Btu/s = 1.054 X 103 Wcalorie/second 1 cal/s - 4.184 X 10° Wfoot-pound force/second 1 ft - Ib f/s = 1.356 X 10° Whorsepower (electric) 1 hp = 7.46 X 102 Whorsepower (metric) (= ps) 1 ps = 7.355 X 102 Whorsepower (550 f t • Ibf/s) 1 hp = 7.457 X 102 W

* Factors are given exactly or to a maximum o f 4 significant figures

M ultip ly by to obtain

Density

pound mass/inch3 1 lbm /in3 = 2.768 X 104 kg/m3pound mass/foot3 1 lb m /ft3 1.602 X 101 kg/m3

Energy

British thermal un it 1 Btu = 1.054 X 103 Jcalorie 1 cal = 4.184 X 10° Jelectron-volt 1 eV s* 1.602 X 10"19 Jerg 1 erg = 1.00 X Ю -’ Jfoot-pound force 1 f t - lb f = 1.356 X 10° Jkilowatt-hour 1 kW-h = 3.60 X 106 J

Pressure

newtons/metre2 1 N /m 2 1.00 Paatmosphere* 1 atm = 1.013 X 105 Pabar 1 bar = 1.00 X 10s Pacentimetres of mercury (0°C) 1 cmHg = 1.333 X 103 Padyne/centimetre2 1 dyn/cm 2 = 1.00 X 10"' Pafeet o f water (4°C) 1 ftH 20 = 2.989 X 103 Painches o f mercury (0°C) 1 inHg = 3.386 X 103 Painches o f water {4°C) 1 inH20 = 2.491 X 102 Pakilogram force/centimetre2 1 kgf/cm 2 = 9.807 X 10" Papound fo rce /foo t2 1 lb f / f t2 = 4.788 X 101 Papound force/inch2 (= psi)b 1 lb f / in 2 = 6.895 X 103 Pato rr (0°C) (= mmHg) 1 to rr = 1.333 X 102 Pa

Velocity, acceleration

inch/second 1 in/s = 2.54 X 101 mm/sfoot/second (= fps) 1 ft/s = 3.048 X 1 0 '1 m/sfoot/m inute 1 ft /m i n = 5.08 X 10"3 m/s

m ile/hour (= mph) 1 mile/h4.470 X 1 0 '1 1.609 X 10°

m/skm/h

knot 1 knot = 1.852 X 10° km/hfree fa ll, standard (= g) = 9.807 X 10° m/s2foot/second2 1 ft/s 2 = 3.048 X 1 0 '1 m/s2

Temperature, thermal conductiv ity, energy /area- time

Fahrenheit, degrees — 32 “ р - з г Т 5 J ° cRankine ° R J 9 I K1 B tu - in / ft2 -s- °F = 5.189 X 102 W /m -K1 B tu /ft-s - °F = 6.226 X 101 W /m -K1 cal/cm-s-°C = 4.184 X 102 W /m -K1 B tu /ft2 -s = 1.135 X 10“ W/m21 cal/cm2-min 6.973 X 102 W/m2

Miscellaneous

fo o t3/second 1 f t 3/s = 2.832 X 1 0 '2 m3 /sfo o t3 /m inute 1 f t 3/m in = 4.719 X 1 0 '4 m3/srad rad = 1.00 X 10’ 2 J/kgroentgen R = 2.580 X 10"4 C/kgcurie Ci = 3.70 X 1010 disintegration/s

eatm abs: atmospheres absolute; atm (g): atmospheres gauge.

b lb f / in 2 (g)I b f/in 2 abs

(= psig) : gauge pressure;(= psia): absolute pressure.

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