residues of higher order and holomorphic vector fields

Annal~ of Global Analy~i~ and Geometry 12: 109-122, 1994. (~) 1994 Kluwer Academic Publishers. Printed in the Netherlands.

109

R e s i d u e s o f H i g h e r O r d e r a n d H o l o m o r p h i c V e c t o r

F i e l d s

DANIEL LEHMANN

Abstract: Let X, and X2 be two holomorphic vector fields on a manifold V with complex di- mensionp. Assume that they have the same singular set E. For all Cl = (cl)it .(c2)i . . . . (cp)ip, (il H- 2i2 H-.." + pip = p), it is known (after Chern-Bott) that each of the vector fields defines a "residual" characteristic class Cy(V, Xl)(resp. Cs(V, X2)) in H2P(V, V - E), which is a lift of the usual characteristic class C[(V) of the tangent bundle. The difference CI(V, X2) - CI(V, X1 ) belongs then to the image of cO in the exact sequence

�9 .. --+ H 2 p - I ( V - E) -~ H2p(V,V - Z) ~. H2p(V) --4 H 2 p ( V - Z) - ~ . . . .

In fact, there exists a canonical lift CI(V, X1,X2) of this difference in H2P-I(V - ~]): we will call it the residual class of order 2 (associated to I, X1 and )(2). This class is localized near the points where X1 and X2 are colinear: we will explain this precisely in terms of Grothendieck residues. The formula that we obtain can be interpreted as a generalization of the purely algebraic identity, obtained from the general one as a byproduct:

, ~ _ ~ .. . z~) ~(n,,...,n,>) ~(~,,---,~,,) ,-,~, <,,, , ~ - o, . , - E - - - - - ~ - - ~ - ~ , 1-Ii=~ Zi ~[L=I O~i 1-Ij,j~,(~, - o, )

where (~ , - - . , ~,,) and (~,,..., ~p) denote two families of no~-zero complex numbers, such that all denominators in this formula do not vanish. (This identity corresponds in fact to the case where X1 and X2 are non-degenerate at the same isolated singular point.)

If the ~ ' s (1 g i < p) depend now differentiably (resp. holomorphically) on a real (resp. complex) parameter t then, denoting by ~ the derivative with respect to t, and assuming all numbers lying in a denominator not to be 0, we can deduce from the above identity the following derivation formula:

' = ~ ~( (" , ) , . . . , (~ , ) ) ( ~ ( ~ , , . . , ~ , ) ~ ~ ' ~ ,

~ - ~ - - r i i j --~(~)-v �9 \ I I ,=,~, ) ,=,

Key words: Holomorphic vector fields, Grothendieck residues, higher order MSC1991: 57R20 ,57R25

1. I n t r o d u c t i o n

Let V be a complex connected manifo ld of complex dimension p > 2. There does no t always exist a holomorphic vector field on V wi thou t s ingular i ty : af ter Bot t ([B1]), a necessary condi t ion for tha t , if V is compac t , is t ha t all real Chern numbers (CI (V) , V) of V are zero, whatever be the mul t i - index I = ( i l , i 2 , " ' , i p ) such tha t

110 D. LEHMANN

[I[ = p, with CI = (cl) i' (c2) i2 . . . (%)i, belonging to the 2p-dimensional real cohomology H 2p and 1I[ = i l + 2i2 + �9 �9 - + pip. If x denotes now a holomorphic vector field on V with singular set* YI., the Chern numbers (CI(V), V) will "localize" near 1~, and will give rise to a "residue theorem". The corresponding residues have been computed in the following cases: - by Bott, in the case where X has only isolated non-degenerate singularities ([B1]), and more generally in the case where the singular set ~ is made of smooth complex compact submanifolds and the action of X on the normal bundles of these submanifolds is non-degenerate ([B2]), - by Baum-Bot t ([BB]), in terms of Grothendieck residues, in the case where X has only isolated (may be degenerate) singularities.

In fact, the data of X without singularity off I~ gives rise to a canonical lift CI(V,X) E H2p(V,V - ~) of CI(V) E H2p(V). This class CI(V,X) will be called the residual characteristic class of CI(V) with respect to X. (It corresponds to the above invoked residue in H0(E) = ~ H o ( I E ~ ) by Alexander-Lefschetz duality, denoting by ~ the connected components of E.)

Assume now that we are given 2 holomorphic non-vanishing vector fields X1 and X2 on V - I ~ : the difference CI(V, X2) - C t ( V , X1) belongs obviously to the image of 0 in the exact sequence

. . . -~ H2p_I (v _ ~) 0 H2P(V, V - ~) --% H2p(V) -+ H2p(V - ~) -+. . . .

We will prove that there exists in fact a canonical lift Cr(V, X1,X2) of this difference to H 2 p - I ( v - X]), which vanishes when X1 and X2 are everywhere linearly independent on V - ~. In the general case, denote by F (a singular complex curve in general) the set of points where X1 A X2 = 0. Then CI(V, X1, X2) will localize near F N (V - X;), by means of a "residue formula" (cf. Propositions 1 and 2). It is these residues that we will call "of order 2".

Precise computations will be given in terms of Grothendieck residues (cf. Theo- rems 1 to 4), and some examples computed. We get in particular, as an amazing byproduct, a curious - purely algebraic - identity (cf. Theorem 5), which is not so easy to check directly (except in very low dimensions), and which has as a corollary a derivative formula for Bott ' s residues.

We will make use of the integration over a Cech-de R_ham complex. This is a machinery already used in [L], and we will refer the reader to this paper: see Section 3 for general definitions, elementary results, and notations for integration and duality in the Mayer-Vietoris complex MV*(LI) (particular case associated to the covering /A = (U, V - ~) of V), and Sections 1 and 6 for integration for more general Cech- de Rham complexes. We apologize for this reference, making the present paper not self contained, but to repeat the description of the used machinery would take too much place.

I had very helpful discussions, with F. Ruiz-Gomez, P. Molino, M. Soares, and in particular with J.F. Mattei who raised the problem. Many thanks to all of them!

* A singularity of X is a point where X either vanishes or is not defined.

RESIDUES OF HIGHER ORDER AND HOLOMORPHIC VECTOR FIELDS 111

2. B a c k g r o u n d a n d A b s t r a c t N o n s e n s e

Let A~ : I*(G) -+ fi*DR(M ) denote the Chern-Weil homomorphism associated to a connection w on a smooth bundle E --+ M with s tructural group G, I* (G) being the algebra of polynomials on the Lie algebra of G which axe invariant under the adjoint representat ion of G. Let f f denote a homogeneous ideal of I*(G). We say tha t w is a i f -connect ion, if 3" C Ker A,~. A set C of ,] '-connections on E is said to be fl-convex, if, for all families (w0, Wl , . - . , w~) of r + 1 connections in C, the connection

I" D = ~ i = 0 tiwi over E x A r ~ M x A ~ defined natural ly by affine combinat ion of the w i ' s still is a i f-connection. (A ~ denotes the r-dimensional affine simplex ~[--0 ti --= 1, tl > 0 in R r + l . ) Then, denot ing by

ZX~o~ 1 ..... : F ( G ) -~ % - ~ ( M )

the i terated difference opera tor t defined by Bot t ([B3]), one has:

f f C Ker Aw o~l ..... �9

E x a m p l e 1. Let E ~ V be a holomorphic vector bundle of complex rank q, whose base V, not necessarily compact , has complex dimension p. Let X be a holomorphic vector field on V, wi thout singularity on V - E, and

O x : F ( E ) -~ r ( E )

a C-ac t ion of X on E, i.e., after Bot t ([B2]), a C-linear endomorphism of the space F ( E ) of C ~ satisfying:

Ox(I , r ) = Yex(~) + ( x f)~r

for all f E C a ( V ) and all a E F(E) . A connection w on EIv-z, (or the associated derivative law V) will be called

special (with respect to ~)x, or to X if there is no ambiguity about (9 as in the par t icular case below), if it satisfies:

Vycr = 0 whenever Y is ant iholomorphic (or s imply belongs to T 0'l)

and a is holomorphic.

L e m m a 1. For M = V - E, G = GL(q, C) , I f = the ideal of I*(G) = R [ c l , c 2 , . . . , Cq] generated by monomials CI = (ci) i ' .(c2) i 2 ' ' . (Cq) i~ such that ii + 2i2 + .. �9 + qiq = p, the set C of special connections is then if-convex.

P a r t i c u l a r case . E = T c ( V ) the complex tangent bundle to V, p = q, O x ( Z ) = [X, Z]. This is the case that we want to s tudy more carefully in the following sections.

E x a m p l e 2. Let H be a Lie subgroup of G, and assume that an H-reduc t ion of the G-principal bundle associated to E -+ M has been given. The set C of all connections on E --~ M preserving this reduct ion is then J - c o n v e x for 3" = Ker (I*(G) --+ I*(H)).

t Recall that this operator is defined as the composition of A,~ with integration along the fiber A r of the projection M x A r ~ M.

112 D. LEHMANN

P a r t i c u l a r cases . G = SO(2r), H = SO(2r - 1), :7 is generated by the Pfaffian (Euler class). G = U(q), H = U(q - k), J is generated by the Chern classes cq , . . . , Cq-k+l. G = U(q), H = SO(q), J is generated by the Chern classes Codd: the corresponding residues of order 2 (see definition after Lemma 3 below) are the (generalized)Maslov classes; (see IV] for bibliography on the subject).

E x a m p l e 3. Let E -~ M be the normal bundle to some foliation on M, of codi- mansion q. The set C of Bott connections ([B3]) is then J -convex for J = the ideal of elements of dimension > 2q in the real smooth case ( r e s p . > q in the holomorphic situation or in the case of riemannian foliations ([M], [P])). (Be careful to the following convention: the dimension of an element in IS(G) is twice the degree of this same element as a polynomial on the Lie algebra of G.)

Let V be a smooth manifold, neither assumed to be complex nor compact for the moment, Z a closed subset of V, and U a "regular" neighbourhood of Z, i.e. an open neighbourhood admitt ing Z as a deformation retract (the existence of such a neighbourhood being the sole assumption on E).

Let E -+ V be a smooth vector bundle with structural group G, and let J be a homogeneous ideal of I~(G). A necessary condition for the existence of an J - connection on E is, of course, the vanishing of J ' ( E ) , i.e. the vanishing of all real characteristic classes ~(E) for ~o E J .

L e m m a 2. I f C denotes a J-convex set of connections on EIv_~, denote by w any connection on E l y_ ~ belonging to C, and by wo any connection on EIu. For any

E J , the element (0, A~o(~o), A ~ o ( ~ ) ) is a cocycle in the relative Mayer-Vietoris complex MV~(14, V - ~). Its cohomology class in H*(V, V - ~) depends only on C, and not on the particular choices of w and wo.

This relative cohomology class will be called the residual characteristic class of E (with respect to C and relative to ~), and will be denoted by ~o(E,C). Its natural image in H*(V) is obviously the usual characteristic class ~(E) .

L e m m a 3. Let C1 and C2 be two different J-convex sets o/ connections on E ] v - z . Let wl E C1 and w2 E C2. For any ~ E J , the form A~,,~ 2(~o) is a cocycle in the de Rham forms on V - E. Its cohomology class [A~l~(~o)] e g * - l ( Y - E) depends only on C1 and C2, and not on the particular choices Of Wl and w2. This cohomology

class vanishes/fC1 NC2 • 9. Its image under H * - I ( v - ~) o H*(V, V - E) is equal to the difference ~o(E, C2) - ~(E, C1). This cohomology class will be called the residual characteristic class of order 2 of E (with respect to C1 and C2, and relative to ~), and will be denoted by ~(E , C1, C2).

R e m a r k . Because HdimV-l(V, V - ~) = 0, the map HdimV- l (v ) --+ --~ Hd imV- l ( v - ~ ) is injective, so that the higher order residue ~(E, CI,C2) lifts naturally to HdimV-I(V) if ~(E, C2) - ~ (E , C1) = 0.

The proofs of the Lemmas 2 and 3 are straightforward, using 1) So t t ' s formula ([BAD:

r

d o A~o~a...~ = ~"(--1)JAwo...~....~ , J

j=O

RESIDUES OF HIGHER ORDER AND HOLOMORPHIC \'ECTOR FIELDS 113

2) the fact (see [L], Section 3) that the long exact sequence of the pair (V, V - ~) is induced in cohomology by the short exact sequence

0 --+ MV*(Sr V - E) -+ MV*(IA) -~ Q*DR(V -- ~) ---+ O.

3. L o c a l i z a t i o n o f R e s i d u a l C h a r a c t e r i s t i c C l a s s e s o f O r d e r 2 fo r a

P a i r o f H o l o m o r p h i c V e c t o r F i e l d s

Let X1 and 22 be two holomorphic vector fields on V (complex dimension p, not necessarily compact) , and ~.. their common singular set (or, more generally, the union of their singular sets).

Denote by Z a smooth singular 2p - 1 cycle in V - ~., and by

(C~(Xl, X2), Z> = f z a~, ~ (C~)

the evaluation on Z of the residual characteristic class of order 2 CI(X1, X2) associated to CI ([II = P, Wl and w2 being connections near Z), special (as defined in Example 1 of Section 2 above) relative to XI and X2, respectively.

We want to give an explicit formula for computing this number, "localizing" it near the intersection F N Z where F denotes the (analytical) set of points where Xi and X2 are colinear. Denote by (Wh)h the family of connected components of P N Z, and by 7~ a (2p - 1)-dimensional submanifold with boundary of Z, containing Wh in its interior.

P r o p o s i t i o n 1. (CI(X1,X2),Z) .= O, if X1 and X2 are linearly independent on (hence near) Z.

( c , ( x l , x 2 , z > = ~ h ( f ~ a ~ , ~ ( c , ) - fo~ Z~o~,~.(c:)), where ~0 denotes any connection near O'7"h simultaneously special for X1 and X2 (such a connection always existing there).

If X1 and )(2 are linearly independent near Z, then wo may be defined everywhere near Z, and one can take wl -- w2 (-- wo), hence the first part of the proposition.

In the general case, let W denote the covering (V - • - P, N) of V - ~, where N denotes any neighbourhood of F - ~ in V - ~ containing all 7~. Then ~1 and ~2 may be defined on the whole V - ~, while wo may be defined on V - ~ - F. Hence, by general machinery relative to the Mayer-Vietoris complex MV*(W), one has: D(A~,o~,,~,~ (C~), 0, 0) = (A~,~,~ (CD, A~,,~,~ (C~), 0)- (0, A~,,~,~ (CD, A~,o~,,~ (CD). (No- tice that A~o~ , (Ci) = 0 (rasp. A~o~ , (CI) = 0), by Lamina 1, because wo and Wl (rasp. ~vo and w2) are both special for X1 (rasp. X2)).

Hence (0, Aw~= (CI), A ~ o ~ 2 (CI)) is a cocycle in M V ' ( W ) cohomologous to the image in MV*(W) of the de Rham form A~,,~2(CI ). Integration on MV*(W) com- pletes the proof of the proposition.

R e m a r k s . 1) This .Proposition 1 allows an interpretation of these higher order residual characteristic classes: (CI(XI,X2), Z>(resp. fTh /kw,w2 (CI) - fo~ A~o~i~2 (CI)) measures in some sense the linear dependence of X~ and X~ nearZ (resp. near Wh). 2) In f~ct, the relative cohomology class of (0, A~,,~,~ (O~), A~,o~,,~, ~ (O~)) in

114 D. LEHMANN

H*-I (V - E, V - E - F) is itself well defined (i.e. independent of the choice of the connections w0, wl and w2).

A particular case of the above Proposition 1 is the following: Denote by p~ an isolated point of E, by S a a small (2p - 1)-dimensional sphere around p~ not inter- secting E and not containing any point of E other than p~ in its interior, by (/3~)h the family of branches of F through pe (F is supposed to be a singular complex curve without any other singular point near p~ than p~ itself), by W~ the intersection B~ I-I Sa assumed to be a circle, by Th a a (2p - 1)-dimensional submanifold with boundary of Sa, containing W~ in its interior, and by CI(Xr the Baum-Bott index of Xc (with c = 1, 2) at pa. Then:

P r o p o s i t i o n 2.

CI(X2 ,pa ) -CI (XI ,Pc~)=~h ( /T A~I~(CI) - ~Th /k~owlw2(Cf)),

where wo denotes any connection near OTh a simultaneously special for X1 and X2, Wl and w2 being special connections relative to X1 and X2, respectively, defined near Sa or only near Th a.

The proof is a straightforward application of Proposition 1 and Lemma 3. We will assume, from now on, that F is a complex singular curve (this is the

generic situation!), without singularity off E, whose branch through Wh is denoted by /3h. We will assume also that Z is transverse to each branch/3h, such that each intersection Wh --- I3h N Z is a circle. (Notice that it is always possible to move Z for realizing this transversality condition without changing its homology class in V - E.) We will make now the following

H y p o t h e s i s (*). For every h, there exist complex coordinates (x, y2 , . . . ,yp) defined near Wh, such that A1A2 r 0 on 7h, and B~ = 0 on Wh, where = A c s + E =2 = 1, 2).

T h e o r e m 1. Under the Hypothesis (~'), we have:

/p~ CI(Mh) dx A dy2 A ' ' ' A dyp z > = - ,

h ,. . . ,p

where we used the following notations and conventions:

= A - - ; - A - 7 '

1 D(A1,B2, . . . . . . ,B~) 1 D(A2,B 2, ,B~) Mh ---

A1 D(x ,y2 , . . . , yp ) A2 D(x ,y2 , . . . , yp ) '

the integration being made on a real p-dimensional closed submanifold h R2,...,p p h ~ = 2 R~ of 07 "h defined in the following way: since X1 and X2 are linearly inde-

pendent on OTh, one at least of the functions Z~ does not vanish on this manifold; therefore the open sets O~ in 07"h defined by Z~ ~ 0 make an open covering r of OTh; we have taken for (Rh, . . . , R h) any system of cells adapted to this covering, such as defined in (ILl, Section 1) under the name of ~syst~me d'alvdoles".

Proof of the theorem. We will make use of Proposition 1 above. We will denote


by the same symbols the connections and their representative forms (with matricial coefficients) with respect to the trivialization (~0;, ;F~'y2,""" , ~ ~ ~ ) near W h. Let Jc =

D(Ac,B~,...,B~) (C = 1, 2). By the definition of special connections, we have: wc(Xc) = D(x,y2 ,.--,y.) - J c �9 But, since Ar ~ 0 on the integration domains, there is no inconvenience to

0J 0 choose wc such that c(~-~v~) = 0, which means that these connections are both

special for any ~ , hence: wc = - - ~ J c d x .

L e m m a 4 . Awl~ 2 ( C I ) - - 0 f o r [ I [ = p .

In fact, the connections wl and w2 are both special for the same F~;" (Actually, because the connection forms are all containing dx, the conclusion of the lemma remains valid for every I such that [I[ >_ 2.)

From this lemma, and from Proposition 1, we can deduce, for this choice of connections wl and w2 :

<c1(xl,x~),z> = - ~ [ ~ o ~ , ~ ( c r ) . ~Jah "7"h

To complete the proof of the theorem, it is therefore enough to prove :

f = f J0 J~ P Th rq, ..,, I-i~=2 Z~

This is an immediate application of Theorem 5 of Section 6 in [L], once we have proved the

Lernrna 5. The forms A~o~1~2(CI) on O%, and Cl(Mh)dzAdy2A...AdYPri~=2 Z~ on the in-

tersection of the open sets O~ in O"]-h, are cohomologous, when both are seen as belonging to the total Cech-de Rham complex CDR*(O) associated to the covering O.

0 b 0 Proof. Near O~, the vector fields XI, X2, ~ , " . , b-~-y~, ' " ", ~ are linearly indepen-

dent; therefore, there exists a connection w~ near O~ which is special simultaneously for X1, X2, and all O (~ # A)'

For any k-simplex L = (10"-'lk) in the nerve of O, write:

AO,1,2,L(CI) --- A~o~2~o...~;k (CI), and

A~j,L(C~) = A~,~j~;o ~; (C~), (i,j e {0,1,2})

and define 7 E CDR2p-3(O) as the family (3,L)L given by:

7L = ( -1 ) [ 2 ]A0,L2,L , where k denotes the dimension IIL[[ of L.

Then, the total differential D0' of 3' in" CDR* ((9) is given by:

k

o3" = _ + + ( - i )

k

-I- ~ (--I)[~]+'~A0,1,2,(L_t,,)

116 D. LEHMANN

= - + f o r I IL I I >__ I ,

and DVz = A1,2j - A0,2J + Ao,I,I -- A0,1,2 for ]ILII = 0. Bu t AI,2,L ---- 0 except for L = (2, 3 , - - . ,p) , i.e. except for ILl = p - 2. In fact,

for ILl < p - 2, there exists some X such tha t all connections w l , w 2 , ~ (l e L) are s imul taneous ly special for 0

A0,1,L = 0, because wo,wl,w~ (1 E L) are all s imul taneously special for X1. By a similar a rgument , A0,2,L ---- 0.

Hence, in the compu ta t ion of DO', all te rms vanish, except --Ao,z,2 in each expres- sion (DO')I, and

(--1)[:~]+ZA1,2,(2,3,...,p) in (D0')(2,3,...m).

Therefore , -Ao, l ,2 and (-1)[~]+lA1,2,(2,3,...m) are cohomologous in CDR~'(O). I t remains to prove:

L e m m a 6.

"] -CI(Mh)dx A dy2 A . . . A dyp. ( -1 ) [5 A1,2,(2,3,...,p ) = 1-I~ Z~

Recall the formula wc -- - AAT Jr dx (c : 1, 2), hence w2 - wl ---- M dx. On the other hand, the condition, for w~, to be s imul taneously special for X1, X2,

and all a (Iz r A) implies:

1 (_T~_(_B~jI+ B~3"2)dx+ Mh d y e ) .

T h e affine combinat ion D of these p + 1 connections near (C1~=2OA) x AP is then equal to

for some QA, with (tz + t2 + ~A t l ) - 1, and tl , t2, t~ all _> 0. I ts curva ture can be wri t ten:

dt~ A dy~ ~ M ~--= (-as1 JI-d$2-~2J2-}-~dt'AQA) Adx-l-(~ ZA ) hq-QAd:r,, A

where Q does not contain any t e rm in dr. Since dr2 -- - d t l ( m o d u l o dt~), and from the fact tha t there are only p - 1 distinct d/~A, Ac~(CI) = CI(~,[2,'" ,~) can be wri t ten:

-F someth ing with at most p - 1 te rms in dr.

By integrat ion over AP, and using the equali ty

, at1 Adt~ A . . . Adt; = p!


one gets:

1 [~]A -CI(Mh)dx A dy2 A ..- A dyp. ( - - ) 1,2,(2,3,.-.,p) "= I'IA ZA

[]

4. C a s e o f a C 2 - A c t i o n

In this section, we will assume that a complex Lie group G, of complex dimension 2, acts holomorphically over V, that X1 and X2 are the fundamental vector fields Xhl and Xh2 associated to two elements hi and h2 linearly independent in the Lie algebra of G, and that each branch /3h is smooth. (Take for instance G = C. 2 ; we will say in this case that X1 and X2 arise from a C2-action.)

Assume, more generally, that the vector fields Xl and X2 are tangent to the branches Bh, and that there exist, for every h, complex coordinates (x, Y2,'" ' , Yp) defined near Wh such that the branch Bh may be defined by the equations y,~ = 0 (2 < A _< p) . Suppose Z small enough to be included in the domains of all these charts (this will often be possible in the case, for instance, where p~ is an isolated point of ~, and Z = S~).

It is now possible to choose 7~ and the cells R h such that P~,...,p fibers over Wh by the map (x, y) --+ x, with (iv - 1)-dimensional tori as fibers. The integration f/~ ... p

is then equal to the composition fwh ~ 9 c, where ~ denotes the integration along the

fibers. Therefore, fp~,...,p ~#dxAdy..A...Adyp is just the integration along Wh of the usual

[ ~ d y 2 A - . . A d y p ] Grothendieck residues Z2, Z3," ", Zp along the fibers at the various points

x 6 W h .

T h e o r e m 2. Let Pa be an isolated point of ~, and S~ a small (2p - 1)-sphere around it, as in Proposition 2. Then, for Ct equal to cp, one has:

<c,(Xl,X2),pol = E ( z ( x 2 , • h

where I(Xc, B h) denotes the indez at p~ of the vector field Xc restricted to Bh~.

Proof. Denote by

[ det(Mh) dy2 A. . . A dyp ] Z2, Z3,''',Z,

the Grothendieck residue along the fiber over x E W h. Since Zx vanishes along W 2, there exist functions M~ such that Z~ = ~ = 2 M~y~,. Notice that M~(x, o) is also equal to the (A, #) coefficient of the matrix (Mh)(~.o), while the (1, A) coefficients are

all zero at (x, o) for 2 _< A _< p, and the (1,1) coefficient is equal to , where we

have written f = -~. Therefore, we may write det(Mh) = ~ ?' det(M~) + ~ , yuD~, for some functions Du. Hence, by standard properties of the Grothendieck residues,

[det(Mh) d y 2 A ' " A d y v ] [ L dg2A" A ] Z2, Z3, " " Zp = f .. dyp ,

' z Y 2 , Y 3 , " ' Y P z

118 D. LEHMANN

and

f~ CI(Mh) dx A dy2 A " - A dyp = (_2iTr)p_ 1/w$ d_f ~...,. I-I~=2 z~ f "

The theorem follows, since ~,l~fw~ d_fi = I(X2, ~ha) -- -]'(Xl, ~h) (observe that X2 =

fX l along Wa h, and that det(Mh) = (-2iTr)Pcp(M)).

N o n - d e g e n e r a t e cases

It is easy to check that, for any holomorphic vector field Y, the restriction of

~2[X2, Y] --~1[X1, Y ]

to /3h depends only on the restriction Y[6h of Y to /3h, and depends linearly on it (with respect to the holomorphic functions over /3h). Let Oh denote this linear endomorphism. The matrix which represents Oh with respect to the trivialization ( ~ , 0 o ~ , ' " , 3~vp) is precisely Mh(restricted to/3h). This proves that CI(Mh)[bh has an intrinsic meaning, independent of the choice of the local system of coordinates (x, Y2, �9 �9 �9 Yp).

Furthermore, since X1 and X2 are tangent to /3h, the operator Oh preserves obviously the submodule of vector fields tangent to /3h, hence passes to the quotient and defines a linear operator O~ on the normal bundle to ~h (represented by the (p - 1) x (p - 1) lower right block M~ of Mh).

We will say that the pair (X1, X2) is

normally non-degenerate for Bh, if e~ is invertible along Bh, totally non-degenerate for Bh, if Oh is invertible along Bh.

T h e o r e m 3. If the pair (X1, X2) is normally non-degenerate along Bh, then:

/R~h CI(Ma) dxAdy2A ' ' 'Ady" ( - 1 ~ p-1 ,...,, I'I~=2ZA = \2izr] f ('~'''''~dx'\ detM~ J

This is a consequence of the formula f~ . . . p = fwh o :~: since ZA vanishes along

Bh, it can be written: Z~ = X~ M;yv, where M~(x, O) = (0~,~ = -(A,/1) term k Y ~ / y = 0

in the matrix Mh. r l" C'(M) ~ d A = ( ~ ) Therefore, T k ~ z ~ J Y2 "'" A dyv (~_~)p-1 , by a standard computa-

h tion of the Grothendieck residues along the fibers.

T h e o r e m 4. If the pair (X1, X2) is totally non-degenerate along Bha, then:

f dy2 A . . . A -lfw Cl(Mh) df CI(Mh)dXA . _ d y p = P~...p I-[~=2 Zx 2i~r h cp(Mh) f '

where f denotes the non-vanishing holomorphic function over I3h such that X2 = fX1.

In fact, cp(Mh) = (f--~)v det((Oh)), and det((@h)) is equal to ~det(O~), since

is an eigenvector field of Oh over Wh, corresponding to the eigenvalue -~.

RESIDUES OF H IG H E R O R D E R AND HOLOMORPHIC V E C T O R FIELDS 119

5. E x a m p l e s

i) Ac t ion o f C 2 on CP(p): Let (a0, a l , " - , a p ) and (b0, b l , ' . - , bp) be two families o fp distinct complex numbers (al ~ aj and bi ~ bj for i ~ j) , such that, for any 3 distinct indices i, j, k, the determinant ( ai - ak ) ( bj - bk ) - ( aj - ak ) ( bl - bk ) is different from 0. Denoting by IX0, X 1 , " . , Xp] the point in CP(p) with homogeneous coordinates (X0,Xl,"", Xp), let us define an action of C 2 over CP(p) by writing:

(Z, Z / ) [ X o , X l , . . . , Xp] = [ea~ eb~ ealZeblz'Xl,..., eapZeb~ Xp].

The corresponding fundamental vector fields ~ and ~z' may be written in the affine open set Xo ~ 0 (denoted by U0), with affine coordinates xi = x~ (1 g i < p):

P ~ b P co --co = - a0) , , a n d =

coZ i=i i=i

In U0, these vector fields vanish only at the origin, and they are linearly independent off the different axis B~ defined respectively by x j = O, ( j ~ i). The contribution of the branch B~ to the second order residue may be defined using coordinates (x l , x 2 , ' . ' , x p ) for (x, y 2 , ' " , y p ) : writing c~i -= ai - ao , t i bi - bo, and 5ij = c l l t j - ajfll (Sii = 0), one gets:

. _ _ 1 ZA = (~A1 xA and M 1 ----

alti Xl' altiXi

0 0 .-. 0 \ 0 ~2,1 0 0 : : : �9 �9

o o o ~p,~

Therefore, the contribution of B~ is -(2i~r)P 61(0'~2'1'~311'""6P'1) and alfl l$2,153,1.--sp,1 '

P C I ( 5 1 . ' " , & . ' " , ~ p 0 - - a . . . . - - - - ,

i=1

(Notice that we get 0 for CI = cp; this is of course not surprising, since the non- degenerate vector fields ~ and ~z, 0 have the same index 1 at the origin.)

Permuting the roles of 0 and i, the computation at the origin of each of the affine sets Ui defined by X i ~ 0 is exactly the same. (In this example, the pair (~z, 0 ~v) is normally non-degenerate (but totally degenerate) along each branch at each singular point.)

2) C o m m u t i n g l inear vector fields in CP: From the preceding computation and from Proposition 2, we get:

T h e o r e m 5. Let ~ be a symmetric polynomia l in p variables (p >_ 2), homogeneous of degree p, with complex coejficients. Let (al, a 2 , ' " , ap) and (/31, f12," "' , tip)

120 D. LEHMANN

be two famil ies of p non-zero complex numbers, such that all differences ~ - -~-a~ are

also ~ 0 for i # j . Then the following (purely algebraic) identi ty holds~ :

~ ( / ~ l , ' ' ' , / ~ p ) ~(O~l, ' ' ' , ( :Zp) L ~(B~'i -- " ~ ' i ' ' ' ' ' ' ~ i -- -~')cxl

I IL i #i = I-ii=l O:i i=1 l'~j,jei ( ~ -- c~j)ai

p O p 0 Take in fact X1 = ~ i = 1 aixiO'-~7, and X2 = ~ i = t f~ixiO~T �9 If the a i (1 _< i _< p) depend now dif ferent iably (resp. ho lomorphica l ly ) on a real

(resp. complex) p a r a m e t e r t, we deduce from T h e o r e m 5, denot ing by ' the der ivat ive wi th respec t to t, and assuming all numbers in a denomina to r to be not 0:

C o r o l l a r y .

) = E : �9

-- i=1 l'Ij,j#i (cq )

3) Non-commut ing linear vector fields in C2: Let X1 = A ~ + B ~ and X2 =

C;9~ + Z;~-~y, where A, B, C, S denote l inear forms in (x, y) in C2: A ( x , y) = ax + a'y,

B ( x , y ) = bx + gy , C ( x , y ) = cx + dy , E ( x , y ) = ex + ely. The express ion Q = A E - B C is a quad ra t i c form.

Let us assume for ins tance t ha t Q has r ank 2: there exist l inear forms K and L such t h a t Q = K L . Hence, F is the union of the 2 complex lines K = 0 and L = 0. By a l inear change of coordina tes , we may assume t h a t these 2 lines are the axis y = 0 and x = 0, i.e. we may assume: ae - bc = O, a'e' - g d = 0 , and ae I + ale - b e / - Uc # O. Since ae - bc = 0, and because we assume fu r the rmore t h a t XI andX2 do not vanish b u t for x = y = 0, there exist non-zero complex numbers k, l, k' and l ' such t ha t the

( ) mat r i ces a b k and it have r ank 1. Therefore , we may use the �9 c e l c ' e '

a U a t C C t b b t e e t coord ina tes (u ,v ) defined by: x = ~ + ~ v = 7 u + T r v , y = u + ~ v = T u + F v , , O i x o _ ~ + l , y O . for which X1 = kx~-ff + k Y3"~ and X2 =

Near the b ranch W1 defined by y = 0, we m a y take (u, v) for the coord ina tes used in Theo rem 1, and we get:

( ) ( ) ( ) D ( k x , k 'y) a a' D( lx , l 'y) c d M1 = 1 0 ~ D ( u , v ) = b b' ' O ( u , v ) = e e' ' x 0 k l

:~ We have in fact the more general identity

~ ( ~ 1 , - . . , ~ ) ~ ( ~ 1 , . . . , ~ , ) x ~ , ~ " ' - ~ ' ~ - ~ )

[I,L1 r I L l '

where ~0 denotes a symmetric polynomial in q variables (q _> i) with complex coefficients, homogeneous of degree p (p integer > 2), (a l, a2 , . . . , a v) and (~I, 3 2 , ' ' ' , 3p) two families of p complex numbers, and (A1, A2,-. ' , Aq) and (p1,#2, '" , t%) two families of q complex numbers, all denominators being assumed of course to be not O. The proof of this, based on similar methods, will be published elsewhere.

RESIDUES OF HIGHER ORDER AND HOLOMORPHIC VE CT OR FIELDS 121

Z = y_(l' k I kk I

- ~ ) , and d u A d v - ( a t / _ b a , ) d x A d y . T X

The contr ibut ion of this first branch W1 to CI(X1, X2) at the origin of C 2 is then:

0 for 6 ' / = c 2 , k'(btl - e'k) 2

l ( a t / - ba')(kl' - lk') for Cr = ( c0 2.

The computa t ion for the second branch W2 defined by x = 0 is similar.

4) Non-linear vector fields in C 2, tangent to the branches : Take

X1 = ax k O + by I -~YO 0 n 0 and X2 = cxm + ey N' where a, b, c, e denote 4 non-zero complex numbers, and k, l, rn, n 4 integers > 1. Let us now s tudy what happens near the branch W1 defined by y = 0. We have:

:r b I--t n--1 . M1 = 0 la~Z2 - n e-y- - r ~,,

The pair (X1, X2) is normally non-degenerate along W1 if at least one of the 2 integers I and n is equal to 1 and if ae - bc ~ 0 in the case they are bo th equal to 1. It is total ly non-degenerate if fur thermore m 7~ k. We may therefore use the formulas of Theorems 2, 3 and 4. The contr ibution of this first branch W1 to Cx(X1, X2) at the origin of C 2 is then:

rn - k = I (X2 , W1) - I (X1 , W1) for CI = c2, and

[ a ( k - m ) + b ] 2 if k = I = l , n > 1, ] ab

2(k - m) if k 7 ~ 1, l = 1, n > 1, for CI = ( e l ) 2.

etc.

The computa t ion for the second branch is similar.

R e f e r e n c e s

[B1] BOTT, R.: Vector fields and characteristic numbers. Michigan Math. J. 14 (1967), 231-244.

[B2] BOTT, R.: A residue formula for holomorphic vector fields. J. Differential Geom. 1 (1967), 311-330.

[B3] BOTT, R.: Lectures on characteristic classes and foliations . Springer Lecture Notes 279 (1972).

[BB] BAUM, P.; BOTT, R.: Singularities o'f holomorphic foliations. J. Differential Geom. 7 (1972), 279-342.

ILl LEHMANN, D.: Rilsidus des sous variiltils invariantes d'un feuilletage singulier. Ann. Inst. Fourier 41 (1991), 211-258.

[M] MOLINO, P.: Classes caracteristiques et obstructions d'Atiyah pour les fibrils prin- cipaux feuilletils, et Connexions transverses projetables. C. R. Acad. Sci. Paris 272 (1971), 779-781, 1376-1378.

122 D. LEHMANN

[P] PASTERNACK, J.: Foliations and compact Lie group actions. Comment. Math. Helv. 46 (1971), 467-477.

IV] VAISMAN, I.: Symplectic geometry and secondary characteristic classes. Progr. Math. 72, Birkhs (1987).

D. LEHMANN GETOD1M, CNI:tS, UA 1407 Universit~ de MontpeUier II Case 051 Place E. BataiUon 34095 Montpellier cedex France

(Received July 21, 1993)

residues of higher order and holomorphic vector fields

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