:ERBNCEIC/91/3

INTERNATIONAL CENTRE FORTHEORETICAL PHYSICS

MONADS, CONNECTIONS AND COMPACTIFICATIONS

INTERNATIONALATOMIC ENERGY

AGENCY

UNITED NATIONSEDUCATIONAL,

SCIENTIFICAND CULTURALORGANIZATION

Hoang Le Minh

1991 MIRAMARE - TRIESTE

IC/91/3

International Atomic Energy Agency

and

United Nations Educational Scientific and Cultural Organization

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

MONADS, CONNECTIONS AND COMPACTIFICATIONS

Hoang Le Minh *

International Centre for Theoretical Physics, Trieste, Italy.

ABSTRACT

Using the monad construction of stable vector bundles and semi-stable torsion free sheaves

on the complex projective plan, we describe natural hermitian complex connections for the vector

bundles and interpret the sheaves on the boundary of compactifications of their moduli spaces in

terms of singular connections with isolated singularity.

1 IntroductionLet (X>Ojr(l)) be a non-singular projective algebraic surface with an ample line bundle,•MJJ = Mx (r, ci, ej)£ be the moduli space of ^-stable vector bundles over X with rank k , the

Chern classes ci and ej . It is known that M% is quasi-projective and can be compsctitied byattaching the S-equivalence classes of semi-stable torsion-free sheaves with the same invariants([12]). Denote the compactined moduli by Mx = Ax (r, ei, e2) . In ([1]) M. Maruyamaconstructed another compactification of M$ under the assumption :

Xu rational, = 0 and d[Kx,Ox(l)) < 0

where d is thp degree with respect to Ox(l), Kx is the canonical line bundle of X. This newcompactification, denoted by Mx , i« dominated by Mx and thus more economic. In thecase when r = 2 and c\ = 0, Mx is identified with the Donaldson analytic-topologicalcompactification ([10]) of the moduli spaces of anti-self-dual SU(2)- connections on the un-derlying real-analytic 4-manifold X (with a "general" Riemannian metric). In fact, due tothe results of Donaldson et al. (see [2,3] and references therein) the space Xx(2,0,cj)J isidentified with the moduli space Mx{n) of irreducible Einstein-Hermitian connections ona CM complex hermitian vector 2-bundle with the second Chern class n. Then Mx[n) istopologically compactified by attaching a subset of Uj}_, Xx(n - r f ) X SJ(X) , where Sd{X)is the d"1 symmetric product of X.

The aim of this work is to establish a relation between these compactiricatiotis, especiallyin the case X = CPS .

Firstly we shall construct an explicit mapping *: .Mjr(2,0,ci)£ >-» Mx(n) , using themonad description of stable vector bundles on CP S ([4]), and then give some interpretationof the torsion-free sheaves on the boundary of the compactification Mx in terms of singularhermitian connections with isolated singularities.

Notat ion and Conversion. For a coherent sheaf 7 on X, rk(T) is the rank, Ci(7)denotes the i** Chern class of f, W(J) = ITfX, 7), ti{7) = dim H*{7). We shall not distin-guish locally free sheaves and the associated holomorphic vector bundles.

MIRAMARE - TRIESTE

January 1991

Permanent address: Department of Mathematics, University of Hochiminh City, Vietnam.

2 Monads and connections

2.1

A complex of locally free sheaves and morphisms on C P " of the form

— 0 (1)

with the cohomologie* (0, T,0), is said to be a monad. This means that a = a[T) is injectiveand 0 - 8(T)\B surjective as sheaf morphisms, and

f = I ma

is the cohomology of the monad. The monad (1) haa a display, i.e. a commutative diagramof exact rows and columns

(2)

Let us fix some local holomorphic frames of the vector bundles, defined by the diagram(2) and consider the morphisms there as matrices (with the holomorphic entries), a» well asthe related hermitian structures. This means that if (e) = (ei,--- ,«,) IS the holomorphicframe of 7, then an hermitian metric h — hf \B defined in (<) by a non-degenerate hermitian(r x r)-matrix h(e) and if ( / ) is another frame, with j € GL(r,C) as the transition matrix,i.e. / = eg, then h(f) = g'h(t)g, where pt = ' j is the transpose of the complex-conjugatedmatrix. There exists an hermitian C°° complex connection V* on 7, which is defined uniquelyby the hermitian metric and holomorphic structure ([3]). The associated connection form andcurvature (l.l)-form in the local holomorphic frame are defined by the following matrices ofcomplex differential forms

0 -»

0 -»

0i

T-i - M

II i «7-1 -i fo

i0

0I

- 71

10

— 0

-• 0

(3)

Let be the homogeneous coordinates on

the standard affine covering and (xi = «j/zo,' = 1, • • •, If) - the affine coordinates in UQ.Then for the holomorphic tangent and cotangent bundles of C P " , locally in UQ :

di ~ d/dxj and dx^ are the local holomorphic frames respectively.The curvature form if decompose* in these frames as ij = 2 i^dxidtj. With respect to

the hermitian metric hj one haa

BTC] = h-ldjdih - h'l3jh.h-xdih

The standard Fubini-Study metric J,J and the associated tensor j * ' are given in (Uo) bythe formulae (see e.g. [6]§3.3)

'i), I*! (5)t = 0

The Einstein-Hermitian condition for {7,hT) means that the following identity

where A = const, holds ([3]p.99)

2.3For an exact sequence of holomorphic bundles and morphisms

0 _ f JU 7 ±* 7" — 0 (7)

giving a hermitian metric h = hj on / , there are uniquely defined hermitian metrics h! andV on the subbundle 7' and quotient bundle 7" • Indeed let us consider locally <p and \ji asmatrices, whose entries are holomorphic functions, then the induced hermitian structures aregiven as

h' = <ph<p , h" = (V'h-V1 ' )"1

There exists a canonical C00 - splitting of (7) by the C°° - morphisms

such thatp'l^1 — 0 , ip'ip = id ^' = idjn

= 0 ([6]§2.4). Explicitly in theand it is uniquely defined by the orthogonal relationlocal Frames one has

v' = h'-V*, $ = h-VvSplit all the rows and columns in the display (2) by the canonical way as above and denote

the splitting C°° -morphisms with the prime mark. As it is not hard to see, there exists aC°° - injective morphism <fi : 7 —» 7® (not holomorphic !) and its associated $' : ̂ ) —• 7which are denned uniquely and puts 7 into 7b as a direct C°° subbundle with respect to ho-Then there are the following relations between the morphisms

= 0 (8)

Suppose ho = const. Then one can compute explicitly all the related hermitian structures,which are induced on t.ie bundles of (2), locally in terms of the matrices of the morphisms.Denote the induced metrics on 7-\ and 7\ by h_i and h\ respectively. Then, in particular,the curvature of the cohomology bundle 7 is given by the following formula

Proposition 2.1 (|7]Prop.2.l)

Remark. After some modifications one can derive the general local formula for the curvatureform of the induced connection of a general monad of the form (1) with respect to the non-flat(i.e. ho * const) hermitian connection on To as follows

2.3

The monad (1) is said to be normalized with ct = 0, if

JJ °i F,- ® 0(i) , where F; , t = —1,0,1 are finite dimensional complex vector spaces

dim F_i = dimFi < dimFoWe shall describe this clua of monads more explicitly by homogeneous coordinates and

in the standard affine covering (4). One has

0 = 0(7") €Hom(/_i ,&) 2 Hom(F_i,F0) ® H°(CPN, 0(1))

= ££L0A,*j where Aj eHom(F_i,F0) and z, form abasia of V* = H°{CPN, 0(1)).Let Idn denote the identity (n x n)~matrix

7 *" , A = (Ao,---,AN)

the column and row block matrices, associated with the morphisms 7-\ —* F_i ® V ® 0 andF _ i ® V ® 0 —»Fo®O respectively.

Then a can be given by the matrix a = AZ, Similarly

N ( Bo \0 = £ B , * | = lZB, where B= : , Bb e Hom(Fo.Fi)

i=o \ 9* )

Let us fix some local holamorphic frames of the bundles and describe the induced hermi-tian structures by the associated hermitian matrices. Since

Z , hi =

it IB clear that the formula in the Prop.2.1 becomes

(9)

Now we can compute the Einstein condition explicitly in the local afline subset Uo, usingProp.2.2 and formula (S)

Proposition 2.2

1*1

Proof. Notice that dtZ = Id™ on the i'*-row ), thus keep in mind that $a = 0

and a'ho<f> — 0 one has

N

••.1=1

1*1* - " : IA] ),N N

»--\A]•=1 1=1

1=0

Analogously for the other half

' ( ) ^ = J ^ * ' ( )

Therefore our formula is proved. Q.E.D.Remark. From the Prop.2.2 one can prove that the local formula of the Eiustein condition

does not depend on the choice of the standard affine subset. In other words if in Uo one has<(V") - Aid? , A = const, then the identity holds also in all U, from the standard affinecovering of CPN .

2.4

We begin this paragraph with a lemma from the theory of matrices

Lemma 2.3 If P it a non-rinjuiar complex matrix matrix, then there exists a non-singularmatrix of the form X = f(P) ,viktrt /(A) t» a polynomial with the complex coefficients, suchthat <J = P

For tae proof see [13],ch.V.

Corollary 2.4 Let P - Q* Q , when Q .s a non-singular complex symmetric or skew-symmetric matrix and X as in LcmmaS.S. Then X it hermitian and one has

QXk = lXkQ , for all k = 0 ,± l ,±2 , - • • (10)

Proof. The fact that X is an hermitian matrix is obvious. Moreover 'X — /(QQ^) thus

QX = Q/(QtQ) = ^ fkQ(QtQ)k = ^ f ^ g g t ^ Q = /(QQt)Q =* XQ

From that̂ there follows the identity (10) for k > 0. Now it ia easy to verify the followingequalities

QX = Q.X-'.X2 = = 'XQ =* = 'X

Similarly one has QT( *X) ]Q = X. The following equalities are verified immediately

QX-' = 'X(qt)"1 = 'X(qt)" lQ-'Q =

Therefore (10) is true for all it < 0. Q.E.D.Remark. If P and X a* in COT .2.4, then obviously for every matrix Y of the form Y —

j(X), where g(X) is a polynomial (over a field), one has

«t 'YQ = XYX

Let us return now to the expression, determining the Einstein condition in the Prop.(2.2).Suppose that there is a normalized monad, satisfying the following condition on the mor-phisms a and S in term* of the associated matrices A and B

(*) A and B are non-singular, BA is skew-symmetric

Then we have the following theorem

Theorem 2.5 Let a given normalited monad, satisfy the condition (*). Then tkere it ahtrmitian metric ho = contt on 7Q tueh that the naturally induced hermitian connection onthe cohomology bundle 7 it Einttein-Hermitian.

Proof Let us denote h = A^hoA and Q = BA, then obviously Bh^lB^ = Qk-'Q* and theright side of the formula in Prop.(2.2), related to the Einstein condition becomea

df 'ZQh l (11)

Therefore in order to prove the existence of a Einstein-Hermitian metric on 7, it is sufficientto find some hermitian matrix A, satisfying the matrix equation

h(zthZ ® = 0

Now we can use the Cor.2.4 to prove that it is the case.Indeed the matrix (Z'hZ ® Id/v+i)"1 can be represented as a polynomial of its inverse,

i.e there exists a polynomial R(A) (with coefficients in the field of rational complex functions)such that

{Zhz ® Idtf-u)-1 = R(Z^hZ) & IdN

Set h = ^ . Let us consider the matrix D = ^^(Z^hZ) Since

(AIdB - D) »Idjy+i = W ' f ^ A I d v + i - H)Z) ® Idw+i

the matrix D ® Id.v+i can be represented as a polynomial of the matrix H. Thus one canuse the remark after Cor.2.4 to verifv that the equality (7) holds. Q.E.D.

Remark. Because every nun-correlation bundle £ on on CPim+i = P{V) is the coho-mology of a normalized monad of the form

0 — 0 ( - l ) — V — 0(1) —. 0

which u defined by a non-degenerate symplectic form on V = C 3 m + ! (see [4]), it is easy toprove that there is a naturally Einatein-Hermitian metric and together with it an associatedEinstein-Hermitian connection on f.

3 Stable vector bundles andtorsion-free sheaves on CP2

3.1 Preliminaries

Let F be a coherent sheaf on C P ! . Letcharacteristic. For 7 with rk{7) > 0 set

x{7) = TiZoi-l)'h*{?) denote the Euler

The Riemann-Roch formula for 7 has the form

where P(X) = 1 + | X +

X(7) = r(PW - A)

is the Hilbert polynomial for the trivial rank 1 sheaf 0.

Let 7' — )iom(7,0) and 7" - double dual to 7• The canonical homomorphism 7 —> 7"extends to the exact sequence

T(7) 7" (12)

Then 7 is torsion sheaf «==> 7 = 1(7) and 7 is torsion-free <==• T(7) = 0.A torsion-free sheaf 7 is said to be fi-»table (n-tcmi-ttable, ttable, temi-etablc respectively)

if for any torsion-free sub»heaf 0 5 7' S 7 with torsion-free quotient 7j7', one has

< n{7) (an < tin. Kn < w)« an =< tt{7) or ti{D = H{7) ic A(7') > i ( T ) respectively)

A sheaf has zero rank if and only if it is a torsion sheaf.For a torsion sheaf £, M(£) = h*(£) = 0 and C = 0 (see below).

A torsion-free rank 1 sheaf is by definition ^-stable. Any non trivial torsion-free of rank1 sheaf with c2 = m has the form J((D), where D denotes an effective divisor of degree ej,£ = xx + • • • + xm is a 0-cycle ([5] p. 120). Hence if 7 \» torsion-free of rank 1 and ci = 0,one has e3 > 0 and 7' — 0. There is a birational morphism of algebraic schemes

If 7 is semi-stable of rt > 1, then there exists a Jordan-Holder filtration

such that gri(7) = 7i/7i-\,i = ! , • • • , ( » stable with the same invariants.The sheaf

does not depend on the choice of the filtration. Two semi-stable sheaves 7 and 7' with thesame invariants by definition are S-equivalent, if gr(7) — gr[7').

Let Rx denote the moduli space of semi-stable sheaves; its points represent the S-equivalence classes of semi-Btabte sheaveB. Mx has a structure of an Hausdorff compactcomplex variety; X, Mot >C, M£ are open subspaces of Mx , consisting of points cor-responding to stable sheaves, stable vector bundles, /j-stable sheaves and /j-stable vectorbundles respectively.

8

Finally, for any non trivial, semi-stable torsion-free sheaf 7 with

rk{7) - r , ci(?) = 0 and ej = n

then h°(/) = 0 and by the Sene duality h*{T) = dim(Hom(7,0(-3)) = 0 ([11]). TheRiemann-Roch formula gives x(7) = -^{7) = -n + r Hence the necessary condition for 7to be semi-stable and not trivial is ej > r. Therefore, if 7 i* of rank 1 torsion-free and nottrivial, then cj(/) > 1.

On the other hand Mx(r,Ci,ci)S ^ 6 ,r > 1 •<==>• cj > r > 1.

3.2 Semi-stable ibeavei and monads

It is well known that the category of coherent sheaves on CPN admits a description (bymeans of the derived category theory) in terms of complexes of simple type (Beilinsnn spectralsequences, see [4]). We would like to explore it to investigate the Btable vector bundles andsemi-stable torsion-free sheaves on CP1 , where many of the given constructions can becarried out explicitly.

Consider the category of the monads of the form (1) on CP3 , which is an abeliancategory with naturally defined sub-, quotient objects and morphisms . Let

rk(7) *£ rk{70) - r*(*-l) - rfc(ft) , HI C l ( / 0 ) -

Definition 3.1 ([11]) The monad (l) is said to be (semi-)stable if for every sub-monad0 * T * 7. one has

and in the case of equality.

) < x{7.)/rk(7.) respectively).

In the following we are interested mainly in normalized monads, i.e. monads with the zero firstChern class of the standard form (aee Ij2).

Proposition 3.2 If a normalized monad on CP1 is semi-stable, then the sheaf morphisma(T'), contidtrti a* a vector bundle morphitm ie injectivt exactly outside a finite number ofpoints. The cohomology of a (tcmi-)stabic normalized monad is a (scmi-)ttaHc tortio t-frctsheaf with ei = 0. Conversely, every (semi-)stable torsion-free sheaf with zero first Chernclass is the cohomology of a n ormalized (»tmi-)»tablc monad .

The proof of this Proposition is similar to that of ([ll]Prop.2.3), with some minor modi-fications which are easy to be done.

Remark. The singularity set of a semi-stable monad, i.e. the set of points where themorphism a doe* not have maximal rank, coincides exactly with the singularity set ([4])SingfT) of its cohomology 7. Outside these points, 7 is locally free and can considered as aholomorphic vector bundle.

Let us consider now a normalized monad, associated with a semi-stable sheaf 7 with thezero first Chern class.

If Sing(7) = t, i.e. 7 ia locally free and represents a holomorphic vector bundle, thenevery hermitian metric ho — hja on the middle term of the monad induces a natural hermitianmetric on / and thus endows 7 with a hermitian connection.

In the case Sing(T) f 0 one can define a natural connection on the T", which whenrestricted on CF1 \Sing(7) gives rise to a singular connection (with isolated singularityin Sing(T)). Our aim is to examine the associated Einstein condition and behavior of thesingularity.

The following lemma lists some standard facts, concerning cohomologies and the totalChern classes, associated with the torsion-free rank 1 sheaf Jm and ihe torsion sheaf Ot of apoint x on CP1 .

Lemma 3.3 a)

» i l i l

h°(Oa(k)) = 1 , hl(O,(k)) = = 0 , V k

{23 ifJfc>0

- i t ' - 3 iffc<o

if * > od)

e)c(Ot(Jfc)) = 1 - dt2 , c(J((k)) = 1 + kt + dt1

where t is a generator of the cohomology ring H*(CPS,Z) = Z[t]/t*.

Corollary 3.4 Suppose £ denotes a simple O-cycle of degree d > 0, i.e. a set of d distinctpoints on CP2 . Then

h°(J((k)) = 0 ,

A°(Jt(*)) - ftVf(*)) =

if k < -2

ha{Ot(k)) = d, hl(O((k)) = h*[O((k)) = 0 , Vk

10

= ° if

Proof . BecauBe 0([k) = © j ^ Z i , where *j,t = 1,2,- •- are points on the support of £ and

O (13)

the formulae in the corollary follow directly from that of the Lemma.Now the standard Koezul complex, associated with a point on CP1 has the form

0 -^ 0 ( -2 ) -» 0 ( - l ) $ 0 ( - l ) — 0 - • 0 , - 0

and splits into the two short exact sequences, one of them is (13) with Jfe = 0 (for one point)and the other is

0 -> 0 ( -2 ) - 0 ( - l ) e 0 ( - l ) -> Jx - 0

The Serre duality and Bott formulae for standard cohomologies on CPS

if k > - 2

if Jfc < - 2

0 if oi

then given the desired formulae immediately.The equalities for the total Chern classes are verified in the similar way. Q.E.D

3.3 Quasi-bundles, quagi-trlvlal sheavesand quasi-equivalence

The following notations are due to Tjurin ([5])

Definition 3.5 A torsion free sheaf 7 on a complex surface X is said to be quasi-bundle ifit fits into the followir g exact sequence

0 - 7 ~* £ -» £ - 0

where £ is a locally free sheaf, £ ^ 0 is a torsion sheaf of codimSupp{£.) = 2 withoutnilpotents. (e.g. the structure sheaf of a simple O-cycle and its tensor products with lintbundles)

A quasi-bundle 7 is quasi-trivial, if £ at above is trivial.Two quaei-bundles are quasi-equivalent, if they are defined by the same £ and £.

The following Proposition follows immediately from the Definition.

Propoaitlon 3.6 If a quari-bundle 7 is defined by £ and £ as in Def.S.5 then one has

11

a) rk[7) = rk{£)h) \f 7 is quasi-trivial then C\{7) = 0 and cj(J) = c2(f) - cj(£)c) 7" ~ £d) Sing{?) =

Let us consider a quasi-trivial sheaf of rank r with ci = 0, cj = n given by

0 -» 7 - 7" - 0»r A

Q.E.D.

Where { = X\ + • • • + xn is a simple 0-cycle. The surjective morphism f is the direct sum ofthe local surjective morphisnu (3*r ^ 0,,(l) , which is defined uniquely by the linesHence the quasi-trivial 7 is defined uniquely by a O-dimensional cycle

in the product XxCP'"1, where £i = (li, kerpi) is defined up to the action of PGL(r - l).From the eohomologv formulae in the Lei i.3,3 it immediately follows that

Proposition 3.T Let 7 is a /i-rtablc qttasi-bttndle of rank r, ci = 0 and cj = n > r. Thenthere ii the following commutative diagram, related with the associated monads of 7 and£ — 7" and the Kotzut complex of 0(

£ : 0 -* R

7 : 0 — B

7- o - h

01

rO(£(-2))®0(-:1

r>(7(-2))«0(-:I

r1(f(-2))®0(-:i0

0I

L) -» ^(xsn'jiso - iI

1) - . H1(7S>Cii)®0 — I

iI) - H*{£®tf)®0 -> I

I0

0i

*°(£(-l))®0(li

iP(£(-l))®0(\

i0

)

) - o

) - . 0

{14}

Q.E.D.Remark. Notice th»t the morphism /)(£.) is not surjective. In fact one has lmfi[£,) — J({1)and thus hl{£.) - 0((l). Hence a(f ) when considered as a vector bundle morphism is notinjective exactly on the support of £, i.e. £.

4 Monads and singular connections

4.1 Maruyama-Donaldgon compactif lcatlon

Let us recall the Maruyama construction of the new compactification for the moduli space ofstable vector bundles on a rational surface, in particular for the case of CP1 and zero firstChern class.

Let 7 be a semi-stable sheaf which C\ = 0. Corresponding to the identity endomorphismin EndfH'ff)) °L Ext^H1^) ® 0, 7) there is & "univewal extension"

7 - V{7) - Hl{7) 9 0

12

(15)

The sheaf U(7) is semi-stable of rank n and Chern classes ci = 0, c3 = n. If T and V areS-equivalent, then ( / ( / ) and U(F) are S-equivalent, too. Therefore it defines a morphismbetween the moduli apace* • : M[r,0,n) —> M(n,0,n). Moreover if 7 is a ^-stable vectorbundle and n > r, then U(f) is a stable, but not ji-stable, vector bundle and the map $,when restricted on M£ is an immersion ([1] Th.3.8). Then the new compactification, denotedby M, is the closure of the image of M% by *.

Denote by CM = M{r,0,n) \ M(r,0,n)g the corona, i.e. boundary of the natural com-pactirkation. We would like to describe the sheaves in CM, which go to the boundary of thenew compactification CM = M \ * ( M% ) and interpret them as complex connections withisolated singularity. For this aim we divide the corona, into the following disjoint subsets

C(»)fl = {T € M I T is semi-stable but not stable }CM M = {7 € M | 7 is stable but not /.-stable }

C^M = {?eM | f is -̂stable and Sing{7) yt 0 }

One has

Similarly one defines the C("'.M, C('^M and CMM. From the above, if n > r then

4.3 The rank 2 case

In the cue of rk(7) = 2, there is a rather explicit description of the corona as a set ofgeometrical points, namely

u • • • u S"(CP*) (16)CM(2,0, n) S MS(2,O,n - 2) x J'(CP J ) u XS(2,0,n -

We shall construct in a canonical way the semi-stable sheaves in CM, which go to the coronaCM under the morphism 4 and interpret them in terms of complex connections with isolatedsingularities.

First of all, by the Prop.3.2 any semi-stable torsion-free sheaf 7 of rank 2, ei = 0 isassociated with a semi-stable normalized monad of the form

where F-! = H1{7'(-2)), Fo = H' (7® fl1), F, = H ' ( / ( - l ) ) .If 7 is locally free, then (see [4]) there exists a non-degenerate symplectic form q : Fo •• *

FJ and an isomorphism u : F l j —> Fj such that

fi = « ° 'aoq

Locally in some appropriate holomorphic frames, the morphisms a, 0 and q are representedby the matrices A, B, and Q, where Q is a non-singular complex skew-symmetric and

B = 'QA

A direct calculation for the cohomologies of display (2) shows that

KerB = H°(?) = 0

so B is injective and A is surjective { A and B have maximal rank n).We shall treat the case cj = 2 and ej > 2 separately in what follows.

4.3 c, = 2

If a locally free ^-stable sheaf 7 £ M £ (2,0,2), then it can be given as the cohomology of anormalized monad (3.1) The matrices A and B in this case are square and thus non-singular.Therefore the construction of Sect .2 can be applied to yield a canonical Einstein-Hermitianconnection on 7 •

Note that in this case , M = M and C<'>W(2,0,2) = C("'>i(2,0,2) = 9, because everystable sheaf of rk 2, ei = 0 and Cj = 2 must be locally free and ^-stable (an easy exercise).

For the corona of the compacti Beat ion one has

Proposition 4.1 The canonical morphitm of "univertal" exttntiont

,2)) -.CAUC("'.M(2,0,2)

it bijectxve and realties the corona at the algebraic schema S1 ( CPJ ) .

Proof. Due to the Maruyama description of the image of 9, every sheaf in the image isdefined up to S-equivalence by the singular set of a rank 1 torsion-free sheaf J, which in thecase ci{J) = 0 is defines J uniquely. Thus * on M(l,0,2) is injective. The fact that * issurjective is obvious. Q.E.D.

H< nee, in order to interpret a sheaf 7 G CM, it is sufficient to look to noi trivial torsion-free sheaves J( of rank 1, where f = xi + Xj and take the universal extension of J( to obtainthe semi-stable rank 2 sheaf 7• Then one can, together with F, get an associated monad. Nowrepeating the construction of induced hermitian connections in §2 one can endow any suchBheaf with a singular hermitian connection, having singularities exactly on the singularity setf. To be precise

Definition 4.2 Let £ be a simple 0-cycle and 7 ; i singular connection (vntk valves in acomplex vector bundle E) having only isolated tingularittet in £, then V f is said to be havingsimple isolated poles in f, if for any holomorphic curve T, pasting through a paint i € *, therestriction of the connection form w | r onT is a complex vector-valued 1-form with isolatedsimp •• pole in z.

1 mo singular connections V( and V'(, with itolated simple poles are gauge equivalent, ifthey have the same singularity set : £ = f and there is a gauge transformation

<b € £{E) C Aut(E)

such that they are gauge equivalent by the gauge transformation <p

outside the singularity set £.

Proposition 4.3 There is an equivalence between the following categories :

a) The category of fi-stable torsion fret sheaves of rank 1, ei = 0, cj = 2.b) The category of quasi-trivial sheaves of rani S, c\ = 0,ej = 2 up to qvasi-equivalence.c) The category of couples { £, V() of a simple 0-cycle of degree 2 and a singularhermitian connections on a trivial rank t hermitian complex vector bundle E,having isolated simple poles on ( up to unitary gauge equivalence.

13 14

Proof. Without the loss of generality one can suppose that the points Xi and xt in £ iregiven by the homogeneous coordinates ( 1 : 0 : 0 ) and ( 0 : 1 : 0 ) respectively. Then thestandard exact sequence, defining £ has the form

0 -* J( - 0 - 0t = 0 , , €> C , - 0 (18)

By tensoring (IS) wilh (9(1) and considering the diagram of monads associated withthe "universal extension" exact sequence (15) of Jf, one se«« that the image 7 = *(^{) >sgiven by the following normalized monad

where a =

000

V o

and ft can be given by the formula /? = *Zqo where qo is the canonical

non-degenerate symplectic form on C* (see §2.3). The corresponding matrix A is of rank 4.It is not hard to see that a as a morphism of the vector bundles is not injective exactly atthe points x\ and i j -

Now 7 is a quasi-t'ivial sheaf, defined by the exact sequence

where the morphism (p is given by the pair of lines (*o = 0, ii = 0 ) . From the diagram (14)it follows that if one takes the canonical hermitian metric ho on C6 , the induced metric h on£ = 7" is degenerate of the form

h (

Thus the associated hermitian connection with the connection 1-form

has the singularity at the points Xi and z3.Suppose now T is aholomorphic curve, passing through n or i j . If w is a local parameter

on F near the point i\ or 13 then the restriction of u on T :

dwr = — •«•(•«)

where jr(w) is a C00 complex function and Jr(0) ̂ 0 (whereas u; = 0 represents the pointXi). Hence the singularity of the hermitian connection is isolated in £ and are simple poles.

Note that outside the singular set £, the hermitian connection satisfies the Einstein con-dition, as was mentioned before.

The rest of the proof is trivial so we omit it. Q.E.D.

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4.4 ej > 2

In this case the sets C^H and C^ M are generally not empty. Set

Ci*1' = {J € C^M | 7 has d distinct singular points, 1 < d < n ~ 2}

Then we have

Proposition 4.4 The boundary of the compactificution C.M(2,0,n) for n > 2 correspond* tothe following qttati-bundlrt

a) fi-atable quati-bundles from C|, , up to quari-equivalence.b) Stable quati-trivial theavtt from C'"' M with maximal n singularity points, tipto quasi-equivalence

Proof. For any 7 £ Cj1 , from the exact sequence in Def.3.5, which defines 7 as a quasi-bundle one has

T" = £, ei(£) = ei(7) - d(= #(Siiff(/)) = d and

= 0, Supp(C) = Sing(7)" is ^-stable with ei = 0 and cj = n - d.

This means that the /j-stable quasi-bundles from Cj exhaust all the geometric points of theform MJ (2,0,n - rf) x S f̂ C P l ), 1 < d < n - 2 on the corona CM under the "universalextension". Up to the quasi-equivalence these sheaves are uniquely defined (see above).

Now suppose 7 is a rank 2 quasi-bundle with maximum singularities and d = 0,ci = n.Then the image of 7 in CM under the map * corresponds to » point in the set Sn( CP ! ),By a result of Artamkin ([14] Cor.9.3) there exists a stable quasi-trivial sheaf in the quaai-equivatence class of 7• Thus the correspondence in part b) is proven.

Note that / carTjiot be /i-siable because 7 has already maximum possible singularitiesand 7" is trivial. Q.E.D.

Theorem 4.5 The semi-stable torsion fret sheaves in the boundary of the Maruyama-Donal-dson compactificntien CM for the moduli space XJ (2,0,n) on CP 1 , where n > 2 can beinterpreted as the coupled ( f, V { / where £ = x\ H xd i» a. simple 0-cycle of the degreed, 1 < d < n and V f t> a singular Einstein-Hermitian connection on a rank S hermitiancomplex vector handle urith Cj = n — d, having only simple isolated poles in £, up to unitarygauge equivalence.

The proof of this Theorem follows in the same manner as in the rank 2 case.Remark. The Donaldson's analytic topological compactification PM(n) for the moduli

space M(n) of anti-self-dual Yang-Mills SU(2)-connections (with the anti-instanton numbern) under the interpretation by semi-stable torsion free sheaves has the following algebro-geometric sense :

A point on the boundary of PM(n), i.e. a torsion free sheaf 7, corresponds to an anti-self-dual connection V 7 , having the delta-function singularity for the curvature |Fv|P at afinite number of points : Sing{7) — (ii,'**,atj).

Among the anti-self-dual connections, corresponding to ̂ -stable vector bundles in neigh-borhoods of 7, there is a subsequence of connections, which converges outside the set Sing(7)(in an appropriate topology) to a regular tnti-self-dual connection V B (up to a gauge equiv-alence; this is the Uhlenbeck't theorem of removable singularities for the Yang-Mills fields).

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The Vac i» nothing but the canonical anti-self-du&l connection, corresponding to 7". Then' '* can be considered as the limit

Obviously the (anti-)instanton number of V«, is n - d, hence V,M p M(n - d).This was the original Donaldson description of the compactification of M(n) as a set of

geometric points.

Acknowledgements

The author would like to thank Professor Abdui Salam, the International Atomic EnergyAgency and UNESCO for hospitality at the International Centre for Theoretical Physics inTrieste, Italy, where this work was done.

References

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[2] Donaldson S.K. Anti-sclf-dvat Yang-Mills connections over complex algebraic »tir-facet and stable vector bundles. Proc. Lend. Math. Soc. , vol. 50 (1985), p. 1-26

[3] Kobayashl S. Differential geometry of complex vector bundles. Publication Math.Soc. Japan, vol. 15, 1987

[4] Okonek C , Schnlelder M., Spindel H. Vector bundles on complex projectivespaces. Progress in Math., vol. 3, Birkhauser 1980

[5] TJurin A.N. Algebraic geometric aspects of tmooth structure. I. The Donaldson'spolynomial/. Russian Math. Surveys, vol. 44 (1989) no 3, p. 113-178

[6] Hoang Le Mlab. On induced hermitian metric for kolomorphic vector bundles.Preprint ICTP, IC/89/272

[7] Hoang L« Mlnh The monad construction and Hermitian-Einstein metrics. PreprintICTP, IC/89/273

[8] Matlin Yu.I. Gauge field theory and complex geometry. Springer-Verlag 1988

[9] Griffiths Ph. , Harris J. Principles of algebraic geometry. John-Wiley k Sons 1978

[10) Donaldson S.K. Connection*, cohomology and the intersection forms of 4-manifolds,i. of Diff. Geom., vol. 24 (1986), p. 275-341.

[11] Drezet J.M., Le Portier J. Fibris stabltt tt fibre's exceptioneU «tir P j . Annales Sc.Ec. Norm. Sup., vol 18 (1985),p. 193-244.

[12] Gieseker On the moduli of vector bundles on an algebraic surface. Ann. of Math.,vol. 2/106 (1977), p. 45-60.

[13] Gantmaeher F.R. The theory of matrices. Chelsea Publ., NY. 1960

[14] Artamkln I.V, On deformation* of torsion free sheaves on algebraic surfaces. IivestjaAkad. Nauk. USSR, vol. 54 (1990) No 3, p.435-468. (in Russian)

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