IndranilBiswas Acriterionfortheexistenceofaflatconnectiononaparabolicvectorbundle

(de Gruyter 2002

A criterion for the existence of a flat connection on a parabolic vector bundle

Indranil Biswas

(Communicated by A. Sommese)

Abstract. We define holomorphic connection on a parabolic vector bundle over a Riemann surface and prove that a parabolic vector bundle admits a holomorphic connection if and only if each direct summand of it is of parabolic degree zero. This is a generalization to the para- bolic context of a well-known result of Weil which says that a holomorphic vector bundle on a Riemann surface admits a holomorphic connection if and only if every direct summand of it is of degree zero.

2000 Mathematics Subject Classification. 14H60, 32L05

1 Introduction

A theorem due to A. Weil says that a holomorphic vector bundleEover a compact connected Riemann surface admits a holomorphic connection if and only if each di- rect summand ofEis of degree zero [8]. Note that giving a holomorphic connection onEis equivalent to giving a flat connection onEcompatible with its holomorphic structure.

Let Sbe a finite subset of a compact connected Riemann surface X. LetE b e a parabolic vector bundle overXobtained by putting a parabolic structure on a vector bundleEover the divisor S. A holomorphic connection onE is a logarithmic con- nection

D:V !KXnOXðSÞnV

with the property that the residue of Dover anysAS is semisimple and it is com- patible with the parabolic data forE overs(the details of the definition are in Sec- tion 3).

We prove thatEadmits a holomorphic connection if and only if each of its direct summands is of parabolic degree zero (Theorem 3.1).

The proof of Theorem 3.1 is carried out using the correspondence between para- bolic bundles and vector bundles equipped with an action of a finite group estab-

lished in [3]. In fact, Theorem 3.1 follows from the corresponding result on such bun- dles which has been established in Theorem 2.3.

In Section 4 we generalize Theorem 3.1 to higher dimensional projective mani- folds (see Proposition 4.2), and obtain a criterion for the existence of a holomorphic connection on a parabolic vector bundle with parabolic structure on a normal cross- ing divisor.

Thanks are due to the referee for comments that helped in improving the paper.

2 Flat connections on vector bundles with group actions

LetYbe a compact connected Riemann surface. LetKY denote the canonical bundle ofY.

Aholomorphic connectionon a holomorphic vector bundleVoverYis a first order di¤erential operator

D:V !K_YnV ð2:1Þ

satisfying the Leibniz identity which says that DðfsÞ ¼ fDðsÞ þqfns, where f is any locally defined holomorphic function on Yandsis any local holomorphic sec- tion ofV. IfqV:V !W_Y^0;1nV is the Dolbeault operator defining the holomorphic structure ofV, thenDþq_V is a flat connection onV. Conversely, for any flat con- nection onVcompatible with its holomorphic structure (that is, theð0;1Þ-part of the connection coincides withq_V), theð1;0Þpart of it is a holomorphic connection onV.

Fix a finite subgroupGHAutðYÞof the automorphism groupY.

AG-linearized vector bundle W over Yis a holomorphic vector bundle equipped with an action ofGcompatible with the obvious action ofGonY[5]. In other words, Gacts on the total space of Wand for anygAGthe action ofg is a vector bundle isomorphism of W with ðg¹ÞW. Given two G-linearized vector bundles W1 and W2, aG-homomorphismfromW1 toW2is aOY-linear homomorphismh:W1!W2

that commutes with the actions ofG, that is,hg¼ghfor allgAG.

AG-holomorphic connection onW is a holomorphic connection that is preserved by the action of G. In other words, for aG-holomorphic connection Dand for any gAG, the isomorphism ofWwithðg¹ÞW, defined byg, takes the connectionDto the connectionðg¹ÞD.

Proposition 2.1.If the vector bundle W admits a holomorphic connection,then it admits aG-holomorphic connection.

Proof. The space of all holomorphic connections on Wis a convex space. Given a holomorphic connectionDonW, consider the average

D⁰:¼ 1 aG

X

gAG

gD

whereaGdenotes the order of the groupG. The holomorphic connectionD⁰is clearly aG-holomorphic connection. This completes the proof of the proposition. r

A G-linearized vector bundle W will be called decomposable if there are two G-linearized vector bundlesW₁ andW₂, with rankðW1Þ;rankðW1Þ>0, such thatW is isomorphic, as a G-linearized vector bundle, to W₁lW₂. We will call W to be indecomposableif it is not decomposable.

Lemma 2.2.If W is an indecomposableG-linearized vector bundle of degree zero,then W admits aG-holomorphic connection.

Proof. In view of Proposition 2.1 it su‰ces to show that W admits a holomorphic connection. We will recall the obstruction for the existence of a holomorphic con- nection.

For a holomorphic vector bundleVoverY, let Di¤_Y¹ðV;VÞdenote the vector bun- dle defined by the sheaf of di¤erential operators of order one onV. So we have the symbol homomorphisms:Di¤_Y¹ðV;VÞ !TYnEndðVÞ. Let

AtðVÞHDi¤¹_YðV;VÞ

denote the subbundle which is the inverse image ofTYnIdV. So we have theAtiyah exact sequence

0!EndðVÞ !AtðVÞ !TY !0: ð2:2Þ

A holomorphic connection onVis a holomorphic splitting of the exact sequence (2.2) [1], [4].

Note that the space of all extensions ofTY by EndðVÞis parametrized by H¹ðY;KYnEndðVÞÞ ¼H⁰ðY;EndðVÞÞ: ð2:3Þ We will recall a few properties of the extension class for (2.2).

Letb_VAH¹ðY;KYnEndðVÞÞbe theAtiyah classrepresenting (2.2), and let b_VAH⁰ðY;EndðVÞÞ

correspond tob_Vby the isomorphism (2.3).

LetIdenote the identity automorphism ofV. We have

b_VðIÞ ¼degreeðVÞ ð2:4Þ

which is a consequence of the construction of Chern classes from Atiyah classes [1, Theorem 6].

IfFis a holomorphic subbundle ofV, then AtðVÞcontains a subbundleF defined by the sheaf of di¤erential operators that preserve the subbundleF. In other words, we have a commutative diagram

0 ! End_FðVÞ ! F ! TY ! 0

??

?y

??

?y

0 ! EndðVÞ ! AtðVÞ ! TY ! 0

where End_FðVÞHEndðVÞis the subbundle that preservesF.

Therefore,b_Vis in the image ofH¹ðY;K_YnEnd_FðVÞÞ. This implies that

b_V AkernelðcÞ; ð2:5Þ

wherec:H⁰ðY;EndðVÞÞ !H⁰ðY;EndFðVÞÞis the obvious homomorphism.

Take anytAAutðYÞ, and let

t:H¹ðY;KYnEndðVÞÞ !H¹ðY;KYnEndðtVÞÞ

be the isomorphism induced byt. Letb_tV AH¹ðY;KYnEndðtVÞÞbe the Atiyah class fortV. The identity

b_tV ¼tðb_VÞ ð2:6Þ

is obviously valid.

LetWb e aG-linearized vector bundle overY. The groupGhas a natural action on H¹ðY;K_YnEndðWÞÞ. Let

bAH¹ðY;KYnEndðWÞÞ

represent the Atiyah exact sequence ofW. From (2.6) it follows immediately that bAH¹ðY;KYnEndðWÞÞ^G:

In other words,bis fixed by the action ofG.

The canonical nature of the isomorphism (2.3) ensures that it commutes with the actions ofGonH⁰ðY;EndðWÞÞ andH¹ðY;K_YnEndðWÞÞ. Therefore, ifb corre- sponds tobby (2.3), then by settingV¼W in (2.3)

bAðH⁰ðY;EndðWÞÞÞ^G:

In other words,bis determined by its evaluations onH⁰ðY;EndðWÞÞ^G.

Take a sectionfAH⁰ðY;EndðWÞÞwhich is invariant under the action ofG. Since Yis compact and connected, the characteristic polynomial of fðyÞAEndðWyÞdoes not depend on y.

Consider the decomposition of W obtained from the generalized eigenspace de- composition forf. Sincefis left invariant by the action ofG, this is a decomposition ofWinto a direct sum ofG-linearized vector bundles.

Assume thatWis indecomposable. This implies thatfðyÞhas only one eigenvalue, sayl. So,

f⁰:¼flId_W

is a nilpotent endomorphism ofW. Iff⁰00, then there is a proper subbundleFof W, withF00, which is preserved byf⁰. Now settingV ¼W in (2.5) we conclude

that bðf⁰Þ ¼0. Finally, if degreeðWÞ ¼0, then from (2.4) it follows thatbðfÞ ¼0.

This completes the proof of the lemma. r

AG-linearized vector bundleW1 is called adirect summandofW if there is aG- linearized vector bundleW2 such thatWis isomorphic, as aG-linearized vector bun- dle, toW1lW2.

It is easy to see that a holomorphic connection onWinduces a holomorphic con- nection on each direct summand of it. IfW1 andW2 both admit holomorphic con- nections, then obviouslyW1lW2also admits a holomorphic connection. Therefore, the following theorem follows from Lemma 2.2 and Proposition 2.1.

Theorem 2.3. AG-linearized vector bundle W admits a G-holomorphic connection if and only if every direct summand of it is of degree zero.

In the next section we will use this theorem in the context of parabolic bundles.

3 Connection on a parabolic bundle

We first recall the definition of a parabolic vector bundle [7]. Let Xbe a compact connected Riemann surface, andSHX be a finite subset. Aparabolic structureover Son a holomorphic vector bundleEoverXconsists of the following data:

(1) a strictly increasing filtration

0¼F₀^sHF₁^sHF₂^sH HF_l^s_s ¼Es

for eachsASknown as thequasi-parabolic filtration;

(2) a sequence of real numbers

1>l₁^s>l₂^s> >l_l^s_sd0;

wherel_i^scorresponds to the subspaceF_i^s.

Aparabolic vector bundle is a vector bundle equipped with a parabolic structure.

As in [7], we will assume theparabolic weightsl_i^sto be rational numbers.

If we denote by E the above defined parabolic vector bundle, then theparabolic degreeofE is defined to be

par-degðEÞ:¼degreeðEÞ þX

sAS

X^ls

i¼1

l_sⁱdimðF_i^s=F_i1^s Þ:

Given a parabolic vector bundle E as above, any subbundle of E has an induced parabolic structure. Also, ifE⁰is another vector bundle with parabolic structure, then ElE⁰has an obvious parabolic structure constructed from the parabolic structures onEandE⁰. See [7] for the details.

We will now define holomorphic connection in the context of parabolic bundles.

Recall that a logarithmic connection on a vector bundleVoverXwith singularity overSis a first order di¤erential operator

D:V !K_XnOXðSÞnV

satisfying the Leibniz identity [4]. The Poincare´ adjunction formula says that the fiber of the line bundleOXðSÞover anysASis identified with the tangent spaceTsX ats.

In other words, the fiber ðKXnOXðSÞÞ_s is identified with C. Given a logarithmic connectionD, consider the composition

V !^D KXnOXðSÞnV ! ðKXnOXðSÞnVÞ_s¼Vs:

It is easy to see that this homomorphism of sheaves defines an endomorphism of the fiber Vs. This endomorphism is called the residue ofDats[4], and it is denoted by ResðD;sÞ.

LetEbe a parabolic structure onEas described above. Aholomorphic connection on E is a logarithmic connection Don E, singular overS, satisfying the following conditions:

(1) for any sAS, the residue ResðD;sÞpreserves the filtration of E_s and it is semi- simple;

(2) the action of ResðD;sÞon F_i^s=F_i1^s is multiplication by the corresponding para- bolic weightl_i^s. (Since ResðD;sÞpreserves the filtration, it acts on each quotient F_i^s=F_i1^s .)

We will call a parabolic bundleE⁰to be adirect summandofEif there is another parabolic bundle E¹ such that E is isomorphic to E⁰lE¹. So, in particular E is isomorphic toE⁰lE¹, whereE⁰andE¹are the underlying vector bundles forE⁰and E¹respectively. Note that ifE²andE³ are two subbundles ofEwithE¼E¹lE³, then it is not necessary thatE¼E²lE³, whereE²andE³have the induced para- bolic structures fromE.

Theorem 3.1.A parabolic vector bundle Eadmits a holomorphic connection if and only if every direct summand of E is of parabolic degree zero.

Proof.Given a parabolic bundleEoverX, in [3] a (ramified) Galois covering p:Y !X

is constructed. LetGdenote the Galois group for p. FromE, aG-linearized vector bundleWonYis constructed. See [3, Section 3] for the details.

LetWdenote theG-linearized bundle overY(for the automorphism groupG) con- structed in [3, Section 3] fromE. Now [3, (3.12)] says that

par-degðEÞ ¼degreeðWÞ

aG : ð3:1Þ

Also, there is a one-to-one correspondence between subbundles ofEandGinvariant subbundles ofW[3, p. 318].

Assume that every direct summand of E is of parabolic degree zero. Since sub- bundles of Eare in one-to-one correspondence with the G invariant subbundles of W, using (3.1) it follows that every direct summand of theG-linearized vector bundle Wis of degree zero. Therefore, we conclude from Theorem 2.3 thatWadmits aG- holomorphic connection.

Take aG-holomorphic connectionDonW. Fix a point yAp¹ðXnSÞ. Let r:p₁ðY;yÞ !GLðn;CÞ

be the monodromy representation of the flat connection ‘:¼Dþq_W, where n¼ rankðWÞ and q_W is Dolbeault operator defining the holomorphic structure of W.

Since‘is left invariant by the action ofGonW, the representationrclearly descends to a representation ofp₁ðXnS;pðyÞÞ. This gives a connection on the restriction ofE toXnS. It can also be seen directly that the condition that‘isGinvariant ensures that it descends to a flat connection onEoverXnS.

This connection onE overXnS extends to a connection onE [2, Lemma 4.11].

Indeed, Lemma 4.11 of [2] says thatG-invariant forms onYdescend as logarithmic forms onX. From this it follows immediately that the above holomorphic connection onEoverXnSgives a holomorphic connection onE.

Conversely, if E has a holomorphic connection, then W has a G-holomorphic connection. Indeed, the pullback of a holomorphic connection onEis a holomorphic connection on the restriction ofWtoYnp¹ðSÞthat is left invariant by the action of Gon Wj_Ynp1ðSÞ. It is easy to see that this connection extends toW overY. We re- marked earlier that direct summands of E are in one-to-one correspondence with direct summands of W. Therefore, using (3.1) it follows that if E admits a holo- morphic connection, then any direct summand of E is of parabolic degree zero.

This completes the proof of the theorem. r

A polystable parabolic vector bundle of parabolic degree zero clearly has the prop- erty that any direct summand of it is of parabolic degree zero. Such a parabolic bundle admits a holomorphic connection which is unitary [7, Theorem 4.1]. Moreover, such a connection is unique.

If we have a parabolic vector bundleEwhose parabolic weights are real numbers, but not necessarily rational, then the proof of Theorem 3.1 is not valid. However, if it is possible to generalize the method of [1] to prove Theorem 3.1 directly, then the restriction on the rationality of the weights can be dropped.

4 Connections on higher dimensional varieties

Let Ybe a connected complex projective manifold of dimension d. A holomorphic connection on a holomorphic vector bundle WoverYis a first order holomorphic di¤erential operator

D:W !W_Y¹ nW;

whereW_Y¹ denotes the holomorphic cotangent bundle ofY, satisfying the Leibniz rule (as in (2.1)). The basic di¤erence between holomorphic connections on a Riemann surface and those on a higher dimensional variety is that the connectionDþq_W need not be flat if d >1. However, the curvature of the connectionDþqW is always a holomorphic section ofW_Y² nEndðWÞ.

The higher dimensional Atiyah exact sequence is constructed as follows. Let Di¤_Y¹ðW;WÞbe the coherent sheaf of di¤erential operators and

s:Di¤¹_YðW;WÞ !TYnEndðWÞ

the symbol map. Note that there is a natural inclusionTY,!TYnEndðWÞdefined bys7!snIdW. The inverse images¹ðTYÞis called the Atiyah bundle and is de- noted by AtðWÞ. Since the kernel ofsis EndðWÞ, the vector bundle AtðWÞfits into an exact sequence

0!EndðWÞ !AtðWÞ !TY!0 ð4:1Þ

as in (2.2), which is called the Atiyah exact sequence. Clearly this construction co- incides with the one in (2.2) if d¼1. Giving a holomorphic connection on W is equivalent to giving a holomorphic splitting of the Atiyah exact sequence [1].

Fix an ample line bundleLonY. Take a holomorphic vector bundleWonY. In [1, Proposition 21], the following criterion for the existence of a holomorphic con- nection onWis proved under the assumption thatdd3. The vector bundleWad- mits a holomorphic connection if and only if for every integerm, there is an integer cdmand an e¤ective smooth divisor ConYwithOYðCÞisomorphic toL^nc such that the restriction Wj_C ofW toCadmits a holomorphic connection (note the as- sumptiondd3). In [1, Proposition 22] an example is given showing that the above criterion is not valid ford¼2.

LetGHAutðYÞbe a finite subgroup such that the quotientM:¼Y=Gis a smooth projective manifold. Let

p:Y !M ð4:2Þ

be the quotient map. Assume thatLGpx, wherexis some ample line bundle onM.

Note that since pis a finite map, the pullback of any ample line bundle on MtoY remains ample.

LetWb e aG-linearized vector bundle overY. As in the case of Riemann surfaces, b y aG-holomorphic connection onWwe mean a holomorphic connection that is left invariant by the action of G on W. Assume that d ¼dimCYd3. The following proposition follows from the criterion of [1] for the existence of a G-holomorphic connection onW.

Proposition 4.1.The vector bundle W admits aG-holomorphic connection if any only if for every integer m there is an integer cdm and a smooth divisor C⁰Ajx^ncjsuch that p¹ðC⁰Þis a reduced smooth divisor on Y and the restriction Wj_p¹_ðC⁰_Þof W to p¹ðC⁰Þ admits a holomorphic connection.

Proof.In Proposition 2.1 it was proved thatWadmits aG-holomorphic connection if and only if it admits a usual holomorphic connection.

Givenm, takecandC⁰as in the statement of the proposition. So, the inverse im- age p¹ðC⁰Þ is a smooth divisor on Y with the property that OYðp¹ðC⁰ÞÞ is iso- morphic to L^nc, where cdm. Now setting C¼p¹ðC⁰Þ in the above criterion of Atiyah for the existence of a holomorphic connection we see that the condition in the proposition ensures thatWadmits a holomorphic connection.

Conversely, a holomorphic connection onWinduces a holomorphic connection on the restriction of W to any smooth divisor. It is easy to see that there is a positive integer k0 such that for everykdk0, the general divisor C⁰ in the complete linear system jx^nkj has the property that the inverse image p¹ðC⁰Þ is a reduced smooth divisor ofY. In particular, there is one such divisor for eachkdk₀. In other words, if Wadmits a holomorphic connection, then given any integerm, there is a pairðc;C⁰Þ satisfying the conditions in the statement of the proposition. This completes the proof

of the proposition. r

Now we consider parabolic vector bundles on higher dimensional varieties.

LetMbe a connected smooth projective manifold of dimension at least three and D0a normal crossing divisor onM. This means thatD0is a reduced e¤ective divisor each of whose irreducible components is smooth and furthermore the components intersect transversally.

LetVbe a parabolic vector bundle overM, with parabolic structure overD0, such that all the parabolic weights are rational numbers and the quasi-parabolic filtration is defined using filtrations by subbundles on the irreducible components ofD0. (See [3] for the elaboration on this condition.) The bijective correspondence between para- bolic vector bundles and G-linearized vector bundles on curves that we used in the proof of Theorem 3.1 remains valid for higher dimensions [3]. In particular, the para- bolic vector bundleV corresponds to aG-linearized vector bundleW on a smooth projective varietyYwithM¼Y=G[3, Section 3]. The existence of a smooth projec- tive manifold Y with the required properties is ensured by the covering lemma of Kawamata [6, Theorem 1.1.1] (see [3] for the details how the covering lemma is used in this context).

LetNbe an integer such that all the parabolic weights ofV are integral multiples of 1=N. The Galois covering depends on the choice ofN. Let

D0¼X^l

j¼1

Dj

be the decomposition of the divisor into irreducible components. The Galois covering pfor the parabolic vector bundleVhas the property that for each jA½1;l, there is an integerk_jsuch that p¹ðDjÞ ¼k_jNðp¹ðDjÞÞ_red, that is, the multiplicity of the non- reduced divisor p¹ðDjÞis divisible byN(see [3, Section 3]).

Fix an ample line bundlexonM. LetC⁰be an e¤ective smooth divisor onMthat intersectsD₀transversally. The parabolic vector bundleVcan be restricted to such a

divisor to obtain a parabolic vector bundle over C⁰. This is done by restricting both the underlying vector bundle forV and the quasi-parabolic filtration. The transver- sality condition onC⁰is required to ensure that the restriction of the quasi-parabolic filtration remains a quasi-parabolic filtration. The parabolic divisor for this restricted parabolic vector bundle Vj_C0 is C⁰VD0. The parabolic weights of the restricted parabolic vector bundle are defined by the parabolic weights ofV.

We will call a divisorC⁰onMto begood for V ifC⁰ intersects the parabolic di- visor D0 transversally and p¹ðC⁰Þis a reduced smooth divisor. It should be noted that the condition thatC⁰ is good forV depends on the choice of the Galois cover- ing. We emphasize that given V, the Galois covering is fixed once and for all. It is easy to see that there is an integerc0with the property that for anycdc0, the general memberC⁰Ajx^ncjis good forV.

A holomorphic connection on the parabolic vector bundleVis defined exactly as for parabolic bundles on a Riemann surface.

Now using the bijective correspondence constructed in [3] between parabolic vector bundles andG-linearized vector bundles the Proposition 4.1 yields the following prop- osition.

Proposition 4.2.A parabolic vector bundle V admits a holomorphic connection if any only if for every integer m there is an integer cdm and a divisor C⁰Ajx^ncjgood for V

such that the parabolic vector bundle obtained by restricting V to C⁰ admits a holo- morphic connection.

Proof. If for everymthere is pairðc;C⁰Þwith the above property, then Proposition 4.1 says that the G-linearized vector bundle W corresponding to V admits a G- holomorphic connection. This connection on Winduces a holomorphic connection onV.

On the other hand, ifVadmits a holomorphic connection, then the corresponding G-linearized vector bundle W admits a G-holomorphic connection. Now recall the earlier remark that there is an integer c0 with the property that for any cdc0, the general member C⁰Ajx^ncjis good forV. In particular, there is at least one divisor C⁰Ajx^ncjwhich is good forV. If we take cdmaxfm;c0g, the corresponding pair ðc;C⁰Þsatisfies the condition in the proposition. This completes the proof of the prop-

osition. r

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Received 1 July, 2001; revised 16 October, 2001

I. Biswas, School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India

Email: [email protected]