On vector bundles over moduli spaces trivial on Hecke curves
aa r X i v : . [ m a t h . AG ] A p r ON VECTOR BUNDLES OVER MODULI SPACES TRIVIAL ON HECKECURVES
INDRANIL BISWAS AND TOM ´AS L. G ´OMEZ
Abstract.
Let M X ( r, ξ ) be the moduli space of stable vector bundles, on a smooth complexprojective curve X , of rank r and fixed determinant ξ such that deg( ξ ) is coprime to r . If E is a vector bundle M X ( r, ξ ) whose restriction to every Hecke curve in M X ( r, ξ ) is trivial,we prove that E is trivial. Introduction
Moduli spaces of vector bundles on a complex projective curve have a long history. Apartform algebraic geometry, the context in which these moduli spaces were introduced, theyalso arise in symplectic geometry, geometric representation theory, differential geometryand mathematical physics. Line bundles and higher rank vector bundles on these modulispaces play central role in their study. On the other hand, these moduli spaces contain adistinguished class of rational curves known as Hecke lines. They can be characterized asminimal degree rational curves on the moduli spaces [Ty] (also proved in [Su]). These Heckecurves play important role in the geometric representation theoretic aspect of the modulispaces and also in the computation of cohomology of coherent sheaves on the moduli spaces.Here we study restriction of vector bundles on moduli spaces to the Hecke lines. Todescribe the result proved here, fix a smooth complex projective curve X of genus at leasttwo. Let ξ be line bundle on X and r ≥ ξ ). Let M X ( r, ξ ) bethe moduli space of stable vector bundles on X of rank r and determinant ξ . We prove thefollowing (see Theorem 5.4): Theorem 1.1.
Let E be a vector bundle on M X ( r, ξ ) such that its restriction to every Heckecurve on M X ( r, ξ ) is trivial. Then E is trivial. Our motivation to study this problem comes from the result that says that a vector bundleon a projective space P N is trivial when the restriction of it to every line is trivial (in fact,it is enough to check it for lines through a fixed point, cf. [OSS, p. 51, Theorem 3.2.1]). Inthis article we are replacing P N by M X ( r, ξ ), and lines in P N by Hecke curves in M X ( r, ξ )(which are also rational curves of minimal degree).To prove Theorem 1.1 we crucially use a theorem of Simpson which says that a semistablevector bundle W on a smooth complex projective variety admits a flat holomorphic con-nection if c ( W ) = 0 = c ( W ). It is relatively straightforward to deduce that the vectorbundle E in Theorem 1.1 is semistable and c ( E ) = 0. Almost all of our work is devoted inproving that c ( E ) = 0. Mathematics Subject Classification.
Key words and phrases.
Moduli space, Hecke curve, semistability. Cohomology of moduli space
Let X be a smooth complex projective curve of genus g , with g ≥
2. Fix an integer r ≥ ξ on X such that deg( ξ ) is coprime to r . Let M = M X ( r, ξ )be the moduli space of stable bundles on X of rank r and degree deg( ξ ). This moduli space M is a smooth projective variety of dimension ( r − g − P on X × M ; two different Poincar´e bundles on X × M differ by tensoring with a line bundlepulled back from M . It is known that Pic( M ) = Z [Ra, p. 69], [Ra, p. 78, Proposition3.4(ii)]. The ample generator of Pic( M ) will be denoted by O M (1). The degrees of anytorsionfree coherent sheaf F on M is defined to bedeg( F ) := (cid:0) c ( F ) ∪ c ( O M (1)) ( r − g − − (cid:1) ∩ [ M ] ∈ Z . Let U be a rank r vector bundle on X × T such that for every point t ∈ T , the restriction U t := U | X × t is stable and has determinant ξ . Let φ : T −→ M = M X ( r, ξ )be the corresponding classifying morphism. DefineDet U := (cid:0) det( Rp T ∗ U ) (cid:1) − := (cid:0) det( R p T ∗ U ) (cid:1) − ⊗ (cid:0) det( R p T ∗ U ) (cid:1) − −→ T , where p T : X × T −→ T is the natural projection. Then, by [Na, Proposition 2.1], φ ∗ O M (1) = (Det U ) r ⊗ ( ^ r U p ) d + r (1 − g ) (2.1)where p ∈ X is any point, and U p = U | p × M . Applying this to the Poincar´e bundle U = P ,it follows that (deg P p ) · d ≡ r (2.2)(see [Ra, p. 75, Remark 2.9] and [Ra, p. 75, Definition 2.10]). Using the slant productoperation, construct the integral cohomology classes f := c ( P ) / [ X ] ∈ H ( M, Z ) , a := c ( P ) / [ p ] ∈ H ( M, Z ) (2.3)and f := c ( P ) / [ X ] ∈ H ( M, Z ) , where [ X ] ∈ H ( X, Z ) and [ p ] ∈ H ( X, Z ) are the positive generators.The following result is standard. Proposition 2.1. • The integral cohomology of M has no torsion. • The rank of H ( M, Z ) is 1. The cohomology class f in (2.3) generates the Q –vectorspace H ( M, Q ) . • For r ≥ , the rank of H ( M, Z ) is , while rank( H ( M, Z )) = 2 for r = 2 . The Q –vector space H ( M, Q ) is generated by ( f ) , a and f , (2.4) where a and f are defined in (2.3) . (Note that f = 0 if r = 2 .) N VECTOR BUNDLES TRIVIAL OVER HECKE CURVES 3
In [AB, p. 578, Theorem 9.9] it is proved that H ∗ ( M, Z ) is torsionfree. See [AB, p. 582,Proposition 9.13] for the second statement. For the third statement, see [AB, p. 543, Propo-sition 2.20], [JK, p. 114, Section 2], [BR, p. 2, Theorem 1.5].The cohomology class a in Proposition 2.1(3) depends on the choice of Poincar´e bundle P .In the following lemma we show that c ( P ) / [ p ] in (2.3) can be replaced by c (End( P )) / [ p ],which does not depend on the choice of Poincar´e bundle. This would simplify our latercalculations. Lemma 2.2.
The cohomology classes ( f ) , b := c (End( P )) / [ p ] and f also generate H ( M, Q ) .Proof. In view of Proposition 2.1(3), it suffices to prove that a in (2.4) can be expressed asa function of the classes (2.4). The slant product H k ( X × M, Z ) ⊗ H ℓ ( X, Z ) −→ H k − ℓ ( M, Z ) ( η, c ) η/c satisfies the following natural condition [GH, p. 264, (29.23)]: For morphisms f : X ′ −→ X and g : M ′ −→ M , (( f × g ) ∗ η ) /c = g ∗ ( η/f ∗ ( c )) . In particular, if i : x ֒ → X is a point and η ∈ H k ( X × M, Z ), then ξ/ [ i ( x )] = ( i × id M ) ∗ ξ/ [ x ] = ξ | i ( x ) × M ∈ H k ( M, Z ) . (2.5)Now consider c ( P ) / [ p ] in (2.3). We have c (End( P )) / [ p ] = − rc ( P ) / [ p ] + (cid:0) c ( P ) (cid:1) / [ p ] . Using (2.5) it follows that (cid:0) c ( P ) (cid:1) / [ p ] = c ( P p ) . Note that Proposition 2.1(2) says that c ( P p ) = kf for some k ∈ Q . Consequently, we have a = − b + k ( f ) r , which proves the lemma. (cid:3) Hecke transformation on two points of the curve
Definition 3.1 ([NR, p. 306, Definition 5.1 and Remark 5.2]) . Let l, m be integers. A vectorbundle F over X is ( l, m )– stable if, for every proper subbundle G of F ,deg( G ) + l rk G < deg( F ) + l − m rk F .
Let F be a (0 , r and determinant ξ ( x + x ) = ξ ⊗ O X ( x + x )for fixed points x , x ∈ X . We are going to perform Hecke transformations on F over thesetwo points x , x . The parameter space will be P × P := P ( E ∨ x ) × P ( E ∨ x ) ∼ = P r − × P r − For a point x ∈ X let i x : P × P −→ X × P × P , ( y, z ) ( x, y, z )be the inclusion map. Let p P × P : X × P × P −→ P × P be the natural projection. I. BISWAS AND T. L. G ´OMEZ
For integers a, b , the line bundle O P ( E ∨ x ) ( a ) ⊠ O P ( E ∨ x ) ( b ) on P ( E ∨ x ) × P ( E ∨ x ) will be denotedby O ( a, b ). Consider the vector bundle U on X × P × P defined by the short exact sequence0 −→ U −→ p ∗ X F −→ ( i x ) ∗ p ∗ P × P O (1 , ⊕ ( i x ) ∗ p ∗ P × P O (0 , −→ . (3.1)Using the fact that F is (0 , U is a family of stable bundles on X . Indeed, for ( p , p ) ∈ P × P , if a subbundle G of the vector bundle U ( p ,p ) := U | X × ( p ,p ) on X contradicts the stability condition, then the subbundle of F generated by G contradictsthe (0 , F (see [NR, p. 307, Lemma 5.5]).From (3.1) it follows that( ^ r U ( p ,p ) ) ⊗ O X ( x + x ) = ^ r F = ξ ⊗ O X ( x + x ) . This implies that V r U ( p ,p ) = ξ . Let ψ : P × P −→ M = M X ( r, ξ ) (3.2)be the corresponding classifying morphism.If the point p in (2.1) is different from x and x , then U p (as in (2.1)) for the family U in(3.1) is evidently trivial. Therefore, from (2.1) it follows that ψ ∗ O M (1) ∼ = O ( r, r ) . (3.3)We assume that the Poincar´e bundle is normalized by imposing the condition (see (2.2))0 < d ′ := deg( P p ) < r . (3.4)By the universal property of the Poincar´e bundle, there exist integers a , a such that(id X × ψ ) ∗ P = U ⊗ p ∗ P × P O ( a , a ) . (3.5)Once we restrict the isomorphism in (3.5) to p × M , it follows from (3.3) and (3.4) that a = a = d ′ . Hence, denoting L := p ∗ P × P O ( d ′ , d ′ ),(id X × φ ) ∗ P = U ⊗ L .
We now calculate the Chern classes c ( U ⊗ L ) = d P + rd ′ ( D + D ) c ( U ⊗ L ) = (1 + ( r − dd ′ ) P ( D + D ) + d ′ r ( r − D + D ) c ( U ⊗ L ) = − P ( D + D ) + (cid:0) d ′ ( r −
2) + d ′ d ( r − r − (cid:1) ( D + D ) ++ (cid:0) d ′ r ( r − r − (cid:1) ( D + D ) c (End( U )) = − r P ( D + D ) , where D ∈ H ( X × P × P , Z ) (respectively, D ∈ H ( X × P × P , Z )) is the pullback ofthe first Chern class H ∈ H ( P × P , Z ) (respectively, H ∈ H ( P × P , Q )) of the linebundle O (1 ,
0) (respectively, O (0 , P is the pullback of the class of a point in X .We calculate the pullback of the generators using the pullback formula for the slant prod-uct: ψ ∗ f = ψ ∗ ( c ( P ) / [ X ]) = ((id X × ψ ) ∗ c ( P )) / [ X ]= c ( U ⊗ L ) / [ X ] = (1 + dd ′ ( r − H + H ) . N VECTOR BUNDLES TRIVIAL OVER HECKE CURVES 5
Analogously, we get that ψ ∗ f = (1 + dd ′ ( r − ( H + H ) ,ψ ∗ b = 0 , (3.6) ψ ∗ f = − ( H + H ) + (cid:0) d ′ ( r −
2) + d ′ d ( r − r − (cid:1) ( H + H ) . Hecke transformation on moving point
It this section we shall construct a family of vector bundles parametrized by Hecke lineswith a moving point.Let W be a (0 , X with determinant ξ ( x ) = ξ ⊗ O X ( x ) , (4.1)for a fixed point x ∈ X , and let W ։ Q be a rank 2 torsionfree quotient. Let X be a copy of X , i.e., X is a curve with a fixedisomorphism with X ; the parameter space that we are going to construct involves severalcopies, so we shall employ this notation to distinguish between them. The points of X willparametrize the points used to perform a Hecke transformation. Consider the projectivebundle P ( Q ∨ ) π (cid:15) (cid:15) X X A point in y ∈ P ( Q ∨ ) over x = π ( y ) ∈ X gives to a 1-dimensional quotient W x ։ Q x ։ C of the fiber over x , so P ( Q ∨ ) is, in a natural way, the parameter space of a family ofHecke transformations with respect to a moving point. We shall write this family explicitly.Consider the Cartesian diagram: P ∆ i / / (cid:15) (cid:15) X × P ( Q ∨ ) id X × π (cid:15) (cid:15) ∆ / / X × X (4.2)where the morphism at the bottom is the diagonal embedding ∆ = X −→ X × X , t ( t, t ). Note that P ∆ is a P –bundle over ∆. In fact it is canonically identified with P ( Q ∨ ) once we invoke the natural isomorphism between the diagonal ∆ and X = X .From this identification between P ∆ and P ( Q ∨ ), let O P ∆ (1) −→ P ∆ be the line bundlecorresponding to the tautological line bundle O P ( Q ∨ ) (1).There is a canonical short exact sequence of sheaves on X × P ( Q ∨ ):0 −→ F −→ p ∗ X W −→ i ∗ O P ∆ (1) −→ i is the inclusion of P in X × P ( Q ∨ ); here we consider F as a family ofvector bundles on X parametrized by P ( Q ∨ ). Using the condition that W is (0 , F y := F | X × y on X is stable for every point y ∈ P ( Q ∨ ). I. BISWAS AND T. L. G ´OMEZ
Indeed, if a subbundle S of F y contradicts the stability condition, then the subbundle of W generated by S contradicts that (0 , W [NR, p. 307, Lemma 5.5].We are going to calculate the Chern character of the vector bundle F in (4.3). Thefollowing notation for the Chow classes on X × P ( Q ∨ ) will be used: • P (respectively, P ) is the pullback of the class of a point in X (respectively, X ). • P is the pullback of the class of a point in X × X . • δ is the pullback of the class of the diagonal in X × X . • D is the pullback of the divisor O P ( Q ∨ ) (1) on P ( Q ∨ ).We can now calculate: ch( W ) = r + (cid:2) ( d + 1) P (cid:3) . Let i ( P ∆ ) ⊂ X × P ( Q ∨ ) be the image of the closed inclusion i in (4.2). To identify ch( O i ( P ∆ ) ),we first do the following calculations on X × X :ch( O X × X ( − ∆)) = 1 − ∆ + ∆ , ch( O ∆ ) = ∆ − ∆ − (2 g ( X ) − p , where p ∈ H ( X × X , Z ) is the class of a point in X × X . It follows thatch( O i ( P ∆ ) ) = p ∗ X × X ch( O ∆ ) = δ − ( g ( X ) − P . Now, ch( i ∗ O P ∆ (1)) = ch( O i ( P ∆ ) ⊗ O P ( Q ∨ ) (1)) = (cid:16) δ − ( g ( X ) − P (cid:17)(cid:16) D + D (cid:17) = δ + (cid:2) δ e D − ( g ( X ) − P (cid:3) + (cid:2) δD g ( X ) − P D (cid:3) . Finally we obtain the Chern character of F :ch( F ) = r + (cid:2) ( d + 1) P − δ (cid:3) + (cid:2) − δD + ( g ( X ) − P (cid:3) + (cid:2) − δD − ( g ( X ) − P D (cid:3) . (4.4)It may be clarified that F in (4.3) is a family of stable vector bundles of degree d parametrized by P ( Q ∨ ), but the determinant is not fixed. Indeed, if y ∈ P ( Q ∨ ) and x = π ( y ), then the determinant of the vector bundle corresponding to the point y is( V r W ) ⊗ O X ( − x ) = ξ ⊗ O X ( x − x ) (see (4.1)). In particular, the family F induces amorphism from P ( Q ∨ ) to the moduli space M X ( r, d ) of stable vector bundles on X of rank r and degree d . But we want a morphism to the fixed determinant moduli space, so we shalltensor this family with an r -th root of O X ( x − x ). Since x ∈ X is a moving point, tohave a family of r -th roots we need to pass to a Galois cover of the parameter space X .Let f : X −→ J ( X ) , x X ( x − x )be the Abel-Jacobi map for X , where x is the point in (4.1) (recall that X = X is acopy of the same curve, but we make this distinction in notation because of the differentroles they will play in the construction). This morphism f corresponds to a family of linebundles on X of degree zero parametrized by X , i.e., a line bundle L on X × X such that L| X × x ∼ = O X ( x − x ). Let w r : J ( X ) −→ J ( X ) , L L ⊗ r N VECTOR BUNDLES TRIVIAL OVER HECKE CURVES 7 be the morphism that sends a line bundle to its r -th tensor power. Consider the Cartesiandiagram T f r / / t (cid:15) (cid:15) J ( X ) w r (cid:15) (cid:15) X f / / J ( X ) (4.5)We note that T is a connected Galois covering of X , because w r is a Galois covering andthe homomorphism f ∗ : π ( X ) −→ π ( J ( X )) induced by f is surjective. It is easy tocheck that, if the morphism f in (4.5) corresponds to a line bundle L on X × X , then themorphism f r corresponds to a line bundle M on X × T such that M ⊗ r ∼ = (id X × t ) ∗ L , where t is the map in (4.5). In other words, after pulling back from X to T , the family L admitsan r -th root namely M .Let Z be defined by the Cartesian diagram Z q / / π T (cid:15) (cid:15) P ( Q ∨ ) π (cid:15) (cid:15) T t / / X (4.6)Finally, we define the vector bundle U = (id X × q ) ∗ F ⊗ (id X × π T ) ∗ M − on X × Z , where F is the vector bundle in (4.3). From the construction of U it is evidentthat U is a vector bundle on X × T which represents a family of vector bundles on X offixed determinant ξ . Also, this is a family of stable vector bundles, because F is a family ofstable vector bundles. Consequently, we have a classifying morphism ϕ : Z −→ M . (4.7)Our objective now is to calculate the class ϕ ∗ b = ϕ ∗ ( c (End( P ))) / [ p ]. Note that theadvantage of working with End( P ) instead of P is that we do not have to worry aboutnormalization of the Poincar´e bundle, and also the tensorization by the line bundle M willnot appear in the calculation. We haveEnd( U ) = (id X × q ) ∗ End( F ) , and c (End( F )) = − ch (End( F )) − c (End( F )) = − ch (End( F ))= − [ch( F ∨ ) ⊗ ch( F )] = 2 r ch ( F ) − ch ( F ) = 2 r δD + (2 r ( g − − d + 1) + (2 − g )) P (see (4.4)). Recall that “slanting with the class of a point is the same thing as restrictionto the slice” (formula (2.5)). It follows that, if p is a point in X and [ p ] ∈ H ( X, Z ) isits homology class, then P / [ p ] = 0 and δD/ [ p ] = [ ̟ ] ∈ H ( P ( Q ∨ , Q ), where [ ̟ ] ∈ H ( P ( Q ∨ ) , Z ) is the positive generator. So, we have c (End( F )) / [ p ] = 2 r [ ̟ ] ∈ H ( P ( Q ∨ ) , Q ) . I. BISWAS AND T. L. G ´OMEZ
Also, c (End( F )) / [ p ] = c (End( F p )), where F p is the restriction of F to the slice p × P ( Q ∨ )and then ϕ ∗ b = ϕ ∗ c (End( P p )) = c (End( U p )) = q ∗ c (End( F p )) = deg( q )2 r = r g r , (4.8)where P p and U p respectively are the restrictions of P and U to the slice p × M .5. Vanishing of Chern classes
Let E be a vector bundle on M = M X ( r, ξ ) such that the restriction of E to every Heckecurve is trivial. Throughout this section, E would satisfy this condition.From the above condition it can be deduced that c ( E ) = 0 . (5.1)Indeed, H ( M, Z ) = Z , and a Hecke curve f : P −→ M induces an injection on thesecond cohomology f ∗ : H ( M, Z ) −→ H ( P , Z ) ∼ = Z . Now, f ∗ c ( E ) = c ( f ∗ E ) = 0, and hence (5.1) holds. Recall that the rank of H ( M, Z ) is3 when r ≥
3, and it is 2 when r = 2, and the generators are given by Lemma 2.2. Lemma 5.1.
Let ψ : P × P −→ M be the morphism (3.2) . The pullback E ′ := ψ ∗ ( E ) isa trivial vector bundle on P × P .Proof. For every line l ⊂ P and every point p ∈ P , the restriction of ψ to l × p ∼ = P isa Hecke curve, so E ′ | l × p is trivial by the hypothesis. A vector bundle on a projective spaceis trivial if it is trivial when restricted to every line on the projective space ([OSS, p. 51,Theorem 3.2.1]). Consequently, E ′ | P × p is trivial, and this is true for every point p ∈ P .Therefore E ′ descends to a vector bundle F on P , i.e., there is a vector bundle F on P such that q ∗ F ∼ = E ′ , where q is the projection of P × P to P . In fact F = q ∗ E ′ .Note that, for any p ∈ P , the restriction E ′ | p × P is isomorphic to F . As before, forevery line l in P , the restriction of ψ to p × l is a Hecke curve, so F is trivial on p × l .Hence F is trivial by the above argument. Consequently, E ′ = q ∗ F is trivial. (cid:3) Lemma 5.2.
Let ϕ : Z −→ M be the morphism in (4.7) . The pullback E Z := ϕ ∗ ( E ) hasChern classes c ( E Z ) = 0 ∈ H ( Z, Z ) and c ( E Z ) = 0 ∈ H ( Z, Z ) . Proof.
The vector bundle E on M has c ( E ) = 0 (see (5.1)), and hence c ( E Z ) = 0.The scheme Z fibers over a curve π T : Z −→ T , and the restriction of ϕ to any fiber is aHecke line (see (4.6)). Hence, the vector bundle E Z is trivial on the fibers of π T . This impliesthat E Z descends to T , i.e., there exists a vector bundle F on T such that E Z = π ∗ T F . Infact, F = π T ∗ E Z .Since F is a vector bundle on a curve, namely Z , it follows that c ( F ) = 0 (as H ( Z, Z ) =0). Therefore, we have c ( E Z ) = π ∗ T c ( F ) = 0. (cid:3) Proposition 5.3.
The second Chern class c ( E ) ∈ H ( M, Q ) of the vector bundle E on M is zero. N VECTOR BUNDLES TRIVIAL OVER HECKE CURVES 9
Proof.
Using Lemma 2.2, we write c ( E ) as a combination of the generators with coefficientsin Q : c ( E ) = α ( f ) + βb + γf . In the case rank r = 2, we have f = 0, so we set γ = 0.Consider the pullback of c ( E ) by the morphism ψ in (3.2). We have ψ ∗ c ( E ) = 0 byLemma 5.1, and hence using (3.6) it follows that ψ ∗ c ( E ) = 0 = α ((1 + dd ′ ( r − ( H + H ) ) − γ ( H + H ) . (5.2)It is easy to check, using (2.2), that 1 + dd ′ ( r − = 0.We have P × P ∼ = P r − × P r − , so if rank r = 2, then H = 0 = H , and hence α = 0.On the other hand, if r >
2, then the classes H + H and ( H + H ) are linearlyindependent in H ( P × P , Z ) ∼ = H ( P r − × P r − , Z ) ∼ = Z , so (5.2) implies α = γ = 0.Summing up, for any rank r ≥ c ( E ) = βb . (5.3)Pulling back (5.3) by the map ϕ in (4.7), and using (4.8) we get that0 = βr g r . Therefore, β = 0, and hence c ( E ) = 0 by (5.3). (cid:3) Theorem 5.4.
Let E be a vector bundle on M = M X ( r, ξ ) satisfying the condition that therestriction of E to every Hecke curve is trivial. Then E is trivial.Proof. The restriction of E to a Hecke curve on M is trivial. From this it can be deducedthat E is semistable. Indeed, if E is not semistable, there is a coherent subsheaf V ⊂ E such that E/V is torsionfree, anddeg( V )rank( V ) > deg( E )rank( E ) = 0 (5.4)(see (5.1)). Now, for a general Hecke curve P ⊂ M , the restriction V | P is torsionfree.Moreover for any smooth closed curve C ⊂ M , and any torsionfree coherent sheaf W on M which is locally free on C , we have deg( W | C ) = deg( C ) · deg( W ). Consequently, from (5.4)it is deduced that deg( V | P ) > P ⊂ M that is contained in the open subset where V is locally free.But the trivial bundle E | P on P does not contain any subsheaf of positive degree. From thiscontradiction we conclude that E is semistable.Since • E is semistable, • c ( E ) = 0 (5.1) and • c ( E ) = 0 (Proposition 5.3), the vector bundle E admits a filtration of subbundles0 = E ⊂ E ⊂ · · · ⊂ E ℓ − ⊂ E ℓ = E such that for every 1 ≤ i ≤ ℓ , the quotient E i /E i − is a stable vector bundle with c ( E i /E i − ) = 0 = c ( E i /E i − ) [Si, p. 39, Theorem 2] (note that ℓ = 1 is allowed).This implies that E admits a flat holomorphic connection [Si, p. 40, Corollary 3.10] (setthe Higgs field to be zero in [Si, Corollary 3.10]); see [BS, p. 4015, Proposition 3.10] for anextension of this result.Since E admits a flat holomorphic connection it is given by a representation of π ( M ) inGL( r, C ), where r is the rank of E . On the other hand, M is simply connected [AB, p. 581,Theorem 9.12]. Therefore, the vector bundle E is trivial. (cid:3) Acknowledgements
The second author thanks Roberto Mu˜noz for discussions. Part of this work was doneduring visits to the Kerala School of Mathematics, Tata Institute of Fundamental Researchand to the International Center for Theoretical Sciences (ICTS, during the program Moduliof Bundles and Related Structures, code: ICTS/mbrs2020/02).This work was supported by Ministerio de Ciencia e Innovaci´on of Spain (grant MTM2016-79400-P, and ICMAT Severo Ochoa project SEV-2015-0554).
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School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road,Mumbai 400005, India
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