A note on the moduli spaces of holomorphic and logarithmic connections over a compact Riemann surface
aa r X i v : . [ m a t h . AG ] F e b A NOTE ON THE MODULI SPACES OF HOLOMORPHIC ANDLOGARITHMIC CONNECTIONS OVER A COMPACT RIEMANNSURFACE
ANOOP SINGH
Abstract.
Let X be a compact Riemann surface of genus g ≥
3. We considerthe moduli space of holomorphic connections over X and the moduli space oflogarithmic connections singular over a finite subset of X with fixed residues. Wedetermine the Chow group of these moduli spaces. We compute the global sectionsof the sheaves of differential operators on ample line bundles and their symmetricpowers over these moduli spaces, and show that they are constant under certaincondition. We show the Torelli type theorem for the moduli space of logarithmicconnections. We also describe the rational connectedness of these moduli spaces. Contents
1. Introduction and statements of the results 12. Preliminaries 42.1. Moduli spaces of holomorphic and logarithmic connections 43. Chow group of the moduli spaces 94. Differential operators on the moduli spaces 145. Torelli type theorem for the moduli spaces 216. Rational connectedness of the moduli spaces 28References 291.
Introduction and statements of the results
Let X be a compact Riemann surface of genus g ≥
3. We consider the moduli space M h ( n ) of rank n holomorphic connections over X , (see section 2 for the definition).In [37] and [38], Simpson constructed the moduli space of holomorphic connectionsover a smooth complex projective variety.Let S = { x , . . . , x m } be a fixed subset of X such that x i = x j for all i = j . We consider the modulispace M lc ( n, d ) of logarithmic connections of rank n and degree d , singular over S ,with fixed residues (see section 2 for the definition). The moduli space of logarithmic Mathematics Subject Classification.
Key words and phrases.
Logarithmic connection, Moduli space, Chow group, Differential oper-ator, Torelli theorem, Rational variety. connections over a complex projective variety singular over a smooth normal crossingdivisor has been constructed in [33].Several algebro-geometric invariants like Picard group, algebraic functions of themoduli space of holomorphic and logarithmic connections have been studied, see [7],[9], [10], [36] [27], and [26].In the present article, our aim is to study the Chow group, global section of certainsheaves, Torelli type theorem, and rationality of these moduli spaces.The structure of the article is as follows. In section 2, we define the notion ofholomorphic and logarithmic connections in a holomorphic vector bundle over X ,and recall their moduli spaces.In section 3, we compute the Chow group of the moduli spaces which is motivatedby the following result in [11]. Let U (2 , O X ( x )) be the moduli space of stable vectorbundles of rank 2 with determinant O X ( x ), where x ∈ X . Then, in [11], the Chowgroup of 1-cycles on U (2 , O X ( x )) has been computed, and it is proved thatCH Q ( U (2 , O X ( x ))) ∼ = CH Q ( X ) . (1.1)Fix a holomorphic line bundle L over X of degree d . Consider the moduli space oflogarithmic connections M lc ( n, L ) of rank n and fixed determinant L as described in(2.10). Let M ′ lc ( n, L ) ⊂ M lc ( n, L ) be the moduli space of logarithmic connections( E, D ) with E stable as described in (2.11). Then, we show the following (seeTheorem 3.3). For every 0 ≤ l ≤ ( n − g − CH l +( n − g − ( M ′ lc ( n, L )) ∼ = CH l ( U ( n, L )) . (1.2)As a consequence for n = 2, we have (see Corollary 3.5),(1) CH g − ( M ′ lc (2 , L )) ∼ = Z .(2) CH Q g − ( M ′ lc (2 , L )) ∼ = CH Q ( X ).(3) CH Q g − ( M ′ lc (2 , L )) ∼ = CH Q ( X ) ⊕ Q .Let L be a holomorphic vector bundle over X of degree zero. Let M ′ h ( n, L ) and U ( n, L ) be the moduli space defined in (3.16) and (3.15) respectively. Then, weshow that (see Theorem 3.9), for every 0 ≤ l ≤ ( n − g − Q l +( n − g − ( M ′ h ( n, L )) ∼ = CH Q l ( U ( n, L )) . (1.3)In section 4, we study the global section of certain locally free sheaves. Let M ′ lc ( n, d ) be the moduli space described in (2.9), and ζ an ample line bundle over M ′ lc ( n, d ). For k ≥
0, let D k ( ζ ) denote the sheaf of differential operators on ζ oforder k . Consider the following natural morphism p : M ′ lc ( n, d ) → U ( n, d ) (1.4)sending ( E, D ) to E . Then we have a morphism e p ♯ : H ( T ∗ M ′ lc , O T ∗ M ′ lc ) → H ( T ∗ U ( n, d ) , O T ∗ U ( n,d ) ) . (1.5)of vector spaces induced from e p : T ∗ U ( n, d ) → T ∗ M ′ lc , ODULI SPACES OF HOLOMORPHIC AND LOGARITHMIC CONNECTIONS 3 where T ∗ U ( n, d ) and T ∗ M ′ lc are cotangent bundle of U ( n, d ) and M ′ lc ( n, d ) respec-tively.Under the assumption that e p ♯ in (1.5) is injective, we show that (see Theorem4.1), for every k ≥
0, H ( M ′ lc ( n, d ) , S ym k ( D ( ζ ))) = C , (1.6)and (see Proposition 4.2) H ( M ′ lc ( n, d ) , D k ( ζ )) = C . (1.7)Under the same assumption, above result is true for the moduli spaces M ′ lc ( n, L )(see (2.11)), M ′ h ( n ) (see (2.2)) and M ′ h ( n, L ) (see (3.16)).In section 5, we prove the Torelli type result for the moduli space of logarithmicconnections and change the notation to emphasis on X , that is, M lc ( X ) = M lc ( X, S ) := M lc ( n, d )and M lc ( X, L ) = M lc ( X, S, L ) := M lc ( n, L ) . First we show that the above moduli spaces does not depend on the choice of S (seeLemma 5.1 and Lemma 5.2), and therefore we remove S from the notation. We showthat (see Theorem 5.6), there are, up to isomorphism, only finitely many compactRiemann surface Y such that M lc ( Y, L ) is isomorphic to M lc ( X, L ).We also show the following (see Theorem 5.8).Let (
X, S ) and (
Y, T ) be two m -pointed compact Riemann surfaces of genus g ≥ M lc ( X, L ) and M lc ( Y, L ′ ) be the corresponding moduli spaces of logarithmic con-nections. Then, M lc ( X, L ) is isomorphic to M lc ( Y, L ′ ) if and only if X is isomorphicto Y .Next, we show the universal property of the morphism (see Proposition 5.9) G : M lc ( X ) −→ P ic d ( X )defined by sending ( E, D ) V n E . Thus, M lc ( X ) determines the pair ( P ic d ( X ) , G )up to an automorphism of P ic d ( X ). In the end of section 5, we present Torellitype theorem for M lc ( X ), that is, let ( X, S ) and (
Y, T ) be two m -pointed compactRiemann surfaces of genus g ≥
3. Let M lc ( X ) and M lc ( Y ) be the correspondingmoduli spaces of logarithmic connections. Then, M lc ( X ) is isomorphic to M lc ( Y )if and only if X is isomorphic to Y .In the last section 6, we talk about rationally connectedness and rationality of themoduli space. This section is motivated by the results in [20]. We show that themoduli spaces M lc ( n, d ) and M h ( n ) are not rational (see Theorem 6.3, and Theorem6.4, respectively). And finally we show that the moduli space M lc ( n, L ) is rationallyconnected (see Corollary 6.7). ODULI SPACES OF HOLOMORPHIC AND LOGARITHMIC CONNECTIONS 4 Preliminaries
We recall the notion of holomorphic and logarithmic connection in a holomorphicvector bundle over a smooth projective curve over C , that is, over a compact Riemannsurface.Let X be a compact Riemann surface of genus g ≥
3. Let E be a holomorphicvector bundle over X . A holomorphic connection in E is a C -linear map D : E → E ⊗ Ω X which satisfies the Leibniz rule D ( f s ) = f D ( s ) + df ⊗ s, (2.1)where f is a local section of O X and s is a local section of E .A theorem due to Atiyah [2] and Weil [39], which is known as the Atiyah-Weilcriterion , says that a holomorphic vector bundle over a compact Riemann surfaceadmits a holomorphic connection if and only if the degree of each indecomposablecomponent of the holomorphic vector bundle is zero. Thus, if a E admits a holomor-phic connection, then deg E = 0 . The slope µ ( E ) of E is defined as µ ( E ) = deg E rk( E ) . A holomorphic connection D in E is said to be semistable (respectively, stable )if for every non-zero proper subbundle F of E which is invariant under D , that is, D ( F ) ⊂ F ⊗ Ω X , we have, µ ( F ) ≤ µ ( F ) < , where µ ( E ) denotes the slope of E .Let M h ( n ) be the moduli space of semi-stable holomorphic connections of rank n .Then M h ( n ) is a normal quasi-projective variety of dimension 2 n ( g −
1) + 2. Let M smh ( n ) ⊂ M h ( n )be the smooth locus of the variety. Let M ′ h ( n ) ⊂ M smh ( n ) (2.2)be the open subvariety whose underlying vector bundle is stable. Then M ′ h ( n ) is anirreducible smooth quasi-projective variety of same dimension as of M h ( n ).We now define the logarithmic connection. Fix a finite subset S = { x , . . . , x m } of X such that x i = x j for all i = j . Let Z = x + · · · + x m ODULI SPACES OF HOLOMORPHIC AND LOGARITHMIC CONNECTIONS 5 denote the reduced effective divisor on X associated to the finite set S . Let Ω X (log Z )denote the sheaf of logarithmic differential 1-forms along Z , see [25] for more details.For the theory of the meromorphic and logarithmic connections, we refer two excel-lent sources [12] and [5].A logarithmic connection on E singular over S is a C -linear map D : E → E ⊗ Ω X (log Z ) = E ⊗ Ω X ⊗ O X ( Z ) (2.3)which satisfies the Leibniz identity D ( f s ) = f D ( s ) + df ⊗ s, (2.4)where f is a local section of O X and s is a local section of E .A logarithmic connection D in E is said to be semistable (respectively, stable )if for every non-zero proper subbundle F of E which is invariant under D , that is, D ( F ) ⊂ F ⊗ Ω X (log Z ) , we have, µ ( F ) ≤ µ ( E )(resp. µ ( F ) < µ ( E )) , where µ ( E ) denote the slope of E .We next describe the notion of residues of a logarithmic connection D in E singularover S . We will denote the fibre of E over any point x ∈ X by E ( x ).Let v ∈ E ( x β ) be any vector in the fiber of E over x β . Let U be an open setaround x β and s : U → E be a holomorphic section of E over U such that s ( x β ) = v .Consider the following compositionΓ( U, E ) → Γ( U, E ⊗ Ω X ⊗ O X ( S )) → E ⊗ Ω X ⊗ O X ( S )( x β ) = E ( x β ) , where the equality is given because for any x β ∈ S , the fibre Ω X ⊗ O X ( S )( x β ) iscanonically identified with C by sending a meromorphic form to its residue at x β .Then, we have an endomorphism on E ( x β ) sending v to D ( s )( x β ). We need to checkthat this endomorphism is well defined. Let s ′ : U → E be another holomorphicsection such that s ′ ( x β ) = v . Then( s − s ′ )( x β ) = v − v = 0 . Let t be a local coordinate at x β on U such that t ( x β ) = 0, that is, the coordinatesystem ( U, t ) is centered at x β . Since s − s ′ ∈ Γ( U, E ) and ( s − s ′ )( x β ) = 0, s − s ′ = tσ for some σ ∈ Γ( U, E ). Now, D ( s − s ′ ) = D ( tσ ) = tD ( σ ) + dt ⊗ σ = tD ( σ ) + t ( dtt ⊗ σ ) , and hence D ( s − s ′ )( x β ) = 0, that is, D ( s )( x β ) = D ( s ′ )( x β ).Thus, we have a well defined endomorphism, denoted by Res ( D, x β ) ∈ End( E )( x β ) = End( E ( x β )) (2.5)that sends v to D ( s )( x β ). This endomorphism Res ( D, x β ) is called the residue ofthe logarithmic connection D at the point x β ∈ S (see [12] for the details). ODULI SPACES OF HOLOMORPHIC AND LOGARITHMIC CONNECTIONS 6
From [23, Theorem 3], for a logarithmic connection D singular over S , we havedeg E + m X j =1 Tr(
Res ( D, x j )) = 0 , (2.6)where, deg E denotes the degree of E , and Tr( Res ( D, x j )) denote the trace of theendomorphism Res ( D, x j ) ∈ End( E ( x j )), for all j = 1 , . . . , m .Let LC ( E ) denote the set of all logarithmic connections in E singular over S . Then LC ( E ) is an affine space modelled over the vector space H ( X, End( E ) ⊗ Ω X (log Z )),that is, if D is any logarithmic connection in E singular over S , then LC ( E ) = D + H ( X, End( E ) ⊗ Ω X (log Z )) . Recall that an endomorphism φ ∈ End( E ( x j )) is said to be a rigid endomorphism iffor every global endomorphism α ∈ H ( X, End( E )) we have φ ◦ α ( x j ) = α ( x j ) ◦ φ, where α ( x j ) : E ( x j ) → E ( x j ) is an endomorphism.In what follows, we fix rigid endomorphisms Φ j ∈ End( E ( x j )), for every j =1 , . . . , m , such that for every direct summand F ⊂ E , we havedeg F + m X j =1 Tr(Φ j | F ( x j ) ) = 0 . (2.7)Here Tr(Φ j | F ( x j ) ) makes sense, because from [8, Theorem 1.3 (1)], for a rigid endo-morphism Φ j ∈ End( E ( x j )), and for every direct summand F of E , we haveΦ j ( F ( x j )) ⊂ F ( x j ) . Let LC ( E ; Φ , . . . , Φ m ) denote the set of all logarithmic connections singular over S with fixed residues Φ j for all j = 1 , . . . , m , that is, LC ( E ; Φ , . . . , Φ m )= { D | D is a logarithmic connection in E with Res ( D, x j ) = Φ j for all j = 1 , . . . , m } Lemma 2.1. LC ( E ; Φ , . . . , Φ m ) is an affine space modelled over H ( X, Ω X ⊗ End( E )) .Proof. Let
D, D ′ ∈ LC ( E ; Φ , . . . , Φ m ). Then, for every local sections s of E and f of O X , we have ( D − D ′ )( f s ) = f ( D − D ′ )( s ) . Therefore, D − D ′ is an O X -linear map. Let U be an open subset of X containingonly x j such that E | U is free of rank n . Let ( e , . . . , e n ) be a frame of E over U .Then s = f e + . . . + f n e n and( D − D ′ )( s ) = f ( D − D ′ )( e ) + · · · + f n ( D − D ′ )( e n ) , where f i ∈ O X ( U ) for 1 ≤ i ≤ n . Evaluating each ( D − D ′ )( e i ) for 1 ≤ i ≤ n , at x j , we have ( D − D ′ )( e i )( x j ) = 0, as both have the same residue Φ j at x j . This willimply that ( D − D ′ )( s ) ∈ Ω U ⊂ Ω U ⊗ O U ( x j ). Thus, ( D − D ′ ) maps E → E ⊗ Ω X ,which corresponds to an element of H ( X, Ω X ⊗ End( E )). ODULI SPACES OF HOLOMORPHIC AND LOGARITHMIC CONNECTIONS 7
Conversely, if D ∈ LC ( E ; Φ , . . . , Φ m ) and θ ∈ H ( X, Ω X ⊗ End( E )), then D + θ is a logarithmic connection singular over S with Res ( D, x j ) = Res ( D + θ, x j ) = Φ j for every x j ∈ S . (cid:3) Notice the difference between vector spaces when residue is fixed and otherwise.We impose some more conditions on the residues Φ j for 1 ≤ j ≤ m to get a ‘wellbehaved’ moduli space of logarithmic connections singular over S with fixed residues.Suppose that the residues (rigid endomorphism) Φ j for every j = 1 , . . . , m satisfiesthe following condition.(P1): For every non-zero subbundle F ⊂ E ,Φ j ( F ( x j )) ⊂ F ( x j ) , and Tr(Φ j | F ( x j ) )rk( F ) = Tr(Φ j )rk( E ) . If we take Φ j = α j E ( x j ) , where α j ∈ C and E ( x j ) is the identity morphism on E ( x j ), for every 1 ≤ j ≤ m , then { Φ j } ≤ j ≤ m satisfies (P1). In what follows, we takeΦ j = α j E ( x j ) for every j = 1 , . . . , m . Lemma 2.2.
Let D ∈ LC ( E ; Φ , . . . , Φ m ) with { Φ j } ≤ j ≤ m satisfying (P1). Then D is semi-stable. Moreover, if (deg E, rk( E )) = 1 , then D is stable.Proof. Let F be a non-zero proper subbundle of E such that D ( F ) ⊂ F ⊗ Ω X (log Z ).Then Res ( D | F , x j ) = Φ j | F ( x j ) , and from [23, Theorem 3], we havedeg F + m X j =1 Tr(Φ j | F ( x j ) ) = 0 . (2.8)From (P1), and (2.8), we getdeg F + rk( F )rk( E ) m X j =1 Tr(Φ j ) = 0 , which is nothing but deg F rk( F ) = deg E rk( E ) . Thus, µ ( F ) = µ ( E ), and hence D is semi-stable.Next, from the last equality µ ( F ) = µ ( E ), since rk( F ) < rk( E ), rk( E ) dividesdeg E , which contradicts the fact that deg E and rk( E ) are coprime. Thus, in thatcase there does not exist such F , and hence, D is stable, is vacuously true. (cid:3) A logarithmic connection D in a holomorphic vector bundle E is called irreducible if for any holomorphic subbundle F of E with D ( F ) ⊂ F ⊗ Ω X (log Z ), then either F = E or F = 0.If D ∈ LC ( E ; Φ , . . . , Φ m ) satisfies (P1), and (deg E, rk( E )) = 1, then D is irre-ducible.Let M lc ( n, d ) denote the moduli space which parametrizes the isomorphic class ofpairs ( E, D ), where, by a pair (
E, D ) we mean that
ODULI SPACES OF HOLOMORPHIC AND LOGARITHMIC CONNECTIONS 8 (1) E is a holomorphic vector bundle of rank n and degree d over X , such that( n, d ) = 1.(2) D is a logarithmic connections with fixed residues Res ( D, x j ) = Φ j satisfying(P1).And two pairs ( E, D ) and ( E ′ , D ′ ) satisfying above conditions (1) and (2) are saidto be isomorphic if there exists an isomorphism Ψ : E → E ′ such that the followingdiagram E Ψ (cid:15) (cid:15) D / / E ⊗ Ω X (log Z ) Ψ ⊗ Ω1 X (log Z ) (cid:15) (cid:15) E ′ D ′ / / E ′ ⊗ Ω X (log Z )commutes. Remark . Now onwards, we assume that the rank n and degree d are coprime.From [33, Theorem 3.5], the moduli space M lc ( n, d ) is a separated quasi-projectivescheme over C . Since { Φ j } ≤ j ≤ m satisfies (2.7), from [8, Theorem 1.3 (2)] the modulispace M lc ( n, d ) is non-empty.As we have observed that every logarithmic connection ( E, D ) in M lc ( n, d ) isirreducible, and the singular points of M lc ( n, d ) correspond to reducible logarithmicconnections [9, p.n. 790], the moduli space M lc ( n, d ) is smooth. Since genus g of X is greater than or equal to 3, the moduli space M lc ( n, d ) is irreducible [38, Theorem11.1].Altogether, M lc ( n, d ) is an irreducible smooth quasi-projective variety of dimen-sion 2 n ( g −
1) + 2. Let M ′ lc ( n, d ) ⊂ M lc ( n, d ) (2.9)be the moduli space of logarithmic connections whose underlying vector bundlesare stable. Then, from [22, p.635, Theorem 2.8(A)] M ′ lc ( n, d ) is an open subset of M lc ( n, d ), and hence an irreducible smooth quasi-projective variety of dimension2 n ( g −
1) + 2.Fix a holomorphic line bundle L over X of degree d , and a logarithmic connection D L on L singular over S with residues Res ( D L , x j ) = Tr(Φ j ) for all j = 1 , . . . , m .Let M lc ( n, L ) ⊂ M lc ( n, d ) (2.10)be the moduli space parametrising isomorphism class of pairs ( E, D ) such that( n ^ E, ˜ D ) ∼ = ( L, D L ) , where ˜ D is the logarithmic connection on V n E induced by D . Then, M lc ( n, L ) isan irreducible smooth quasi-projective variety of dimension 2( n − g − M ′ lc ( n, L ) ⊂ M lc ( n, L ) (2.11)be the moduli space of logarithmic connections ( E, D ) with E stable. ODULI SPACES OF HOLOMORPHIC AND LOGARITHMIC CONNECTIONS 9 Chow group of the moduli spaces
In this section, we determine the Chow groups of the moduli spaces M ′ lc ( n, L ), M ′ lc ( n, d ), M ′ h ( n ) and M ′ h ( n, L ).Before that we recall the definition of Chow group of a quasi-projective schemeover a field (see [29] and [35]).Let X be a quasi-projective scheme over a field K . Let Z k ( X ) be the free abeliangroup generated by the reduced and irreducible k -dimensional closed subvarieties of X , or we can say that free abelian group generated by a k -dimensional closed integralsubscheme of X . An element of Z k ( X ) is called a k -dimensional algebraic cycle on X .Let f ∈ K ( X ) ∗ . Then, we have a divisor div( f ) on X associated to the non-zerorational function f on X .A k -cycle α is rationally equivalent to zero , written α ∼
0, if there are finitenumber of ( k + 1)-dimensional subvarieties (that is, closed integral subschemes) W i of X and f i ∈ K ( W i ) ∗ , such that α = X i div( f i ) . Since 0 = div(1) and div( f − ) = − div( f ), the cycles rationally equivalent to zeroform a subgroup Z k ( X ) rat of Z k ( X ).We define the quotient group CH k ( X ) := Z k ( X ) /Z k ( X ) rat , and call it the Chow group of k -cycles on X . A graded sum is denoted by CH ∗ ( X ) = dim( X ) M k =0 CH k ( X ) . The Chow group of k -cycles on X in the rational coefficients will be denoted byCH Q k ( X ).Let U ( n, L ) be the moduli space of stable vector bundles of rank n with V n E ∼ = L .Then U ( n, L ) is a smooth projective variety of dimension ( n − g − n and deg( L ) = d are coprime.Let x ∈ X , and O X ( x ) the line bundle on X associated with the reduced effectivedivisor x . For n = 2, we have U (2 , O X ( x )) the moduli space of stable vector bundlesof rank 2 over X whose determinant is O X ( x ).In [4], it was shown thatCH Q g − ( U (2 , O X ( x ))) ∼ = CH Q ( X ) M Q . (3.1)In [11], Choe and Hwang computed the Chow group of 1-cycles on U (2 , O X ( x )),and they proved that CH Q ( U (2 , O X ( x ))) ∼ = CH Q ( X ) . (3.2)Let M be a holomorphic line bundle over X of degree d ′ and M = O X ( x ) ⊗ M ⊗ . ODULI SPACES OF HOLOMORPHIC AND LOGARITHMIC CONNECTIONS 10
Then deg( M ) = 2 d ′ + 1. Define a mapΨ M : U (2 , O X ( x )) → U (2 , M )by Ψ M ([ E ]) = [ E ⊗ M ] . The map is well-defined, and in fact, an isomorphism of varieties.Thus, the above results (3.1) and (3.2) are true for U (2 , L ), where deg( L ) is odd.Define p : M ′ lc ( n, L ) → U ( n, L ) (3.3)by sending ( E, D ) E , that is, p is the forgetful map which forgets its logarithmicstructure.For every E ∈ U ( n, L ), p − ( E ) is an affine space modelled over H ( X, Ω X ⊗ ad( E )),where ad( E ) ⊂ End( E ) is the subbundle consists of endomorphism of E whose traceis zero. Actually, p is a fibre bundle and using Riemann-Roch theorem and Serreduality it can be easily computed that the dimension of p − ( E ) is ( n − g − U ( n,L ) denote the holomorphic cotangent bundle on U ( n, L ). Then, for a point E ∈ U ( n, L ), the cotangent space Ω U ( n,L ) ,E at E is isomorphic to H ( X, Ω X ⊗ ad( E )).Thus, the action Ω U ( n,L ) ,E × p − ( E ) → p − ( E )given by ( ω, D ) ω + D is faithful and transitive. In fact, M ′ lc ( n, L ) is an Ω U ( n,L ) -torsor on U ( n, L ).We state two standard lemmas from the theory of Chow groups which we will useto compute the Chow group of moduli spaces.Let Y be a variety over a field K . Let i : F → Y be the inclusion of a closedsubscheme. Let j : U = Y \ F → Y be the inclusion of the complement. Since j isan open immersion, it is flat, and i is a closed immersion, it is proper. Therefore, wehave morphisms of j ∗ : CH k ( Y ) → CH k ( U ) and i ∗ : CH k ( F ) → CH k ( Y ) of Chowgroups and j ∗ ◦ i ∗ = 0, since the cycles supported on F do not intersect U . Thus, wehave what is called localisation sequence . Lemma 3.1. [35, Lemma 9.12]
The following sequence of abelian groups CH l ( F ) i ∗ −→ CH l ( Y ) j ∗ −→ CH l ( U ) → . (3.4) is exact for every l = 0 , . . . , dim( Y ) . Theorem 3.2. [35, Theorem 9.25]
Let π : P ( E ) → Y be a projective bundle withrank rk( E ) = r . Then the map r − M k =0 h k π ∗ : r − M k =0 CH l − r +1+ k ( Y ) → CH l ( P ( E )) (3.5) is an isomorphism, where h ∈ Pic( P ( E )) denote the class of the tautological linebundle O P ( E ) (1) . Theorem 3.3.
For every ≤ l ≤ ( n − g − , we have canonical isomorphism CH l +( n − g − ( M ′ lc ( n, L )) ∼ = CH l ( U ( n, L )) . (3.6) ODULI SPACES OF HOLOMORPHIC AND LOGARITHMIC CONNECTIONS 11
Proof.
From [26, Proposition 5.2], there exists an algebraic vector bundle F over U ( n, L ) with M ′ lc ( n, L ) is embedded in P ( F ) such that the compliment P ( F ) \M ′ lc ( n, L ) is a hyperplane H at infinity. We shall sketch the construction of F ,to connect the pieces of the proof. Since the dual of an affine space (modelled ona vector space) is a vector space, similarly, the dual of a torsor is a vector bundle,we use this fact to construct F . Let E ∈ U ( n, L ). Then, p − ( E ) is an affine spacemodelled on H ( X, Ω X ⊗ ad( E )). Consider the dual p − ( E ) ∨ = { ϕ : p − ( E ) → C | ϕ is an affine linear map } , which is a vector space over C of dimension ( n − g −
1) + 1.For every open subset of U ⊂ U ( n, L ), define a presheaf F ( U ) consists of algebraicfunctions f : p − ( U ) → C whose restriction to each fibre p − ( E ) is an affine linearmap from p − ( E ) → C , that is, f | p − ( E ) ∈ p − ( E ) ∨ . The presheaf is in fact a locallyfree sheaf, let π : F → U ( n, L )be a vector bundle associated to the locally free sheaf. Thus, a fiber F ( E ) = π − ( E )of F at E ∈ U ( n, d ) is p − ( E ) ∨ .Let ( E, D ) ∈ M ′ lc ( n, L ), and define the evaluation map ev ( E,D ) : p − ( E ) ∨ → C by ev ( E,D ) ( ϕ ) = ϕ [( E, D )] . The kernel Ker(ev ( E,D ) ) defines a hyperplane in p − ( E ) ∨ denoted by Π ( E,D ) . Let P ( F ) be a projective bundle defined by hyperplanes in thefibre p − ( E ) ∨ , that is, we have ˜ π : P ( F ) → U ( n, L )induced from π . Define a map ι : M ′ lc ( n, L ) → P ( F )by sending ( E, D ) to the equivalence class of Π ( E,D ) , which is clearly an open em-bedding. Set H = P ( F ) \ M ′ lc ( n, L ) . (3.7)We call H to be the hyperplane at infinity. Then ˜ π − ( E ) ∩ H is a hyperplane in˜ π − ( E ) for every E ∈ U ( n, L ).Next, the hyperplane at infinity H as defined in (3.7) is canonically identified withthe total space of the projective bundle P ( T U ( n, L )), the space of all hyperplanes inthe fibre of the tangent bundle T U ( n, L ).Let E ∈ U ( n, L ) and α ∈ ˜ π − ( E ) ∩ H . Then α represents a hyperplane in π − ( E ),and we denote this hyperplane by ∆ α . Let f ∈ ∆ α ⊂ π − ( E ) = p − ( E ) ∨ . Since p − ( E ) is an affine space modelled over H ( X, Ω X ⊗ ad( E )) ∼ = Ω U ( n,L ) ,E , the differ-ential df ∈ (Ω U ( n,L ) ,E ) ∗ = T E U ( n, L ) , where T E U ( n, L ) is the tangent space on U ( n, L ) at E . Since α ∈ H , the set { df | f ∈ ∆ α } spans a hyperplane in T E U ( n, L ), and hence a point ˜ α ∈ P ( T E U ( n, L )). Thus, weget a map α ˜ α : H → P ( T U ( n, L )) , ODULI SPACES OF HOLOMORPHIC AND LOGARITHMIC CONNECTIONS 12 which is an isomorphism.Put F = P ( T U ( n, L )), Y = P ( F ) and U = M ′ lc ( n, L ) in Lemma 3.1, we get anexact sequence CH l ( P ( T U ( n, L ))) i ∗ −→ CH l ( P ( F )) j ∗ −→ CH l ( M ′ lc ( n, L )) → . (3.8)of abelian groups for every l = 0 , . . . , dim( P ( F )) = 2( n − g − F ) = ( n − g −
1) + 1, and rk( T U ( n, L )) = ( n − g − CH l ( P ( F )) ∼ = ( n − g − M k =0 CH l − ( n − g − k ( U ( n, L )) (3.9)and CH l ( P ( T U ( n, L ))) ∼ = ( n − g − − M k =0 CH l − ( n − g − k ( U ( n, L )) . (3.10)From (3.8), (3.9) and (3.10), we get an exact sequence ( n − g − − M k =0 CH l − ( n − g − k ( U ( n, L )) i ∗ −→ ( n − g − M k =0 CH l − ( n − g − k ( U ( n, L )) j ∗ −→ CH l ( M ′ lc ( n, L )) → , (3.11)which is actually a short exact sequence, because i ∗ is injective. Thus, we have CH l ( M ′ lc ( n, L )) ∼ = CH l − ( n − g − ( U ( n, L )) , (3.12)for every ( n − g − ≤ l ≤ n − g − l , we will get thedesired result, and this completes the proof. (cid:3) Corollary 3.4.
For l = 2( n − g − − , we have CH l ( M ′ lc ( n, L )) ∼ = Z . Proof.
See [26, Proposition 5.3]. (cid:3)
Corollary 3.5.
For n = 2 , we have (1) CH g − ( M ′ lc (2 , L )) ∼ = Z . (2) CH Q g − ( M ′ lc (2 , L )) ∼ = CH Q ( X ) . (3) CH Q g − ( M ′ lc (2 , L )) ∼ = CH Q ( X ) ⊕ Q .Proof. From Theorem 3.3, and equations (3.1) and (3.2), we conclude the Corollary. (cid:3)
Next, let U ( n, d ) be the moduli space of stable vector bundle of rank n and degree d . Consider the following natural morphism p : M ′ lc ( n, d ) → U ( n, d ) (3.13) ODULI SPACES OF HOLOMORPHIC AND LOGARITHMIC CONNECTIONS 13 sending (
E, D ) to E . Then p − ( E ) is an affine space modelled over the vector spaceH ( X, Ω X ⊗ End( E )). Since E is stable, the dimension of the vector space H ( X, Ω X ⊗ End( E )) is n ( g −
1) + 1. In view of [26, Theorem 1.1], we can show a result similarto the Theorem 3.3, which interprets the Chow groups of M ′ lc ( n, d ) in terms of Chowgroups of U ( n, d ). Theorem 3.6.
For every ≤ l ≤ n ( g −
1) + 1 , we have canonical isomorphism CH l + n ( g − ( M ′ lc ( n, d )) ∼ = CH l ( U ( n, d )) . (3.14)Now, we compute the same for the moduli space of holomorphic connections. Fixa holomorphic line bundle L degree 0 on X . Let U ss ( n, L ) denote the moduli spaceof semi-stable holomorphic vector bundles of rank n and determinant V n E ∼ = L .Then the moduli space U ss ( n, L ) is known to be an irreducible normal projectivevariety of dimension ( n − g − U ( n, L ) ⊂ U ss ( n, L ) (3.15)be the open subvariety parametrizing the stable bundle on X . This open subvarietycoincides with the smooth locus of U ss ( n, L ) follows from [30, p. 20, Theorem 1].Fix a holomorphic connection D L on L . Let M h ( n, L ) be the moduli spaceof holomorphic connections parametrising the isomorphism class of the pairs ( E, D )where E is a holomorphic vector bundle of rank n with( n ^ E, ˜ D ) ∼ = ( L , D L ) , and ˜ D is a holomorphic connection on V n E induced from D . Then M h ( n, L ) is anirreducible normal quasi-projective variety of dimension 2( n − g − M smh ( n, L ) ⊂ M h ( n, L )be the smooth locus of M h ( n, L ). Let M ′ h ( n, L ) ⊂ M smh ( n, L ) (3.16)be the subset consists of holomorphic connections whose underlying vector bundle isstable. Then M ′ h ( n, L ) is an irreducible smooth quasi-projective variety of dimen-sion 2( n − g − q : M ′ h ( n, L ) → U ( n, L ) (3.17)be the forgetful map which forgets the holomorphic connection. Then for every E ∈ U ( n, L ), q − ( E ) is an affine space modelled over H ( X, Ω X ⊗ ad( E )). In fact, M ′ h ( n, L ) is an Ω U ( n,L ) -torsor on U ( n, L ).Let Y be an N -dimensional smooth quasi-projective variety. Then, the Picardgroup Pic( Y ) ⊗ Z Q can be identified with CH Q N − ( Y ). Thus, it is enough to computePic( Y ).The morphism q as defined in (3.17) will induces a homomorphism q ∗ : Pic( U ( n, L )) → Pic( M ′ h ( n, L )) (3.18) ODULI SPACES OF HOLOMORPHIC AND LOGARITHMIC CONNECTIONS 14 of Picard groups given by sending a line bundle M to its pull-back q ∗ M . Using thesimilar techniques as in [26, Theorem 1.2], we can show the following. Proposition 3.7.
The homomorphism q ∗ : Pic( U ( n, L )) → Pic( M ′ h ( n, L )) is anisomorphism of groups. Since Pic( U ( n, L )) ∼ = Z , we have Corollary 3.8.
For l = 2( n − g − − , we have CH l ( M ′ h ( n, L )) ∼ = Z . Using the exactly similar steps as in Theorem 3.3, we can prove the following.
Theorem 3.9.
For every ≤ l ≤ ( n − g − , we have canonical isomorphism CH l +( n − g − ( M ′ h ( n, L )) ∼ = CH l ( U ( n, L )) . (3.19)Next, let U ( n ) := U ( n,
0) be the moduli space of stable bundle of rank n anddegree zero. Then U ( n ) is an irreducible smooth projective variety of dimension n ( g −
1) + 1. Again, we have a natural morphism q : M ′ h ( n ) → U ( n ) (3.20)of varieties which forgets the holomorphic connection. Using the same method asabove, we have the following theorem similar to the Theorem 3.6. Theorem 3.10.
For every ≤ l ≤ n ( g −
1) + 1 , we have canonical isomorphism CH l + n ( g − ( M ′ h ( n )) ∼ = CH l ( U ( n )) . (3.21)4. Differential operators on the moduli spaces
In [6], Biswas studied the global sections of sheaf of differential operators on anample line bundle over a polarised abelian variety. Also, in [28], Hitchin variety isdefined and global sections of sheaf of k -th order differential operators, and symmetricpowers of sheaf of first order differential operators on a line bundle over a Hitchinvariety have been studied. The moduli space of stable vector bundles over a compactRiemann surface is an example of Hitchin variety. The moduli space of holomorphicand logarithmic connections are not Hitchin varieties. In this section, we study theglobal sections of certain sheaves over the four moduli spaces M ′ lc ( n, d ), M ′ h ( n ), M ′ lc ( n, L ) and M ′ h ( n, L ) which we have defined in previous section.Let ζ be an ample line bundle over M ′ lc ( n, d ). Let k ≥ k on ζ is a C -linear map θ : ζ → ζ (4.1)such that for every open subset U of M ′ lc ( n, d ) and for every f ∈ O M ′ lc ( n,d ) ( U ), thebracket [ θ | U , f ] : ζ | U → ζ | U defined as [ θ | U , f ] V ( s ) = θ ( f | V s ) − θ | V P V ( s )is a differential operator of order k −
1, for every open subset V of U , and for all s ∈ ζ ( V ), where differential operator of order zero on ζ is just O M ′ lc ( n,d ) -module ODULI SPACES OF HOLOMORPHIC AND LOGARITHMIC CONNECTIONS 15 homomorphism (see [15] and [24] for the definition and properties of differentialoperators).For k ≥
0, let D k ( ζ ) denote the sheaf of differential operators on ζ of order k . In fact, D k ( ζ ) is a locally free sheaf with D ( ζ ) = O M ′ lc ( n,d ) . Given a firstorder differential operator θ on ζ , we get a section of the tangent bundle T M ′ lc ( n, d )denoted by σ ( θ ), where σ is called symbol of a first order differential operator. Forsimplicity, we shall denote T M ′ lc ( n, d ) by T M ′ lc . Thus, consider the symbol operator σ : D ( ζ ) → T M ′ lc . This induces a morphism S ym k ( σ ) : S ym k ( D ( ζ )) → S ym k ( T M ′ lc )of k -th symmetric powers. Now, because of the following composition O M ′ lc ( n,d ) ⊗ S ym k − ( D ( ζ )) ֒ → D ( ζ ) ⊗ S ym k − ( D ( ζ )) → S ym k ( D ( ζ )) , we have S ym k − ( D ( ζ )) ⊂ S ym k ( D ( ζ )) for all k ≥ . (4.2)Thus, we get a short exact sequence0 → S ym k − ( D ( ζ )) → S ym k ( D ( ζ )) S ym k ( σ ) −−−−−→ S ym k ( T M ′ lc ) → . (4.3)Thus, we have S ym k ( D ( ζ )) / S ym k − ( D ( ζ )) ∼ = S ym k ( T M ′ lc ) for all k ≥ . (4.4)From (4.2), we have the following chain of C -vector spacesH ( M ′ lc ( n, d ) , O M ′ lc ( n,d ) ) ⊂ H ( M ′ lc ( n, d ) , S ym ( D ( ζ ))) ⊂ . . . (4.5)Consider the following commutative diagram, T ∗ M ′ lcπ ′ (cid:15) (cid:15) T ∗ U ( n, d ) e p o o π (cid:15) (cid:15) M ′ lc ( n, d ) p / / U ( n, d ) (4.6)where π , π ′ are the canonical projections and e p is induced from p as defined in(3.13). Thus, we have a morphism e p ♯ : H ( T ∗ M ′ lc , O T ∗ M ′ lc ) → H ( T ∗ U ( n, d ) , O T ∗ U ( n,d ) ) . (4.7)of vector spaces induced from e p . Theorem 4.1.
Suppose that e p ♯ in (4.7) is an injective morphism. Then, for every k ≥ , we have H ( M ′ lc ( n, d ) , S ym k ( D ( ζ ))) = C . (4.8) Proof.
Let M lcX := M lc (1 , d ) (4.9)be the moduli space of rank one logarithmic connections singular over S , with fixedresidues Tr(Φ j ) for every j = 1 , . . . , m , for more details, see [36] and [27]. Then thereis a natural morphism of varietiesdet : M ′ lc ( n, d ) −→ M lcX (4.10) ODULI SPACES OF HOLOMORPHIC AND LOGARITHMIC CONNECTIONS 16 sending (
E, D ) ( V n E, e D ), where e D is the induced logarithmic connection on V n E . For any pair ( L, ∇ ) ∈ M lcX ,det − (( L, ∇ )) = M ′ lc ( n, L ) . From [36, Theorem 2], we have H ( M lcX , O M lcX ) = C , and from [26, Theorem 1.4], we haveH ( M ′ lc ( n, L ) , O M ′ lc ( n,L ) ) = C . Combining both the results and using det morphism in (4.10), we haveH ( M ′ lc ( n, d ) , O M ′ lc ( n,d ) ) = C . Thus from (4.5), it is enough to show that for every k ≥
0, the inclusionH ( M ′ lc ( n, d ) , O M ′ lc ( n,d ) ) → H ( M ′ lc ( n, d ) , S ym k ( D ( ζ )))is an isomorphism. From the isomorphism in (4.4), we have the following commuta-tive diagram0 / / S ym k − ( D ( ζ )) (cid:15) (cid:15) / / S ym k ( D ( ζ )) (cid:15) (cid:15) S ym k ( σ ) / / S ym k ( T M ′ lc ) (cid:15) (cid:15) / / / / S ym k − ( T M ′ lc ) / / S ym k ( D ( ζ )) S ym k − ( D ( ζ )) / / S ym k ( T M ′ lc ) / / · · · / / H ( M ′ lc ( n, d ) , S ym k T M ′ lc ) (cid:15) (cid:15) δ ′ k / / H ( M lc ( n, d ) ′ , S ym k − ( D ( ζ ))) (cid:15) (cid:15) / / · · ·· · · / / H ( M ′ lc ( n, d ) , S ym k T M ′ lc ) δ k / / H ( M ′ lc ( n, d ) , S ym k − T M ′ lc ) / / · · · (4.12)In order to prove the theorem, it is enough to show that the connecting homo-morphism δ ′ k , depicted in the above commutative diagram (4.12), is injective for all k ≥
1. Again from the above commutative diagram (4.12), δ ′ k is injective for every k ≥ δ k : H ( M ′ lc ( n, d ) , S ym k T M ′ lc ) → H ( M ′ lc ( n, d ) , S ym k − T M ′ lc ) (4.13)is injective for every k ≥ ζ ) ∈ H ( M ′ lc ( n, d ) , T ∗ M ′ lc ) be the Atiyah class of the line bundle ζ , whichis nothing but the extension class of the Atiyah exact sequence (see [2])0 → O M ′ lc → D ( ζ ) σ −→ T M ′ lc → . (4.14)The Atiyah class at( ζ ) determines the first Chern class c ( ζ ) of the line bundle ζ .Let γ k be the extension class of the short exact sequence (4.3). Since the shortexact sequence (4.3) is the symmetric power of (4.14), the extension class γ k can be ODULI SPACES OF HOLOMORPHIC AND LOGARITHMIC CONNECTIONS 17 expressed in terms of the first Chern class c ( ζ ). Further, let α k denote the extensionclass of the following short exact sequence0 → S ym k − ( T M ′ lc ) → S ym k ( D ( ζ )) S ym k − ( D ( ζ )) → S ym k ( T M ′ lc ) → , (4.15)which is the bottom short exact sequence in the commutative diagram (4.11). Then γ k maps to α k . Thus, α k can also be described in terms of the first Chern class c ( ζ ).Since a connecting homomorphism can be expressed as the cup product by theextension class of the corresponding short exact sequence, the connecting homomor-phism δ k in (4.13) can be described using the first Chern class c ( ζ ) of the line bundle ζ . Indeed, the cup product with c ( ζ ) gives rise to a homomorphism τ : H ( M ′ lc ( n, d ) , S ym k T M ′ lc ) → H ( M ′ lc ( n, d ) , S ym k T M ′ lc ⊗ T ∗ M ′ lc ) . (4.16)The canonical homomorphism υ : S ym k T M ′ lc ⊗ T ∗ M ′ lc → S k − T M ′ lc induces a morphism of C -vector spaces υ ∗ : H ( M ′ lc ( n, d ) , S ym k T M ′ lc ⊗ T ∗ M ′ lc ) → H ( M ′ lc ( n, d ) , S ym k − T M ′ lc ) . (4.17)Thus, we get a morphism˜ τ = υ ∗ ◦ τ : H ( M ′ lc ( n, d ) , S ym k T M ′ lc ) → H ( M ′ lc ( n, d ) , S ym k − T M ′ lc ) , (4.18)Then from the above observation we have ˜ τ = δ k . It is sufficient to show that ˜ τ isinjective.Moreover, we have the natural projection η : T ∗ M ′ lc → M ′ lc ( n, d ) (4.19)and η ∗ η ∗ O M ′ lc ( n,d ) = ⊕ k ≥ S ym k T M ′ lc . (4.20)Thus, we haveH j ( T ∗ M ′ lc , O T ∗ M ′ lc ) = ⊕ k ≥ H j ( M ′ lc ( n, d ) , S ym k T M ′ lc ) for all j ≥ . (4.21)Now, we would like to determine the cohomology group H ( T ∗ M ′ lc , O T ∗ M ′ lc ). Be-cause of the assumption that e p ♯ is injective, we shall first understand the nature ofthe group H ( T ∗ U ( n, d ) , O T ∗ U ( n,d ) ), and for that we use the Hitchin fibration.Let M Higgs ( n, d ) be the moduli space of Higgs bundles over the compact Riemannsurface. Let H : M Higgs ( n, d ) → B n = ⊕ ni =1 H ( X, K iX ) (4.22)be the Hitchin map (see [17]) defined by sending a pair ( E, φ ) to P ni =1 trace ( φ i ),where φ : E → E ⊗ Ω X is an O X -linear map. Notice that the base B n of the Hithcin map H in (4.22) isa vector space over C of dimension n ( g −
1) + 1. We know that the cotangentbundle T ∗ U ( n, d ) of the moduli space U ( n, d ) is an open subset of M Higgs ( n, d ) withcodimension of M Higgs ( n, d ) \ T ∗ U ( n, d ) in M Higgs is at least 2.
ODULI SPACES OF HOLOMORPHIC AND LOGARITHMIC CONNECTIONS 18
Let g : T ∗ U ( n, d ) → C be an algebraic function. Since codimension of M Higgs ( n, d ) \ T ∗ U ( n, d ) in M Higgs is at least 2, g extends by Hartog’s theorem to an algebraic func-tion b g on M Higgs . Since the generic fibres of H are abelian varieties, this will inducean algebraic function e g : B n → C , such that b g = e g ◦ H. Thus, for every algebraic function on T ∗ U ( n, d ), we have associated an algebraicfunction on B n .Now, set B n = d(H ( B n , O B n )) ⊂ H ( B n , Ω B n ) the space of all exact algebraic1-form. Define a map θ : H ( T ∗ U ( n, d ) , O T ∗ U ( n,d ) ) → B n (4.23)by g d ˜ g , where ˜ g is the function on B n which is defined by descent of g as above.Then θ is an isomorphism.Under the assumption that e p ♯ is injective and from (4.21),(4.23), we get a mor-phism ̺ = θ ◦ e p ♯ : ⊕ k ≥ H ( M ′ lc ( n, d ) , S ym k T M ′ lc ) → B n (4.24)which is injective.Now, we restrict H on T ∗ U ( n, d ), and let T H = T T ∗ U ( n,d ) /B n = K er ( d H )be the relative tangent sheaf on T ∗ U ( n, d ), where d H : T ( T ∗ U ( n, d )) → H ∗ T B n is a morphism of bundles. Now, we use the fact that the vector fields on genericfibres H − ( v ) are constant, therefore, the pulled back bundle H ∗ T ∗ B n is identifiedwith T H , therefore, we haveH ( B n , Ω B n ) ⊆ H ( T ∗ U ( n, d ) , T H ) = H ( T ∗ M ′ lc , ( e p ) ∗ T H )and hence from (4.24), we have an injective homomorphism ν : ⊕ k ≥ ̺ (H ( M ′ lc ( n, d ) , S ym k T M ′ lc )) → H ( T ∗ M ′ lc , ( e p ) ∗ T H ) , (4.25)where ( e p ) ∗ T H is push forward of the relative tangent bundle T H under e p defined in(4.6).Consider the morphismH ( T ∗ M ′ lc , ( e p ) ∗ T H ) → H ( T ∗ M ′ lc , ( e p ) ∗ T H ⊗ T ∗ T ∗ M ′ lc )defined by taking cup product with the first Chern class c ( η ∗ ζ ) ∈ H ( T ∗ M ′ lc , T ∗ T ∗ M ′ lc ) . Further, composing the above morphism with the morphism induced from the pairingpairing ( e p ) ∗ T H ⊗ T ∗ T ∗ M ′ lc → O T ∗ M ′ lc , we get a homomorphism ψ : H ( T ∗ M ′ lc , ( e p ) ∗ T H ) → H ( T ∗ M ′ lc , O T ∗ M ′ lc ) (4.26) ODULI SPACES OF HOLOMORPHIC AND LOGARITHMIC CONNECTIONS 19 of vector spaces. Since c ( η ∗ ζ ) = η ∗ ( c ζ ), we have ψ ◦ ν ◦ ̺ ( ω ) = ˜ τ ( ω ) , (4.27)for all ω ∈ H ( M ′ lc ( n, d ) , S ym k T M ′ lc ). Since ν and ̺ are injective homomorphisms,it is enough to show that ψ | ν ( B n ) is injective homomorphism. Let ω ∈ B n \ { } be anon-zero exact 1-form. Choose u ∈ B n such that ω ( u ) = 0. A generic fibre of H (see[3])can be expressed as H − ( u ) ∩ T ∗ U ( n, d ) = A u \ F u , where A u is an abelian variety and F u is a subvariety of A u such that codim( F u , A u ) ≥
2. Now, ψ ( ν ( ω )) ∈ H ( T ∗ M ′ lc , O T ∗ M ′ lc ) and we have the following morphisms inducedfrom the restriction mapsH ( T ∗ M ′ lc , O T ∗ M ′ lc ) → H ( T ∗ U ( n, d ) , O T ∗ U ( n,d ) ) → H ( A u \ F u , O H − ( u ) ∩ T ∗ U ( n,d ) ) . Since ω ( u ) = 0, ψ ( ν ( ω )) ∈ H ( H − ( u ) ∩ T ∗ U ( n, d ) , O H − ( u ) ∩ T ∗ U ( n,d ) ). Because of thefollowing isomorphismsH ( H − ( u ) ∩ T ∗ U ( n, d ) , O H − ( u ) ∩ T ∗ U ( n,d ) ) ∼ = H ( A u , O A u ) ∼ = H ( A u , T A u ) , it follows that ψ ( ν ( ω )) = 0. This completes the proof. (cid:3) Proposition 4.2.
Under the hypothesis of the Theorem 4.1, for k ≥ , we have H ( M ′ lc ( n, d ) , D k ( ζ )) = C . (4.28) Proof.
Consider the following increasing chain of sheaves of differential operators onthe line bundle ζ . O M ′ lc ( n,d ) = D ( ζ ) ⊂ · · · ⊂ D k ( ζ ) ⊂ D k +1 ( ζ ) ⊂ · · · It is enough to show that for every k ≥ ( M ′ lc ( n, d ) , D k − ( ζ )) → H ( M ′ lc ( n, d ) , D k ( ζ )) . Consider the following commutative diagram,0 / / D k − ( ζ ) (cid:15) (cid:15) / / D k ( ζ ) (cid:15) (cid:15) σ k / / S ym k ( T M ′ lc ) (cid:15) (cid:15) / / / / S ym k − ( T M ′ lc ) / / D k ( ζ ) D k − ( ζ ) / / S ym k ( T M ′ lc ) / / σ k is the symbol operator. Above diagram induces a commutative diagram oflong exact sequences as follows. · · · / / H ( M ′ lc ( n, d ) , S ym k T M ′ lc ) (cid:15) (cid:15) δ ′ k / / H ( M ′ lc ( n, d ) , D k − ( ζ )) (cid:15) (cid:15) / / · · ·· · · / / H ( M ′ lc ( n, d ) , S ym k T M ′ lc ) δ k / / H ( M ′ lc ( n, d ) , S ym k − T M ′ lc ) / / · · · (4.30)Now, proof follows from the similar steps as in Theorem 4.1. (cid:3) ODULI SPACES OF HOLOMORPHIC AND LOGARITHMIC CONNECTIONS 20
Under the same hypothesis of the Theorem 4.1 for the corresponding the modulispaces, we have
Theorem 4.3.
Suppose that the hypothesis of Theorem 4.1 holds for the moduli space X , where X denote M ′ lc ( n, L ) , M ′ h ( n ) or M ′ h ( n, L ) . Let ζ be a line bundle over X .Then, for every k ≥ , we have (1) H ( X , S ym k ( D ( ζ ))) = C . (2) H ( X , D k ( ζ )) = C . In [9], global sections of a line bundle on the moduli space of logarithmic con-nections singular exactly over one point of a compact Riemann surface have beenstudied. In this section, we study global sections of line bundles over M ′ h ( n, L ) and M ′ lc ( n, L ).Let L be a line bundle over M ′ h ( n, L ). Then L = q ∗ Θ l (4.31)for some l ∈ Z , where q is the morphism defined in (3.17) and Θ is the canonicalgenerator of Pic( U ( n, L )) ∼ = Z , corresponding to 1, that is, Θ is the generalisedtheta line bundle over U ( n, L ). Theorem 4.4.
For every l < , we have H ( M ′ h ( n, L ) , q ∗ Θ l ) = 0 . (4.32) Proof.
Let C (Θ) be the sheaf on U ( n, L ) whose sections over an open subset U ⊂U ( n, L ) are holomorphic connections in Θ | U . For every U ⊂ U ( n, L ), the space C (Θ) | U of holomorphic connections is an affine space modelled over H ( U, Ω U ( n,L ) ).Therefore, C (Θ) is an Ω U ( n,L ) - torsor . The isomorphism classes of Ω U ( n,L ) - torsors are parametrised by H ( U ( n, L ) , Ω U ( n,L ) ). Sincedim C H ( U ( n, L ) , Ω U ( n,L ) ) = 1 , any two non-zero classes in the cohomology group H ( U ( n, L ) , Ω U ( n,L ) ) are iso-morphic if and only if one is non-zero constant multiple of the other. Now, thecohomology class associated with the two Ω U ( n,L ) -torsors C (Θ) and M sh ( n, L ) arenon-zero. Therefore, the two Ω U ( n,L ) - torsors C (Θ) and M sh ( n, L ) on U ( n, L ) areisomorphic. Thus, (4) is equivalent toH ( C (Θ) , q ∗ Θ l ) = 0 , (4.33)for all l <
0. Now, consider the Atiyah exact sequence0 → O U ( n,L ) ı −→ At(Θ) σ −→ T U ( n, L ) → , (4.34)where At(Θ) is called the Atiyah bundle associated with the holomorphic line bun-dle Θ; see [2]. Let P (At(Θ)) be the projectivization of At(Θ), that is, P (At(Θ))parametrises hyperplanes in At(Θ). Let P ( T U ( n, L )) be the projectivization of thetangent bundle T U ( n, L ). Notice that P ( T U ( n, L )) is a subvariety of P (At(Θ)),and P ( T U ( n, L )) is the zero locus of the of a section of the tautological line bundle O P (At(Θ)) (1). Now, observe that C (Θ) = P (At(Θ)) \ P ( T U ( n, L )). Now we are in the ODULI SPACES OF HOLOMORPHIC AND LOGARITHMIC CONNECTIONS 21 same situation as in [9, p.797, Theorem 4.3], and the same techniques will work inour case too. (cid:3)
Similarly, for any line bundle L over M ′ lc ( n, L ), we have L = p ∗ ˜Θ l , for some l ∈ Z , where p is the morphism defined in (3.3) and ˜Θ is the generalisedtheta line bundle over U ( n, L ). Then we have a natural generalisation of [9, p.797,Theorem 4.3], and the same ideas can be used to prove the following. Theorem 4.5.
For every l < , we have H ( M ′ lc ( n, L ) , p ∗ ˜Θ l ) = 0 . (4.35)5. Torelli type theorem for the moduli spaces
In [7, Theorem 5.2], Torelli type theorem has been proved for the moduli space ofholomorphic connections over compact Riemann surface, and in [10], Torelli typetheorems have been proved for the moduli space of logarithmic connections singularexactly over one point with fixed residue. In this section, we prove the Torelli typetheorem for the moduli spaces M lc ( n, d ) and M lc ( n, L ). We assume that Φ j = α j E ( x j ) , for every j = 1 , . . . , m , where α j ∈ C .We show that the isomorphic class of the moduli spaces M lc ( n, d ) and M lc ( n, L )do not depend on the choice of S . Let T = { y , . . . , y m } be a finite subset of X suchthat y i = y j for i = j . Note that ♯S = ♯T .In this section, we use the following notations M lc ( X, S ) := M lc ( n, d ) , and M lc ( X, S, L ) := M lc ( n, L )to emphasis on S and T . Let M lc ( X, T ) and M lc ( X, T, L ) denote the moduli spacescorresponding to T . Lemma 5.1.
There is an isomorphism between M lc ( X, S ) and M lc ( X, T ) .Proof. Depending on the sets S and T , we have two cases(1) S ∩ T = ∅ . (2) S ∩ T = ∅ . Suppose S ∩ T = ∅ . For every i = 1 , . . . , m , let L i = O X ( y i − x i ) be a line bundleof degree zero. Let D i be the de Rham logarithmic connection on the line bundle L i singular over x i and y i , defined by sending a local section s i of L i to ds i . Then Res ( D i , x i ) = − Res ( D i , y i ) = 1. Define a line bundle L = m O i =1 L i . ODULI SPACES OF HOLOMORPHIC AND LOGARITHMIC CONNECTIONS 22
Then L admits a logarithmic connection induced from { D i } mi =1 , which can be ex-pressed as follows D = m X i =1 L ⊗ · · · ⊗ α i D i ⊗ · · · ⊗ L m . Moreover,
Res ( D , x i ) = − α i and Res ( D , y i ) = α i for every i = 1 , . . . , m . Let( E, D ) ∈ M ( X, S ). Then E ⊗ L admits a logarithmic connection given by D ⊗ L + E ⊗ D . Note that for every i = 1 , . . . , m , we have Res ( D ⊗ L + E ⊗ D , x i ) = 0 , and Res ( D ⊗ L + E ⊗ D , y i ) = α i . Thus, we have a morphismΨ ( L ,D ) : M ( X, S ) −→ M ( X, T )of algebraic varieties sending (
E, D ) to ( E ⊗ L , D ⊗ L + E ⊗ D ), which is anisomorphism.Next suppose that S ∩ T = ∅ . Without loss of generality, we assume that x = y , x = y , . . . , x r = y r for r ≤ m . In this case, we consider the line bundle L = m O j = r +1 O X ( y j − x j ) . Now, using above steps, we can get a logarithmic connection D in L . A morphismsimilar to Ψ ( L ,D ) can be defined, which turns out to be an isomorphism. (cid:3) A similar result is true for the moduli space M lc ( X, S, L ). Lemma 5.2.
There is an isomorphism between M lc ( X, S, L ) and M lc ( X, T, L ′ ) . Thus, for the simplicity of the notations, we write M lc ( X ) in place of M lc ( X, S )and M lc ( X, L ) in place of M lc ( X, S, L ).Now, we shall compute the cohomology group of M lc ( X ) and M lc ( X, L ). First,we recall another description for U ( n, d ) due to Atiyah and Bott [1]. Fix E a C ∞ complex vector bundle of rank n and degree d over the compact Riemann surface X .Let A r ( X ), A r ( X, E ) and A p,q ( X, E ) respectively denote the bundle of C ∞ r -formson X , C ∞ r -forms on X with values in E and C ∞ ( p, q )-forms on X with values in E . A holomorphic structure on E is a C -linear map ∂ E : A ( X, E ) → A , ( X, E ) (5.1)which satisfies ∂ E ( f s ) = ∂ ( f ) s + f ∂ E ( s ) , where f is a smooth function on X and s is a smooth section of E over X . Since X is a compact Riemann surface, A , ( X ) = 0 and hence ∂ E = 0. Thus, a holomorphic ODULI SPACES OF HOLOMORPHIC AND LOGARITHMIC CONNECTIONS 23 vector bundle E over X can be thought of as a pair ( E , ∂ E ), where E denote theunderlying C ∞ complex vector bundle of E , and ∂ E is the operator defined in (4.4).Let H be the set of all ∂ E operators defined on E , and let G c be the group of all C ∞ complex automorphism of E . Then H and G c both are infinite dimensional over C , and the group G c acts on H by the following rule: g.∂ E = g∂ E g − , where g ∈ G c and ∂ E ∈ H .The quotient H / G c is identified with the set of isomorphism classes of holomorphicvector bundles of rank n and degree d . This space is in general need not be Hausdorff.To get a well behaved quotient we will restrict to the case of semi-stable holomorphicvector bundles. Let H ss ⊂ H be the open subset consists of semi-stable holomorphicvector bundles. Then U ( n, d ) is identified with the quotient H ss / G c . Let G be thegroup of C ∞ automorphisms of E which are unitary with respect to a fixed hermitianmetric on E . Then the group G c is the complexification of the group G . From [1], therational cohomology ring of B G is freely generated as a graded commutative algebraby elements a r ∈ H r ( B G , Q ), b jr ∈ H r − ( B G , Q ) for 1 ≤ r ≤ n and 1 ≤ j ≤ g , and f r ∈ H r − ( B G , Q ) for 2 ≤ r ≤ n . The rational cohomology ring H ∗ ( U ( n, d ) , Q ) isgenerated by the set T = { a r | ≤ r ≤ n } ∪ { b jr | ≤ r ≤ n, ≤ j ≤ g } ∪ { f r | ≤ r ≤ n } , (5.2)for more details see [19].Let M ′ lc ( X ) := M ′ lc ( n, d ), and M ′ lc ( X, L ) := M ′ lc ( n, L ). Then Let p : M ′ lc ( X ) −→ U ( n, d )be the morphism defined in (3.13). Since a fibre of p is an affine space modelledover a vector space, which is contractible, we get an isomorphism p ∗ : H i ( U ( n, d ) , Q ) −→ H i ( M ′ lc ( X ) , Q ) (5.3)of rational cohomology groups for every i ≥
0. Thus, we have an obvious
Proposition 5.3.
The rational cohomology ring H ∗ ( M ′ lc ( X ) , Q ) is generated by theset p ∗ T , where T is the set defined in (5.2) . Let Z := M lc ( n, L ) \ M ′ lc ( n, L ). Then, form [10, Lemma 3.1], we have Lemma 5.4.
The codimension of the Zariski closed set Z in M lc ( n, L ) is at least ( n − g −
2) + 1 . In particular, if n ≥ , g ≥ , then codim( Z, M lc ( n, L )) ≥ . Similarly, let p : M ′ lc ( X, L ) → U ( n, L ) be the morphism defined in (3.3). Then p is a fibre bundle with fibres as affine spaces modelled over vector spaces, and sinceaffine spaces with usual topology are contractible, the induced homomorphism p ∗ : H i ( U ( n, L ) , Z ) −→ H i ( M ′ lc ( X, L ) , Z ) (5.4)of cohomology groups, is an isomorphism for all i ≥ Y be a complex algebraic variety. For every i ≥
0, there is a mixed Hodgestructure on the cohomology group H i ( Y , Z ). This result is due to Deligne, formore details see [13], [14]. ODULI SPACES OF HOLOMORPHIC AND LOGARITHMIC CONNECTIONS 24
The isomorphism p ∗ in (5.4) is an isomorphism of mixed Hodge structures. More-over, the cohomology group H i ( M ′ lc ( X, L ) , Z ) is equipped with pure Hodge struc-ture of weight i for every i ≥
0, because U ( n, L ) is a smooth projective variety over C , and from [13] the cohomology group H i ( U ( n, L ) , Z ) is endowed with a pure Hodgestructure of weight i , for every i ≥ A be a smooth complex analytic space. For every integer k ≥
0, the ( k + 1)-th intermediate Jacobian variety J k +1 ( A ) of Y is defined as follows. J k +1 ( A ) := H k +1 ( A , R ) / H k +1 ( A , Z ) (5.5)The space J k +1 ( A ) carries a canonical structure of complex manifolds. We considerthat the moduli space M lc ( X, L ) is equipped with the complex analytic topology.
Proposition 5.5.
The second intermediate J ( M lc ( X, L )) is isomorphic to the Ja-cobian J ( X ) := P ic ( X ) of X .Proof. First we show that the mixed Hodge structure on M lc ( X, L ) is in fact a pureHodge structure. Let Z := M lc ( X, L ) \ M ′ lc ( X, L ) as in Lemma 5.4. Then, we havea long exact sequence of relative cohomology groups,H Z ( M lc ( X, L ) , Z ) → H ( M lc ( X, L ) , Z ) ι ∗ −→ H ( M ′ lc ( X, L ) , Z ) ∂ −→ H Z ( M lc ( X, L ) , Z )where ι ∗ is induced by the inclusion map ι : M ′ lc ( X, L ) ֒ → M lc ( X, L ) and ∂ is theboundary operator. We show that ι ∗ is an isomorphism. From Alexander duality[18, Theorem 4.7, p.381], we have an isomorphismH iZ ( M lc ( X, L ) , Z ) −→ H BM N − i ( Z, Z ) , where N = 2( n − g −
1) is the complex dimension of the moduli space M lc ( X, L )and H BM ∗ is the Borel-Moore homology. In view of Lemma 5.4, we havecodim( Z, M lc ( X, L )) ≥ , therefore the real dimension of Z is at most 2 N −
4. Thus,H BM N − i ( Z, Z ) = 0 , for i = 0 , , , , and hence H Z ( M lc ( X, L ) , Z ) = 0 . This implies that ι ∗ is an injective morphism. Let Γ be a smooth compactificationof M lc ( X, L ), and Z ′ = Γ \ M ′ lc ( X, L ) . Then, from [13, Corollaire 3.2.17], we have surjective morphismH (Γ , Q ) −→ H ( M lc ( X, L ) , Q )of mixed Hodge structures, and sincecodim( Z ′ , Γ) ≥ (Γ , Z ) −→ H ( M ′ lc ( X, L ) , Z ) ODULI SPACES OF HOLOMORPHIC AND LOGARITHMIC CONNECTIONS 25 of Hodge structures. Then we have a commutative diagramH (Γ , Q ) (cid:15) (cid:15) + + ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ H ( M lc ( X, L ) , Q ) ι ∗ Q / / H ( M ′ lc ( X, L ) , Q )and from the above facts, the vertical and diagonal arrows are the surjective mor-phism of mixed Hodge structures induced from their respective inclusion maps. Now,because of the commutativity of the diagram, ι ∗ Q is surjective morphism of mixedHodge structures. Since H ( M ′ lc ( X, L ) , Z ) (being isomorphic to H ( U ( n, L ) , Z )) istorsion free Z -module of finite rank [31, Theorem 3] and ι ∗ is an injective mor-phism, H ( M lc ( X, L ) , Z ) is torsion free. Thus, ι ∗ is a surjective morphism, andhence the mixed Hodge structure on H ( M lc ( X, L ) , Z ) is a pure Hodge structureof weight 3. Therefore, the second intermediate Jacobian J ( M lc ( X, L )) is isomor-phic to J ( M ′ lc ( X, L )), and the later is isomorphic to J ( U ( n, L )). Thus, from [31,Theorem 3], J ( M lc ( X, L )) is isomorphic to J ( X ). This completes the proof. (cid:3) Let Θ be the theta divisor on the Jacobian J ( X ). The pair ( J ( X ) , Θ) is calledprincipally polarised Jacobian. Then, the classical Torelli theorem says that the pair( J ( X ) , Θ) determines compact Riemann surface X up to isomorphism.In view of Proposition 5.3, the moduli space M lc ( X, L ) determines the Jacobian J ( X ) of the compact Riemann surface X . But this does not qualify for the de-termination of X , because two non-isomorphic compact Riemann surfaces can haveisomorphic Jacobian.Nevertheless, from [32, p.125, Corollary 1.2], there are, up to isomorphism, onlyfinitely many compact Riemann surface having a given abelian variety as the Jaco-bian. Altogether, we have Theorem 5.6.
Suppose that the moduli space M lc ( X, L ) is given. Then, thereare, up to isomorphism, only finitely many compact Riemann surface Y such that M lc ( Y, L ) is isomorphic to M lc ( X, L ) .Remark . Let e Θ be the canonical polarisation on the second intermediate Jacobian J ( U ( n, L )). Then, from [31, Theorem 3], we have( J ( U ( n, L )) , e Θ) ∼ = ( J ( X ) , Θ) . In [10, Section 4], Biswas and Mu˜noz constructed the principal polarisation b Θ on thesecond intermediate Jacobian of the moduli space M x lc ( X ) of logarithmic connectionssingular exactly over one point x of the compact Riemann surface X with fixeddeterminant such that the principally polarised abelian variety ( J ( M x lc ( X )) , b Θ) isisomorphic to the principally polarised abelian variety ( J ( U ( n, L )) , e Θ). Imitatingthe similar technique as in [10, Section 4], a principal polarisation can be constructedon M lc ( X, L ).From Lemma 5.2, the moduli space M lc ( X, L ) does not depend on the choice of S . Thus, we have ODULI SPACES OF HOLOMORPHIC AND LOGARITHMIC CONNECTIONS 26
Theorem 5.8.
Let ( X, S ) and ( Y, T ) be two m -pointed compact Riemann surfacesof genus g ≥ . Let M lc ( X, L ) and M lc ( Y, L ′ ) be the corresponding moduli spaces oflogarithmic connections. Then, M lc ( X, L ) is isomorphic to M lc ( Y, L ′ ) if and only if X is isomorphic to Y . Next, we show the Torelli type theorem for the moduli space M lc ( X ). Let G : M lc ( X ) −→ P ic d ( X ) (5.6)be the map sending ( E, D ) V n E . Note that the morphism G is surjective. Forinstance let L ∈ P ic d ( X ). Consider the vector bundle E := L ⊕ O n − X . Then V n E = L , and E is stable because n and d are coprime. Since d , n andΦ j = α j E ( x j ) for j = 1 , . . . , m satisfies (2.7), from [8, Theorem 1.3 (2)] E admits alogarithmic connection with residues α j E ( x j ) at x j ∈ S .Now, we have a natural generalisation of [7, p.431, Lemma 5.1] and [10, p.313,Proposition 5.1]. Proposition 5.9.
Let A be a complex abelian variety, and f : M lc ( X ) −→ A (5.7) a regular morphism. Then there exists a unique regular morphism f : P ic d ( X ) −→ A such that f ◦ G = f, (5.8) where G is defined in (5.6) .Proof. Consider M ′ lc ( X ) := M ′ lc ( n, d ) ⊂ M lc ( X ) as in (2.9). Let p : M ′ lc ( X ) −→ U ( n, d )be the morphism defined in (3.13). For E ∈ U ( n, d ), it has been observed that p − ( E ) is an affine space modelled over the vector space H ( X, Ω X ⊗ End( E )), andhence p − ( E ) is a rational variety. Restricting f to p − ( E ), we get a map f | p − ( E ) : p − ( E ) −→ A, which is a constant map, because any regular morphism from a rational variety toan abelian variety is constant. Since f is constant on each of the fibre of p , we geta morphism ψ : U ( n, d ) −→ A such that the following diagram M ′ lc ( X ) f (cid:15) (cid:15) p / / U ( n, d ) ψ u u ❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧ A commutes. ODULI SPACES OF HOLOMORPHIC AND LOGARITHMIC CONNECTIONS 27
Now, consider the determinant map F : U ( n, d ) −→ P ic d ( X )defined by sending E to V n E . Then F is a surjective map. For any L ∈ P ic d ( X ), F − ( L ) is nothing but the moduli space U ( n, L ). Thus, we get a regular morphism ψ | F − ( L ) : U ( n, L ) = F − ( L ) −→ A. on each of the fibres of F . From [20, Theorem 1.2], U ( n, L ) is a rational variety, andhence the regular morphism ψ | F − ( L ) is constant. Since, ψ is constant on each ofthe fibre of F , we get a morphism f : P ic d ( X ) −→ A such that the following diagram M ′ lc ( X ) f (cid:15) (cid:15) p / / U ( n, d ) F (cid:15) (cid:15) A P ic d ( X ) f o o commutes. This completes the proof. (cid:3) Let M Xlc be the moduli space defined in (4.9). Then, we have a morphism δ : M Xlc −→ P ic d ( X ) (5.9)defined by ( L, D ) L . Then δ − ( L ) is an affine space modelled over H ( X, Ω X ).Then G = δ ◦ det , where G is defined in (5.6), and det : M lc ( X ) → M Xlc defined in (4.10). Thus, wehave a morphism η : G − ( L ) −→ M lc ( X, L ) (5.10)which is a fibration and each fibre is an affine space modelled over H ( X, Ω X ). Sincethe fibre of η is contractible, we have an isomorphism η ∗ : H i ( M ( X, L ) , Z ) −→ H i ( G − ( L ) , Z )of cohomology groups for all i ≥
0. Therefore, we have J ( M ( X, L )) ∼ = J ( G − ( L )) . As mentioned in Remark 5.7, similar steps gives a principal polarisation bb Θ on J ( G − ( L )) such that ( J ( M ( X, L )) , b Θ) ∼ = ( J ( G − ( L )) , bb Θ) . Thus, in view of Lemma 5.1 and using the Theorem 5.8, we get
Theorem 5.10.
Let ( X, S ) and ( Y, T ) be two m -pointed compact Riemann surfacesof genus g ≥ . Let M lc ( X ) and M lc ( Y ) be the corresponding moduli spaces oflogarithmic connections. Then, M lc ( X ) is isomorphic to M lc ( Y ) if and only if X isisomorphic to Y . ODULI SPACES OF HOLOMORPHIC AND LOGARITHMIC CONNECTIONS 28 Rational connectedness of the moduli spaces
In [27], we have shown that the moduli space of rank one logarithmic connectionswith fixed residues is not rational. In this section, we show that the moduli space M lc ( n, d ) is not rational. For the theory of rational varieties, we refer [21].Recall that a smooth complex variety V is said to be rationally connected if anytwo general points on V can be connected by a rational curve in V . The followinglemma is an easy consequence of the definition. Lemma 6.1.
Let f : Y → X be a dominant rational map of complex algebraicvarieties with Y rationally connected. Then, X is rationally connected. Theorem 6.2 ([20], Theorem 1.1) . The moduli space U ( n, d ) is birational to J ( X ) × A ( n − g − , where J ( X ) is the Jacobian of X . Note that J ( X ) is not rationally connected, because it does not contain any ra-tional curve. Therefore, U ( n, d ) is not rationally connected. Proposition 6.3.
The moduli space M lc ( n, d ) is not rational.Proof. It is enough to show that the moduli space M lc ( n, d ) is not rationally con-nected. Let p : M ′ lc ( n, d ) −→ U ( n, d )be the morphism of varieties defined in (3.13). Suppose that M ′ lc ( n, d ) is rationallyconnected. Then, from Lemma 6.1, U ( n, d ) is rationally connected, which is not true.Thus, M ′ lc ( n, d ) is not rationally connected and hence not rational. Since M ′ lc ( n, d )is an open dense subset of M lc ( n, d ), M lc ( n, d ) is not rational. (cid:3) A similar argument gives the following.
Proposition 6.4.
The moduli space M h ( n ) is not rational. Lemma 6.5 ([16], Corollary 1.3) . Let f : X → Y be any dominant morphism ofcomplex varieties. If Y and the general fibre of f are rationally connected, then X isrationally connected. Proposition 6.6.
The moduli space M ′ lc ( n, L ) is rationally connected.Proof. Consider the dominant morphism p : M ′ lc ( X, L ) −→ U ( n, L )defined in (3.3). As observed earlier every fibre of p is an affine space and hence ratio-nally connected. Since U ( n, L ) is rationally connected, from Lemma 6.5, M ′ lc ( n, L )is rationally connected. (cid:3) Corollary 6.7. M lc ( n, L ) is rationally connected.Proof. It follows from the fact that rationally connectedness is a birational invariant,and M ′ lc ( n, L ) is a dense open subset of M lc ( n, L ). (cid:3) Therefore, we have a natural question.
Question . Is the moduli space M lc ( n, L ) rational ? ODULI SPACES OF HOLOMORPHIC AND LOGARITHMIC CONNECTIONS 29
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