Comparison of Poisson structures on moduli spaces
aa r X i v : . [ m a t h . AG ] F e b COMPARISON OF POISSON STRUCTURES ON MODULI SPACES
INDRANIL BISWAS, FRANCESCO BOTTACIN, AND TOM ´AS L. G ´OMEZ
Abstract.
Let X be a complex irreducible smooth projective curve, and let L be analgebraic line bundle on X with a nonzero section σ . Let N denote the moduli spaceof stable Hitchin pairs ( E, θ ), where E is an algebraic vector bundle on X of fixed rank r and degree δ , and θ ∈ H ( X, E nd ( E ) ⊗ K X ⊗ L ). Associating to every stable Hitchinpair its spectral data, an isomorphism of N with a moduli space M of stable sheaves ofpure dimension one on the total space of K X ⊗ L is obtained. Both the moduli spaces M and N are equipped with algebraic Poisson structures, which are constructed using σ . Here we prove that the above isomorphism between M and N preserves the Poissonstructures. Introduction
Let X be a complex irreducible smooth projective curve of genus g . Take an algebraicline bundle N on X such that N ⊗ K − X admits a nonzero section, where K X is thecanonical bundle of X ; fix a section σ ∈ H ( X, N ⊗ K − X ) \ { } . Fix integers r and δ .Let N denote the moduli space of stable pairs of the form ( E, θ ), where E is an algebraicvector bundle on X of rank r and degree δ , and θ ∈ H ( X, E nd ( E ) ⊗ N ). These arecalled Hitchin pairs; when N = K X , they are called Higgs bundles. The moduli space N is nonempty if one of the following four assumptions hold (cf. [Ma, Remark 3.4]):(1) the genus g of X is at least 2;(2) g = 1, N = K X and gcd( r, δ ) = 1;(3) g = 1 and deg( N ⊗ K − X ) >
0; and(4) g = 0 and deg( N ⊗ K − X ) ≥ N has a naturalalgebraic Poisson structure [Bo1], [Ma], which is constructed using σ .In the special case where N = K X and σ is the constant function 1, this N is a modulispace of Higgs bundles. In that case, the Poisson structure is nondegenerate, meaning itis a symplectic structure; this symplectic structure was constructed earlier in [Hi1], [Hi2].Furthermore, there is a natural algebraic 1–form on N such that the symplectic form isthe exterior derivative of it.Let S denote the smooth complex quasi-projective surface defined by the total space ofthe line bundle N . Given any Hitchin pair ( E, θ ) ∈ N , there is a subscheme Y ( E,θ ) ⊂ S of dimension one associated to it. Furthermore, associated to the pair ( E, θ ) there is acoherent sheaf L ( E,θ ) on Y ( E,θ ) which is pure of dimension one. This pair ( Y ( E,θ ) , L ( E,θ ) ) Mathematics Subject Classification.
Key words and phrases.
Spectral data, Hitchin pair, Poisson structure, hypercohomology. is known as the spectral datum associated to (
E, θ ). This construction produces analgebraic map Φ :
N −→ M , (1.1)where M is a moduli space of stable sheaves of pure dimension one on S . This constructionis reversible, and Φ is in fact an isomorphism (see [Hi1], [Hi2], [Si], [BNR]).The earlier mentioned section σ of N ⊗ K − X produces a Poisson structure on the surface S . An algebraic Poisson structure on M is constructed using this Poisson structure on S [Bo2], [Ty].It may be mentioned that when N = K X and σ is the constant function 1, thePoisson structure on S is the canonical symplectic form on the cotangent bundle K X of X . In that case the Poisson structure on M coincides with the symplectic structure on M constructed by Mukai in [Mu]. Also, in this case there is a natural algebraic 1–formon M such that the symplectic form is the exterior derivative of it.In this continuation of the papers [Bo1, Bo2] of the second author, our aim is to provethe following (see Theorem 4.6): Theorem 1.1.
The isomorphism Φ in (1.1) takes the Poisson structure on N to thePoisson structure on M . When N = K X , related results can be found in [HK], [HH], [BM]; there the symplecticform on a moduli space of Higgs bundles is compared with the symplectic form on theHilbert scheme of zero dimensional subschemes of fixed length of the total space of K X (Mukai had shown that this Hilbert scheme has a symplectic structure).2. Hitchin pairs and spectral data
Let X be an irreducible smooth complex projective curve of genus g , and let N be afixed algebraic line bundle on X . A Hitchin pair ( E, θ ) is an algebraic vector bundle E on X together with a morphism θ : E −→ E ⊗ N of O X -modules [Hi1, Ni].A Hitchin pair ( E, θ ) is stable (respectively, semistable ) ifdeg F rk F < deg E rk E (respectively, deg F rk F ≤ deg E rk E )for all subbundles 0 = F ( E for which θ ( F ) ⊂ F ⊗ N .Let N denote the moduli space of all S-equivalence classes of semistable Hitchin pairs(constructed in [Ni]) of fixed rank r and degree δ . Let N ⊂ N be the open subset of stable Hitchin pairs. In the sequel we assume that one of thefollowing four assumptions hold:(1) g ≥ g = 1, N = K X and gcd( r, δ ) = 1;(3) g = 1 and deg( N ⊗ K − X ) > g = 0 and deg( N ⊗ K − X ) ≥ OMPARISON OF POISSON STRUCTURES ON MODULI SPACES 3
This assumption implies the existence of stable Hitchin pairs [Ma, Remark 3.4], and hence N is nonempty.We will now recall the spectral construction , which is a bijective correspondence betweenHitchin pairs ( E, θ ) and certain sheaves L on the total space p : S := V ( N ) −→ X (2.1)of the line bundle N . This construction is in [Hi2, §
5] for smooth spectral curves, [BNR,p. 173–174, Proposition 3.6] for integral spectral curves, and [Si, p. 18, Lemma 6.8] ingeneral (see also [BNR, p. 173–174, Remark 3.7]).Consider the projection p in (2.1). Let x denote the tautological section of p ∗ N on S = V ( N ). For a Hitchin pair ( E, E θ −→ E ⊗ N ), we define the associated sheaf L usingthe following short exact sequence of coherent sheaves on S −→ p ∗ ( E ⊗ N ∨ ) h −→ p ∗ E −→ L −→ , (2.2)where p is the projection in (2.1) and h := p ∗ θ − x , with x being the tensor product withthe above mentioned tautological section x ; the homomorphism E ⊗ N ∨ −→ E given by θ is denoted here by θ also. Throughout, the dual will be denoted by the superscript “ ∨ ”.The homomorphism h in (2.2) is injective (it is an isomorphism over the generic point of S ), so (2.2) is indeed a short exact sequence of coherent sheaves. The spectral curve Y ⊂ S for ( E, θ ) is the subscheme defined by the characteristic polynomialdet( p ∗ θ − x ) = 0 . Equivalently, the spectral curve Y is the subscheme defined by the 0–th Fitting ideal sheafFitt ( L ) ⊂ O S . To see this equivalence, note that (2.2) gives a presentation of L , and, by definition, the0–th Fitting ideal is the ideal generated by the codimension zero minors of the morphism h in (2.2). Since the ranks of the source and target of h are equal, we only have oneminor which is the determinant det( h ) = det( p ∗ θ − x ), in other words the characteristicpolynomial of θ . We have the following diagram Y i / / π ❅❅❅❅❅❅❅❅ S p (cid:15) (cid:15) X (2.3)If the above spectral curve Y is smooth, then there is a line bundle L on Y such that L = i ∗ L , where i is the map in (2.3).The Hitchin pair ( E, θ ) can be recovered from the sheaf L by setting E = p ∗ L (2.4)and θ = p ∗ x , where p is the projection in (2.3). More explicitly, consider the multiplica-tion by the tautological section x of p ∗ N L x −→ L ⊗ p ∗ N I. BISWAS, F. BOTTACIN, AND T. L. G ´OMEZ and then θ = p ∗ x is defined as the push-forward of this homomorphism, using the pro-jection formula θ : E = p ∗ L −→ p ∗ ( L ⊗ p ∗ N ) = ( p ∗ L ) ⊗ N = E ⊗ N (see (2.4)). The above reversible construction gives a correspondence between the Hitchinpairs ( E, θ ) and the coherent sheaves L on S , of pure dimension one, such that p ∗ L iscoherent (equivalently, the closure of the support of L in S = P ( N ⊕ O X ) lies in S ).We shall now recall the definition of stability given by Simpson, [Si], for the sheaves on S that appear in this correspondence. Let L be a coherent sheaf on S , such that p ∗ L isalso coherent. Let P ( L , m ) = χ ( L ⊗ p ∗ O X ( m ))be its Hilbert polynomial. Its degree d is equal to the dimension of the support of L ,and the leading coefficient is am d /d !, where a = a ( L ) is an integer which we call therank of L . We say that L is stable (respectively, semistable ) if for all proper subsheaves0 = L ′ ⊂ L , a ( L ) P ( L ′ , m ) < a ( L ′ ) P ( L , m ) (respectively, a ( L ) P ( L ′ , m ) ≤ a ( L ′ ) P ( L , m ))It is easy to see that if L is semistable, then it has pure dimension (the dimension of thesupport of every nonzero subsheaf is the same as the dimension of the support of L ). Notethat, in the spectral construction, the condition that L ′ is a subsheaf of L translates intothe condition that the subbundle E ′ := p ∗ L ′ ⊂ E := p ∗ L satisfies the condition θ ( E ′ ) ⊂ E ′ ⊗ N . It follows that a Hitchin pair is stable (respectively,semistable) if and only if the corresponding sheaf on S is stable (respectively, semistable)[Si, p. 19, Corollary 6.9].As before, fix the rank and the degree of the vector bundle underlying a Hitchin pair.The above construction of bijection can be carried out for families, so we get an isomor-phism between the moduli space of stable Hitchin pairs N and a moduli space M ofstable sheaves of pure dimension oneΦ : N −→ M . (2.5)More generally, we get an isomorphism between the moduli functor for Hitchin pairs andmoduli functor for corresponding sheaves on S .We are interested in calculating the differential of the isomorphism Φ in (2.5). Moregenerally, we will give an isomorphism between the infinitesimal deformation space of aHitchin pair ( E, θ ) and the infinitesimal deformation space of the corresponding sheaf L on S . 3. Poisson structure on the moduli spaces
The infinitesimal deformation space of L is given byExt ( L , L ) (3.1) OMPARISON OF POISSON STRUCTURES ON MODULI SPACES 5 [Bo2, p. 425, (3.1)]. To calculate the dual of this vector space we will use Serre duality.Recall that L is a coherent sheaf on the surface S = V ( N ) in (2.1) which is not projective.However, it has a projective compactification j : S −→ S = P ( N ⊕ O X ) ;furthermore, the direct image j ∗ L is a coherent sheaf on S , because the closure, in S , ofthe support of L does not meet the boundary S \ S . We haveExt ( L , L ) ∨ = Ext O S ( j ∗ L , j ∗ L ) ∨ ∼ = Ext O S ( j ∗ L , j ∗ L ⊗ K S ) = Ext ( L , L ⊗ K S ) , (3.2)where the second isomorphism is Serre duality on the projective surface S .As in (2.5), let M be the moduli space of stable sheaves on S with the numericalinvariants of L . The tangent space T [ L ] M at the point corresponding to L is canonicallyidentified with Ext ( L , L ), and the cotangent space T ∨ [ L ] M is canonically identified withExt ( L , L ⊗ K S ) (see (3.2)).Henceforth assume that the line bundle N ⊗ K − X on X admits a nonzero section. Wefix a nonzero section σ ∈ H ( X, N ⊗ K − X ) \ { } . (3.3)Since the anticanonical line bundle K ∨ S of the surface S in (2.1) is identified with p ∗ ( N ⊗ K − X ), where p is the projection in (2.1), we have s := p ∗ σ ∈ H ( S, K ∨ S ) , (3.4)where σ is the section in (3.3). This s produces an algebraic Poisson structure on M B : T ∨ M −→ T M (3.5)[Bo2], [Ty]. We will recall below the fiberwise construction of B .Tensoring with the section s produces a homomorphism L ⊗ K S −→ L . Consequently,we obtain a homomorphism T ∨ [ L ] M ∼ = Ext ( L , L ⊗ K S ) −→ Ext ( L , L ) ∼ = T [ L ] M . (3.6)The isomorphism Ext ( L , L ) ∼ = T [ L ] M is a consequence of the fact that the infinitesimaldeformations of L are parametrized by Ext ( L , L ), while the isomorphism T ∨ [ L ] M ∼ =Ext ( L , L ⊗ K S ) follows from (3.2). The homomorphism B ([ L ]) in (3.5) coincides withthe one in (3.6) (see [Bo2, p. 428, Formula (4.4)]).On the other hand, the infinitesimal deformation space of a Hitchin pair ( E, θ ) is givenby the first hypercohomology of a complex H ( E ∨ ⊗ E [ · , θ ] −→ E ∨ ⊗ E ⊗ N ) (3.7)(see [Bo1, p. 399, Proposition 3.1.2], [Ma, p. 271, Proposition 7.1], [BR, p. 220, Theorem2.3]). The dual vector space H ( E ∨ ⊗ E [ · , θ ] −→ E ∨ ⊗ E ⊗ N ) ∨ is calculated using Serre duality for hypercohomologies. Denote by A • the complex A • : E ∨ ⊗ E [ · , θ ] −→ E ∨ ⊗ E ⊗ N I. BISWAS, F. BOTTACIN, AND T. L. G ´OMEZ concentrated in degrees 0 and 1. Serre duality gives an isomorphism H ( A • ) ∨ ∼ = −→ H − ( A •∨ [1] ⊗ K X ) = H ( A •∨ [ − ⊗ K X )[Hu, p. 67, Theorem 3.12]. The dual complex A •∨ : E ⊗ E ∨ ⊗ N ∨ [ · , θ ] t −→ E ⊗ E ∨ is concentrated in degrees − E ⊗ E ∨ and E ∨ ⊗ E by switching thefactors; then the morphism [ · , θ ] t becomes [ θ, · ], so A •∨ : E ∨ ⊗ E ⊗ N ∨ [ θ, · ] −→ E ∨ ⊗ E . (3.8)Finally we shift the complex in (3.8) by [ − − A •∨ [ − ⊗ K X : E ∨ ⊗ E ⊗ N ∨ ⊗ K X [ · , θ ] −→ E ∨ ⊗ E ⊗ K X . (3.9) Remark 3.1.
In [Bo1, p. 402, Proposition 3.1.10] the morphism is actually [ θ, · ] insteadof the homomorphism [ · , θ ] in (3.7). Of course there is an isomorphism between the twocomplexes (multiplication by − N be a moduli space of stable Hitchin pairs such that the rank anddegree of the vector bundle underlying a Hitchin pair are r and δ respectively. Using thesection σ in (3.3) an algebraic Poisson structure B H : T ∨ N −→ T N (3.10)on N is constructed [Bo1, § § B H in (3.10).Tensoring with the section σ in (3.3) induces homomorphisms N ∨ ⊗ K X −→ O X and K X −→ N . Using these we obtain a homomorphism of complexes E ∨ ⊗ E ⊗ N ∨ ⊗ K X [ · , θ ] −→ E ∨ ⊗ E ⊗ K X σ y y σ E ∨ ⊗ E [ · , θ ] −→ E ∨ ⊗ E ⊗ N . (3.11)The morphism of complexes in (3.11) gives a morphism of cohomologies H ( E ∨ ⊗ E ⊗ N ∨ ⊗ K X [ · , θ ] −→ E ∨ ⊗ E ⊗ K X ) −→ H ( E ∨ ⊗ E [ · , θ ] −→ E ∨ ⊗ E ⊗ N ) . (3.12)We have T [( E,θ )] N = H ( E ∨ ⊗ E [ · , θ ] −→ E ∨ ⊗ E ⊗ N ), because the infinitesimal deformationsof ( E, θ ) are parametrized by H ( E ∨ ⊗ E [ · , θ ] −→ E ∨ ⊗ E ⊗ N ) (see (3.7)); this and (3.9)together imply that T ∨ [( E,θ )] N = H ( E ∨ ⊗ E ⊗ N ∨ ⊗ K X [ · , θ ] −→ E ∨ ⊗ E ⊗ K X ) . (3.13)Using (3.13) and (3.7), the homomorphism in (3.12) becomes a homomorphism T ∨ [( E,θ )] N −→ T [( E,θ )] N . OMPARISON OF POISSON STRUCTURES ON MODULI SPACES 7
This homomorphism coincides with B H ([( E, θ )]) in (3.10) [Bo1], [Ma].
Remark 3.2.
If we compare (3.11) with the commutative diagram in [Bo1, Remark 1.3.3],we see that the sign is changed in the vertical left arrow and in one of the horizontal arrows(there is [ θ, · ] in [Bo1, Remark 1.3.3] as opposed to [ · , θ ] here).4. Comparison of Poisson structures
We shall compare the Poisson structures B and B H , constructed in (2.5) and (3.10)respectively, using the isomorphism Φ in (2.5).We first recall that the global Ext can be calculated using locally free resolutions andhypercohomology. Lemma 4.1 ([Gr, Corollary 2 to Theorem 4.2.1]) . Let V and W be coherent sheaves ona scheme T . Let · · · −→ L − −→ L − −→ L − −→ L −→ V −→ be a resolution of V by finitely generated locally free sheaves. Then there is an isomorphism Ext i ( V, W ) ∼ = H i ( H om ( L • , W )) which is functorial on W . Proposition 4.2.
Let W be a coherent sheaf on the surface S in (2.1) . Let x : W −→ W ⊗ p ∗ N be multiplication by the tautological section of p ∗ N on S , where p is the projectionin (2.1) . Then for any ( E, θ ) ∈ N (see (2.5) ) there is an isomorphism ϕ W : Ext ( L , W ) ∼ = −→ H ( E ∨ ⊗ p ∗ W f W −→ E ∨ ⊗ N ⊗ p ∗ W ) , where L = Φ(( E, θ )) (see (2.5) ), f W = θ ∨ ⊗ − ⊗ p ∗ x , and f W is seen as a com-plex concentrated in degrees and . This isomorphism is functorial on W , meaning ahomomorphism V −→ W of coherent sheaves on S induces a commutative diagram Ext ( L , V ) / / (cid:15) (cid:15) H ( E ∨ ⊗ p ∗ V f V −→ E ∨ ⊗ N ⊗ p ∗ V ) (cid:15) (cid:15) Ext ( L , W ) / / H ( E ∨ ⊗ p ∗ W f W −→ E ∨ ⊗ N ⊗ p ∗ W ) Proof.
The short exact sequence in (2.2) gives a locally free resolution L • of L (concen-trated on degrees − L • : p ∗ ( E ⊗ N ∨ ) h −→ p ∗ E .
For any sheaf W on S , Lemma 4.1 gives an isomorphism functorial for W Ext ( L , W ) = H ( H om ( L • , W )) . (4.1)Consider the diagram S f (cid:15) (cid:15) p / / X γ { { ✇✇✇✇✇✇✇✇✇ Spec C I. BISWAS, F. BOTTACIN, AND T. L. G ´OMEZ where f and γ are the structure morphisms. Note that the hypercohomology of anycomplex D • on S can be calculated as the cohomology of the complex Rf ∗ D • (where Rf ∗ is the derived functor between the derived categories). We have Rf ∗ = R ( γ ∗ ◦ p ∗ ) ∼ = −→ Rγ ∗ ◦ Rp ∗ . The morphism p is affine, hence p ∗ is exact and then Rp ∗ = p ∗ . If we apply this obser-vation to the complex H om ( L • , W ), we get H ( H om ( L • , W )) ∼ = −→ H ( p ∗ H om ( L • , W )) . (4.2)In view of (4.1) and (4.2), to finish the proof, it only remains to show that the complex p ∗ H om ( L • , W ) is equal to the complex f W in the statement of the proposition.Applying H om ( · , W ) to the complex L • we obtain a complex concentrated in degrees0 and 1 H om ( L • , W ) : p ∗ E ∨ ⊗ W h ′ −→ p ∗ E ∨ ⊗ p ∗ N ⊗ W , (4.3)where h ′ := p ∗ θ ∨ ⊗ − ⊗ x . By the projection formula, p ∗ H om ( L • , W ) is E ∨ ⊗ p ∗ W = p ∗ ( p ∗ E ∨ ⊗ W ) p ∗ h ′ −→ p ∗ ( p ∗ E ∨ ⊗ p ∗ N ⊗ W ) = E ∨ ⊗ N ⊗ p ∗ W , where p ∗ h ′ = θ ∨ ⊗ − ⊗ p ∗ x , and the proposition follows. (cid:3) Remark 4.3.
In the proof of Proposition 4.2, the main step is the isomorphism in (4.2),because it facilitates the passage of objects defined on S to objects defined on X . Notethat this isomorphism is induced by the push-forward p ∗ using p .Let d Φ : T N −→ Φ ∗ T M (4.4)be the differential of the isomorphism Φ in (2.5). Lemma 4.4.
Let ϕ : Ext ( L , L ) ∼ = −→ H ( E ∨ ⊗ E f L −→ E ∨ ⊗ N ⊗ E ) be the homomorphism given by Proposition 4.2 for W = L (see (2.4) ). Then the followingdiagram is commutative Ext ( L , L ) ϕ −→ H ( E ∨ ⊗ E [ · , θ ] −→ E ∨ ⊗ E ⊗ N ) x α y βT [ L ] M d Φ(Φ − ([ L ])) ←−−−−−−− T [( E,θ )] N where α and β are the infinitesimal deformation maps in (3.1) and (3.7) respectively, and d Φ(Φ − ([ L ])) is the homomorphism in (4.4) at Φ − ([ L ]) ∈ N .Proof. Consider an infinitesimal deformation L ǫ of L , i.e., a sheaf on S × Spec( C [ ǫ ] / ( ǫ ))flat over Spec( C [ ǫ ] / ( ǫ )), that fits in a short exact sequence0 −→ L ǫ −→ L ǫ mod ǫ −→ L −→ . (4.5) OMPARISON OF POISSON STRUCTURES ON MODULI SPACES 9
By applying the functor Hom( L , · ) to this short exact sequence we obtain a long exactsequence · · · −→ Hom( L , L ǫ ) −→ Hom( L , L ) δ −→ Ext ( L , L ) −→ · · · ;the element of Ext ( L , L ) that corresponds to the infinitesimal deformation L ǫ is δ (id),the image of the identity id ∈ Hom( L , L ) by the connecting homomorphism δ .If we set W = L in Proposition 4.2, we have p ∗ W = E and p ∗ x = θ ; then f W = [ · , θ ],and Proposition 4.2 produces the isomorphism ϕ .The element β ◦ ϕ ◦ α ( L ǫ ) ∈ T [( E,θ )] N , (4.6)where L ǫ ∈ T [ L ] M is the element in (4.5), has the following description.Consider the short exact sequence (4.5). Applying p ∗ to it, we obtain an exact sequenceof complexes 0 −→ K • ǫ −→ K • ǫ mod ǫ −→ K • −→ , (4.7)where K • denotes the complex 0 −→ E θ −→ E ⊗ N −→
0. Since K • ǫ in (4.7) isa family of Hitchin pairs on X parametrized by Spec( C [ ǫ ] / ( ǫ )), it corresponds to anelement of T [( E,θ )] N . This element of T [( E,θ )] N coincides with the one in (4.6). Now fromthe construction of the map Φ it follows that d Φ(Φ − ([ L ]))( β ◦ ϕ ◦ α ( L ǫ )) ∈ T [ L ] M coincides with L ǫ ∈ T [ L ] M . (cid:3) We remark that the key to the commutativity of the diagram in the statement of Lemma4.4 is that both ϕ and Φ − are induced by the push-forward p ∗ . Lemma 4.5.
Consider ϕ := ( ϕ L⊗ K S ) − : H ( E ∨ ⊗ E ⊗ K X ⊗ N ∨ [ · , θ ] −→ E ∨ ⊗ E ⊗ K X ) −→ Ext ( L , L ⊗ K S ) , where ϕ L⊗ K S is the isomorphism in Proposition 4.2 for W = L ⊗ K S . Let ( d Φ) ∨ ([ L ]) be dual of the differential of the isomorphism Φ at [ L ] . Then the following diagram iscommutative Ext ( L , L ⊗ K S ) ϕ ←− H ( E ∨ ⊗ E ⊗ K X ⊗ N ∨ [ · , θ ] −→ E ∨ ⊗ E ⊗ K X ) y α ∨ x β ∨ T ∨ [ L ] M ( d Φ) ∨ ([ L ]) −−−−−−→ T ∨ [( E,θ )] N where α ∨ and β ∨ are the natural isomorphisms obtained from (3.2) and (3.9) respectively,and ( d Φ) ∨ is the dual of the homomorphism in (4.4) .Proof. Consider the homomorphism ϕ in Lemma 4.4. For any ω ∈ H ( E ∨ ⊗ E ⊗ K X ⊗ N ∨ [ · , θ ] −→ E ∨ ⊗ E ⊗ K X )and v ∈ Ext ( L , L ), we have ϕ ( ω )( v ) = ω ( ϕ ( v )) ∈ C ; (4.8) recall that H ( E ∨ ⊗ E ⊗ K X ⊗ N ∨ [ · , θ ] −→ E ∨ ⊗ E ⊗ K X ) = H ( E ∨ ⊗ E [ · , θ ] −→ E ∨ ⊗ E ⊗ N ) ∨ (see (3.13) and (3.7)), and Ext ( L , L ⊗ K S ) = Ext ( L , L ) ∨ (see (3.2)). From (4.8) it follows immediately that ϕ coincides with the dual homomor-phism ϕ ∨ .Therefore, every homomorphism in the diagram in this lemma is the dual of the cor-responding homomorphism in the diagram in Lemma 4.4. Hence the lemma follows fromLemma 4.4. (cid:3) The homomorphism B in (3.5) gives an algebraic section B ∈ H ( M , ^ T M ) , (4.9)and the homomorphism B H in (3.10) gives an algebraic section B H ∈ H ( N , ^ T N ) . Consider the homomorphism d Φ in (4.4). We note thatΦ ∗ B H := ( ^ d Φ)( B H ) ∈ H ( M , ^ T M ) (4.10)is a Poisson structure on M . Theorem 4.6.
The isomorphism Φ in (2.5) satisfies the condition Φ ∗ B H = B , where Φ ∗ B H and B are the sections in (4.10) and (4.9) respectively.Proof. The section σ ∈ H ( X, N ⊗ K ∨ X ) gives a section s of K ∨ S = p ∗ ( N ⊗ K ∨ X ), andhence it produces a homomorphism L ⊗ K S −→ L . We apply Proposition 4.2 to thishomomorphism and get the following commutative diagram:Ext ( L , L ⊗ K S ) ∼ = ϕ ′ / / (cid:15) (cid:15) H ( E ∨ ⊗ E ⊗ K X ⊗ N ∨ [ · , θ ] −→ E ∨ ⊗ E ⊗ K X ) (cid:15) (cid:15) Ext ( L , L ) ∼ = ϕ / / H ( E ∨ ⊗ E [ · , θ ] −→ E ∨ ⊗ E ⊗ N ) (4.11)where the left vertical homomorphism is the one in (3.6) and the right vertical homomor-phism is the one in (3.12).Lemma 4.5 shows that the top horizontal homomorphism ϕ ′ in (4.11) coincides with( d Φ) ∨ ([ L ]). We note that ϕ ′ = ( ϕ ) − , where ϕ is the homomorphism in Lemma 4.5.On the other hand, Lemma 4.4 shows that the bottom horizontal homomorphism in(4.11) coincides with d Φ − ([ L ]). The left vertical homomorphism in (4.11) gives B ([ L ]),the Poisson structure on M , while the right vertical homomorphism in (4.11) gives B H ([( E, θ )]) = B H (Φ − ([ L ])), the Poisson structure on N . Consequently, the theoremfollows from the commutativity of the diagram in (4.11). (cid:3) OMPARISON OF POISSON STRUCTURES ON MODULI SPACES 11
Acknowledgements
The first author is supported by a J. C. Bose fellowship. The third author is supportedby Ministerio de Ciencia e Innovaci´on of Spain (grants PID2019-108936GB-C21 and IC-MAT Severo Ochoa project CEX2019-000904-S) and CSIC (2019AEP151 and
Ayuda ex-traordinaria a Centros de Excelencia Severo Ochoa
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