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Hodge conjecture for the moduli space of semi-stable sheaves over a nodal curve

In this article, we prove the Hodge conjecture for a desingularization of the moduli space of rank 2, semi-stable, torsion-free sheaves with fixed odd degree determinant over a very general irreducible nodal curve of genus at least 2. We also compute the algebraic Poincare polynomial of the associated cohomology ring.

curve of genus g−1. Equivalently, a very general nodal curve lies outside countably many, proper closed subsets of the image of the clutching map from M g−1,2 to M g , where M g−1,2 denotes the moduli space of genus g − 1 curves with 2 marked points and M g is the moduli space of stable curves of genus g (see [2, Chapter XII, §10]). Given a torsion-free sheaf E on X 0 , we say that E has determinant L 0 if there is an O X 0 -morphism ∧ 2 (E) → L 0 which is an isomorphism outside the node x 0 . If E is locally free then this means ∧ 2 E ∼ = L 0 . Using [31,Theorem 2], one can check that there exists a moduli space, denoted U X 0 (2, L 0 ), parameterizing rank 2, semi-stable sheaves on X 0 with determinant L 0 . However, the moduli space U X 0 (2, L 0 ) is singular. We show: Theorem 1.1. For X 0 a very general, irreducible nodal curve, there exists a desingularization G 0 of U X 0 (2, L 0 ) (in the sense that G 0 is non-singular and there is a proper birational morphism from G 0 to U X 0 (2, L 0 )) such that the Hodge conjecture holds for G 0 . See Theorem 5.2 for a proof.
One obstacle to simply generalizing the techniques used in the smooth curve case is that an analogous description of the cohomology ring of U X 0 (2, L 0 ) is not available. More precisely, Balaji-King-Newstead in [5] prove that there are Hodge classes α ∈ H 2 (M C (2, L), Z), β ∈ H 4 (M C (2, L), Z) and a surjective morphism Since the Hodge conjecture holds for Jac(C) of a general smooth, projective curve C, they are able to conclude the Hodge conjecture for M C (2, L). Unfortunately such a morphism does not exist if we replace M C (2, L) by U X 0 (2, L 0 ). Moreover, since the Jacobian of a nodal curve is not projective, the statement of the Hodge conjecture does not apply to the Jacobian of X 0 . As a result the classical tools fail in this setup.
Our strategy is as follows. We first embed the very general nodal curve X 0 as the central fiber of a regular, flat family of projective curves π : X → ∆ (here ∆ denotes the unit disc), smooth over ∆ * := ∆\{0} (see [4,Theorem B.2]) for the existence of such a family. Note that, the invertible sheaf L 0 on X 0 lifts to a relative invertible sheaf L X over X . By [31] and [32], there exists a relative Gieseker moduli space with fixed determinant over the family X given by a flat, projective morphism π 2 : G(2, L) → ∆ such that for all s ∈ ∆ * , G(2, L) s := π −1 2 (s) = M Xs (2, L s ). The total space G(2, L) is regular and the central fiber π −1 2 (0), denoted G X 0 (2, L 0 ), is a reduced simple normal crossings divisor of G(2, L) with two smooth, irreducible components such that one of them is a desingularization of U X 0 (2, L 0 ). We denote this desingularization by G 0 . Since for all s = 0, G(2, L) s = M Xs (2, L s ) and we already know that the Hodge conjecture holds true for M Xs (2, L s ) by [5], it is natural to compare the Hodge classes and the algebraic classes on G X 0 (2, L 0 ) using variation of mixed Hodge structures. We prove that the Hodge conjecture holds for both of the smooth components of the central fibre and therefore for a desingularization of U X 0 (2, L 0 ).
As a by-product we obtain the algebraic Poincaré polynomial of G 0 . Recall by [5, (5.1)], we have for any s ∈ ∆ * , the algebraic Poincaré polynomial of M Xs (2, L s ), denoted where X s := π −1 (s) and L s := L X | Xs . Analogously, we prove (Theorems 5.2 and 5.3): Theorem 1.2. The algebraic Poincaré polynomial for G 0 is given by We note that this article is part of a series of articles in which we study related but different questions pertaining to the moduli space of stable, rank 2 sheaves on an irreducible nodal curve (see [6,7,9,10]). The answers to these questions is well-known in the case when the underlying curve is smooth. Therefore we have often employed the theory of limit mixed Hodge structures to study the question for the nodal curve case using analogous results known for the case when the curve is smooth. However, the results in these articles are independent and overlap only in the background material.
Notation: Given any morphism f : Y → S and a point s ∈ S, we denote by Y s := f −1 (s). The open unit disc is denoted by ∆ and ∆ * := ∆\{0} denotes the punctured disc. Unless mentioned otherwise, all cohomology groups are taken with Q-coefficients.
As a consequence, there exist Hodge filtrations on H i (Y ∞ , C) defined by To define a weight filtration on H i (Y ∞ , Z), we use the local monodromy transformations.
Definition 2.4. For any s ∈ ∆ * and i ≥ 0, denote by is an isomorphism for all l ≥ 0. By [28,Theorem 6.16] the induced filtrations on H i (Y ∞ , C) define a mixed Hodge structure (H i (Y ∞ , Z), W • , F • ), called the limit mixed Hodge structure on H i (Y ∞ , Z). Definition 2.6. Recall, for any t ∈ ∆ * , there exist natural specialization morphisms: obtained by composing the natural inclusion of the special fiber Y t into Y with the retraction map to the central fiber Y 0 . Unfortunately, the resulting specialization maps are not morphism of mixed Hodge structures. However, if one identifies H i (Y t ) with H i (Y ∞ ) as mentioned above, then the resulting (modified) specialization morphisms are morphisms of mixed Hodge structures (see [26,Theorem 11.29]). Furthermore, by the local invariant cycle theorem [26,Theorem 11.43], we have the following exact sequence: where sp i denotes the (modified) specialization morphism which is a morphism of mixed Hodge structures as discussed above.
Suppose that Y 0 is a simple, normal crossings divisor and an union of two smooth, irreducible components, say Y 1 , Y 2 . Since Y 1 ∩ Y 2 , Y 1 , Y 2 are non-singular (hence has pure Hodge structure), the Mayer-Vietoris sequence associated to Y 0 , gives rise to the exact sequence: is the zero map (see [26,Proposition B.30]). Hence, the composed morphism. We recall the following useful result which we use repeatedly.
Corollary 2.7. Suppose that the central fiber Y 0 is a simple, normal crossings divisor and an union of two smooth, irreducible components, say Y 1 , Y 2 . Then, we have the following exact sequence of mixed Hodge structures: where f i is the morphism (2.5) and g i is the natural quotient by Proof. See [10, Corollary 2.4] for a proof.
3. Limit mixed Hodge structures of the relative Gieseker moduli space In this section we recall the basic definitions and results on the relative Gieseker moduli space, the associated monodromy action and extend the exact sequence (2.6) in this setup.
3.1. Notation. Let X 0 be an irreducible nodal curve of genus g ≥ 2, with exactly one node, say at x 0 . Denote by π 0 : X 0 → X 0 the normalization map. There exists a flat, projective family of curves π 1 : X → ∆ smooth over ∆ * and central fiber isomorphic to X 0 (see [4,Theorem B.2]). Moreover, X is a regular variety. Fix an invertible sheaf L on X such that L| Xs is of odd degree, say d for all s ∈ ∆. Set L 0 := L| X 0 , the restriction of L to the central fiber. Denote by L 0 := π * 0 (L 0 ).

3.2.
Relative Gieseker moduli space. Recall, that a curve X k is semi-stably equivalent to X 0 if it is the union of the normalization X 0 and a chain of rational curves of length k. See [22, Definition-Notation 2] for the precise definition. We say that a family of curves f : X S → S is semi-stably equivalent to the family π 1 above, if (1) there is a morphism from X S to X inducing a morphism h from S to ∆ such that the resulting diagram is commutative, (2) for any point s mapping to 0 ∈ ∆, the fiber f −1 (s) is semi-stably equivalent to X 0 . For other points s ∈ S, the fiber f −1 (s) is isomorphic to the fiber of π 1 over h −1 (s).

3.3.
Monodromy action on the relative Gieseker moduli space. An important step in proving the Hodge conjecture for G 0 is to study the monodromy action on H 2i (G(2, L) s ) for s ∈ ∆ * . For this purpose, we need to consider the relative version of the construction of the Mumford-Newstead isomorphism [20]. Denote by There exists a (relative) universal bundle U over W associated to the (relative) moduli space G(2, L) ∆ * . Note that U is only well-defined up to pull-back by a line bundle on Note that, the restriction of the universal bundle U to the fiber over any point of the moduli space G(2, L) ∆ * is uniquely determined by the point (and not by the choice of the universal bundle). Hence, the value of the section c 2 (U ) at each point of ∆ * is independent of the choice of U . Since a (local) section of a local system is uniquely determined by its value at a point, the section c 2 (U ) does not depend on the choice of the universal bundle. As a result, the homomorphism Φ ∆ * is independent of the choice of U . By [20, Lemma 1 and Proposition 1], we conclude that the homomorphism Φ ∆ * is an isomorphism such that the induced isomorphism on the associated vector bundles: is monodromy invariant i.e., for all s ∈ ∆ * , the following diagram is commutative: where T Xs and T G(2,L)s are the monodromy transformations on H 1 (X s , Z) and H 3 (G(2, L) s , Z), respectively.

Limit Mumford-Newstead isomorphism.
Let H 1 X and H 3 G(2,L) be the canonical extensions of H 1 X ∆ * and H 3 G(2,L) ∆ * , respectively. Then, the morphism Φ ∆ * extends to the entire disc (by canonicity of the extensions): . Using the identification (2.1) and restricting Φ to the central fiber, we have an isomorphism: where X ∞ (resp. G(2, L) ∞ ) is the base change of X (resp. G(2, L)) to the upper half plane h as described in Definition 2.3. Recall, Φ 0 is an isomorphism of mixed Hodge structures: Proof. See [6, Proposition 4.1] for a proof of the statement.
By the Deligne-Blanchard theorem [12] (the Leray spectral sequence degenerates at E 2 for smooth families), we have Since M X 0 (2, L 0 ) is smooth and simply connected (as it is rationally connected [15, Proposition 2.3.7 and Remark 2.3.8] which implies simply connected [11,Corollary 4.29]), the local systems R j ρ 1, * Q and R j ρ 2, * Q are trivial. Therefore, for any y ∈ M X 0 (2, L 0 ), the natural morphisms are surjective. By the Leray-Hirsch theorem (see [33,Theorem 7.33]), we then have: Using this one can check that: The following holds true: Proof. See [10, Theorem 4.2] for proof of the statement.
Let G(2, L) ∞ denote the base change of the family G(2, L) (defined in §3.2) over ∆ to the universal cover h by the morphism e : h → ∆ * j − → ∆. Using the definitions and results from §2, we can equip H i (G(2, L) ∞ ) with a limit mixed Hodge structure and obtain a specialization morphism: which is a morphism of mixed Hodge structures for all i. Replacing Y 1 (resp. Y 2 ) in §2 by G 0 (resp. G 1 ) from §3.2, we have by Proposition 3.3 that the kernel of the morphism f i is isomorphic to H i−4 (M X 0 (2, L 0 ))(−2) i.e., we have an exact sequence of mixed Hodge structures (see Corollary 2.6):

Limit Hodge conjecture for relative Gieseker moduli space
Recall, that for a very general smooth, projective curve Y , H 2p Hdg (Jac(Y )) ∼ = Q for every p (see [8,Theorem 17.5.1]. It is natural to ask, given a degenerating family of smooth, projective curves, is the sub-space of p-th Hodge classes in the limit mixed Hodge structure associated to the corresponding family of Jacobians, one dimensional for each p? Apriori, this is false as can be seen from the theory of Noether-Lefschetz loci i.e., very generally, the rank of the Hodge lattice jumps. However, in this section we prove that under the genericity assumption of the nodal curve X 0 , we have a positive answer (see Theorem 4.3). This result will later play an important role in the proof of Theorem 5.2.

4.1.
Hodge conjecture in the smooth case. We now recall from [5] the proof of the Hodge conjecture for the moduli space of rank 2 vector bundles with fixed determinant on a very general, smooth, projective curve. We give a brief sketch for the sake of completion.
Let C be a smooth, projective curve and L an invertible sheaf on C of odd degree. Denote by M C (2, L) the moduli of semi-stable, rank 2 sheaves on C with determinant L. By [20, Proposition 1], there exists an isomorphism of pure Hodge structures: By [24, Theorem 1], there exists α ∈ H 2 (M C (2, L), Z) and β ∈ H 4 (M C (2, L), Z) such that the cohomology ring H * (M C (2, L)) is generated by α, β and H 3 (M C (2, L)). Now, there is a natural isomorphism of pure Hodge structures where the first isomorphism comes from the cohomology of Jacobians (see [33, p. 169]) and the second isomorphism is the Poincaré duality. Combining with the isomorphism Φ above, we get an isomorphism of pure Hodge structures: Since cup-product is a morphism of pure Hodge structures, this isomorphism induces a surjective (graded) ring homomorphism which is a morphism of pure Hodge structures on each graded piece (α and β are given weights 2 and 4, respectively): (see [5, (1.2)]). As α and β are linear combination of Chern classes of the universal bundle associated to M C (2, L), they are algebraic cohomology classes (see [24, p. 338]). Then, by [5,Theorem 2] the isomorphism δ C induces a surjective ring homomorphism: of curves which is smooth over the punctured disc ∆ * and the central fiber is the irreducible nodal curve X 0 . There exists a family of Jacobians of curves associated to the family π 1 , i.e.

This induces an isomorphism of local systems
which induces an isomorphism of Hodge bundles from H 1 J ∆ * to H 1 X ∆ * i.e., preserves the Hodge filtration F p (see §2 for notations). Composing the above isomorphism with Φ ∆ * from §3.3, we get an isomorphism of local systems which induces an isomorphism of Hodge bundles from H 1 J ∆ * to H 3 G(2,L) ∆ * . Since the monodromy action on isomorphic local systems is the same, the weight filtrations associated to the two limit mixed Hodge structures are the same. In particular, we have Theorem 4.1. The morphism δ ∆ * induces an isomorphism of mixed Hodge structures: where J ∞ (resp. G(2, L) ∞ ) is the base change of J ∆ * (resp. G(2, L) ∆ * ) to the upper half plane h (see Definition 2.3 for notation) and the mixed Hodge structures on H 1 (J ∞ ) and H 3 (G(2, L) ∞ ) are as defined in §2.
Proof. The proof is identical to the proof of Theorem 3.1 (or [6, Proposition 4.1]) after replacing X in the statement with J. Remark 4.2. Since cup-product is a morphism of mixed Hodge structures (see [26,Corollary 5.45]), we have the following morphism of mixed Hodge structures for each i ≥ 0: where ∪ denotes the cup-product morphism. Choose a point s ′ ∈ h such that e(s ′ ) = s. Since h is contractible, by the Ehresmann's lemma, there is a natural isomorphism (induced by the inclusion of the fiber over s ′ into G(2, L) ∞ ): commuting with the cup-product (cup-product commutes with pull-back by continuous maps) where α s ∈ H 2 (M Xs (2, L s ), Z) and β s ∈ H 4 (M Xs (2, L s ), Z) arise as linear combination of the Chern classes of a universal bundle U s for M Xs (2, L s ) (see [24, p. 337] for an explicit description). As U s is the restriction to the fiber over s of the universal bundle U as in §3.3 for G(2, L) ∆ * , the classes α s and β s are monodromy invariant. Hence, α ∞ and β ∞ does not depend on the choice of s. Then the morphism Φ (i) ∞ induces a (graded) ring homomorphism which induces a morphism of mixed Hodge structures on each graded piece (α ∞ and β ∞ are considered as Hodge classes of weight 2 and 4, respectively). Since H * (M Xs (2, L s )) is generated by α s , β s and H 3 (M Xs (2, L s )) ( [24, Theorem 1]) and φ i s commutes with cup-product, the cohomology ring H * (G(2, L) ∞ ) is generated by α ∞ , β ∞ and H 3 (G(2, L) ∞ ). Since the morphism δ ∞ is an isomorphism, this implies that the morphism Φ ∞ above is surjective.

4.3.
Compactified Jacobians of curves. Let X 0 be an irreducible nodal curve. The compactified Jacobian J X 0 parametrizes rank 1, degree 0 torsion-free sheaves on X 0 (see [18,Theorem 3.3]). The compactified Jacobian is a semi-normal, projective variety. In particular, the normalization map h : J X 0 → J X 0 is the desingularization of J X 0 i.e., J X 0 is non-singular and the pre-image under h of the singular locus of J X 0 consists of two disjoint non-singular divisors, say D 1 and D 2 in J X 0 which are both isomorphic to the Jacobian Jac( X 0 ), of the normalization X 0 of the curve X 0 . Moreover, J X 0 is a P 1 -bundle over Jac( X 0 ). See [23, p. 567] for a nice description.
One can also consider the compactified Jacobian J X 0 as a degeneration of Jacobians of smooth curves. In particular, given a family π 1 of smooth curves degenerating to X 0 as above, there exists a flat, projective family (see [18,Theorem 3.3]) π 5 : J → ∆ such that for any s ∈ ∆ * , π −1 5 (s) ∼ = Jac(X s ) and the central fiber π −1 5 (0) = J X 0 . Blowing up the family J along the singular locus of the central fiber J X 0 followed by an etale base change, we get a flat projective family (this is the semi-stable reduction of π 5 , see [30, Example (2.15)]) π 5 : J ′ → ∆ such that the fibers over ∆ * coincides with that of π 5 above and the central fiber π −1 5 (0) is a reduced simple normal crossings divisor consisting of two irreducible components J X 0 and E, where E is a P 1 -bundle over Jac( X 0 ) and E ∩ J X 0 = D 1 ∪ D 2 .

Limit Hodge conjecture. Denote by H 2p
Hdg (J ∞ ) := H p,p Gr W 2p H 2p (J ∞ ). Using the description of compactified Jacobians in families given above we prove: Theorem 4.3. For a very general nodal curve X 0 as defined in the Introduction, we have H 2p Hdg (J ∞ ) ∼ = Q for each p.
Proof. For the proof of this statement we use the exact sequence (2.6). Notations as in §4.3. Applying Corollary 2.7 to π 5 above, we get the exact sequence of pure Hodge structures: To compute the middle term in this exact sequence we use the Mayer-Vietoris sequence. In particular, we have an exact sequence of pure Hodge structures: Recall, E and J X 0 are P 1 -bundles over Jac( X 0 ). Denote by q E : E → Jac( X 0 ) and q : J X 0 → Jac( X 0 ) the natural bundle morphisms. Let ξ 0 ∈ H 2 (P 1 ) be a generator. By the Leray-Hirsch theorem (see [33,Theorem 7.33]), we have for any closed point y ∈ Jac( X 0 ) and p ≥ 0: Denote by q i : D i ֒→ J X 0 and q E,i : D i ֒→ E the closed immersions. As q • q i and q E • q E,i are isomorphisms, q * i (resp. q * E,i ) maps q * H 2p (Jac( X 0 )) (resp. q * E H 2p (Jac( X 0 ))) isomorphically to H 2p (D i ). Using the decompositions (4.8) and (4.9), we then conclude that the last (restriction) morphism of (4.7) is surjective. Since (by assumption) X 0 is very general, we have H 2p Hdg (Jac( X 0 )) ∼ = Q for every p ≥ 0. Hence, using the decompositions (4.8) and (4.9) we have Then the surjectivity on the right of (4.7) implies that dim H p,p Gr W 2p H 2p ( J X 0 ∪ E) = 2. Using the exact sequence (4.6) it therefore suffices to check that the kernel of f 2p when restricted to H 2p−2 Hdg (D 1 ∪ D 2 )(−1) is one dimensional. Using the projection formula (see [26,Lemma B.26]) and the identification described in the Leray-Hirsch theorem (identifying certain cohomology classes with their restriction to y) we have for any γ ∈ H 2p−2 (Jac( X 0 )), that Taking γ = θ p−1 , where θ ∈ H 2 (Jac( X 0 )) is the class of the theta divisor, we then observe that the kernel of is one dimensional, generated by (a scalar multiple of) the pull-back of θ p−1 (recall from §2 that f 2p is induced by the Gysin morphism). This proves the theorem. The above corollary is what we call the "limit" Hodge conjecture in the family of Gieseker moduli spaces. Also note that H * Hdg (G(2, L) ∞ ) is not a polynomial ring in 3 variables i.e., there are relations between α ∞ , β ∞ and θ ∞ (see [16, p. 408]).
Proof of Corollary 4.5. Recall from Remark 4.2 that Φ (i) ∞ is a morphism of mixed Hodge structures and Φ ∞ , which is the direct sum over all the graded pieces, is surjective. This implies that we have a surjective morphism induced by Φ ∞ : By Theorem 4.3, H * Hdg (J ∞ ) is generated by θ ∞ . Hence, H * Hdg (G(2, L) ∞ ) is generated by α ∞ , β ∞ and (the image of) θ ∞ . This proves the first part of the corollary.
Observe that for a very general s ∈ ∆ * , H * A (G(2, L) s ) = H * A (M Xs (2, L s )) is generated as a ring over Q, by α s , β s and (the image of) a generator θ s of H 2 A (Jac(X s )) (see §4.1). Since α ∞ , β ∞ and (the image of) θ ∞ maps to α s , β s and (the image of) θ s , respectively (upto multiplication by a scalar) by the morphism φ i s as in (4.5) for i = 2, 4, 6, we conclude that H * A (G(2, L) ∞ ) is generated by α ∞ , β ∞ and (the image of) θ ∞ . Using the first part, we conclude H * A (G(2, L) ∞ ) = H * Hdg (G(2, L) ∞ ) as graded rings over Q. This proves the corollary.

Main Results
Assume that X 0 is a very general, irreducible nodal curve as defined in the Introduction. Note that, by the genericity of X 0 , we can choose the family of curves π 1 as in §3.1 such that for a very general s ∈ ∆ * , we have H * Hdg (Jac(X s )) is generated by the cohomology class of the theta divisor θ s ∈ H 2 Hdg (M Xs (2, L s )). We fix such a family π 1 , for this section. Let j 0 : G 0 ֒→ G X 0 (2, L 0 ) and j 1 : G 1 ֒→ G X 0 (2, L 0 ) be the natural inclusions. Define the algebraic cohomology groups on G X 0 (2, L 0 ) as: We first obtain an exact sequence analogous to (3.8) for algebraic classes.
Lemma 5.1. For any i ≥ 0, the algebraic cohomology group H 2i A (G X 0 (2, L 0 )) sits in the following exact sequence: Proof. The exact sequence (5.1) will be induced by the exact sequence (3.8). We first claim that the restriction of the specialization morphism sp 2i to the algebraic cohomology group H 2i A (G X 0 (2, L 0 )) factors through H 2i A (G(2, L) ∞ ). Let γ ∈ H 2i A (G X 0 (2, L 0 )). By definition, for The above sequence shows the surjectivity of j * 0 . Using this and the equality H * A (G X 0 (2, L 0 )) = H * Hdg (G X 0 (2, L 0 )), the Hodge conjecture for G 0 follows. Since G 1 is P 3 -bundle over M X 0 (2, L 0 ), it satisfies the Hodge conjecture. This proves the theorem.
As an easy consequence we can compute the algebraic Poincaré polynomial of G 0 .
Substituting for H(g, t) one immediately gets the corollary.