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Cohomology of line bundles on compactified Jacobians

Let C be an integral projective curve with surficial singularities. We prove that topologically trivial line bundles on the compactified Jacobian of C are in one-to-one correspondence with line bundles on C (the autoduality conjecture), and compute the cohomology of the line bundles. We also show that the natural Fourier-Mukai functor between the derived categories of quasi-coherent sheaves on the Jacobian and on the compactified Jacobian is fully faithful.


Introduction
Let C be a smooth irreducible projective curve over a field k, and let J be the Jacobian of C. As an abelian variety, J is self-dual. More precisely, J × J carries a natural line bundle (the Poincaré bundle) P that is universal as a family of topologically trivial line bundles on J.
The Poincaré bundle defines the Fourier-Mukai functor  ([22]) proved that F is an equivalence of categories; the proof uses the formula where O ζ is the structure sheaf of the zero element ζ ∈ J and g is the genus of C. Formula (1) goes back to Mumford (see the proof of the theorem in [23,Section III.13]). Now suppose that C is a singular curve, which we assume to be projective and integral. The Jacobian J is no longer projective, but it admits a natural compactification J ⊃ J. By definition, J is the moduli space of torsion-free sheaves F on C such that F has generic rank one and χ(F ) = χ(O C ); J is identified with the open subset of locally free sheaves F . It is natural to ask whether J is in some sense selfdual. For instance, one can look for a Poincaré sheaf (or complex of sheaves) P on J × J. One can then ask whether P is, in some sense, a universal family of sheaves on J and whether the corresponding Fourier-Mukai functor F : D b (J) → D b (J) is an equivalence.
In the case when singularities of C are nodes or cusps, such Poincaré sheaf P is constructed by E. Esteves and S. Kleiman in [12]; they also prove the universality of P . In addition, if C is a singular plane cubic, F is known to be an equivalence ( [8,9], also formulated as Theorem 5.2 in [6]).
If singularities of C are more general, constructing the Poincaré sheaf P on J × J is much harder (see Remark (i) at the end of the introduction). However, it is easy to construct a Poincaré bundle P on J × J. It can then be used to define a Fourier-Mukai transform (2) F : Here it is important to work with the derived categories of quasicoherent sheaves, since F does not preserve coherence.
In this paper, we assume that C is an integral projective curve with planar singularities; the main result is that the formula (1) still holds in this case. This implies that (2) is fully faithful. As a simple corollary, we prove the following autoduality result: P is the universal family of topologically trivial line bundles on J, so that J is identified with the connected component of the trivial bundle in the moduli space of line bundles on J. This generalizes the Autoduality Theorem of [11] (see the remark after Theorem C).
Remarks. (i) Suppose that there exists an extension of P to a sheaf P on J × J such that the corresponding Fourier-Mukai transform F : D b (J) → D b (J ) is an equivalence. After the first version of this paper was completed, such an extension was constructed in [3]. Then (2) is a composition of F and the direct image j * : for the open embedding j : J ֒→ J. Since j * is fully faithful, so is (2). Thus our result is natural assuming existence of F.
(ii) Compactified Jacobians appear as (singular) fibers of the Hitchin fibration for the group GL(n); therefore, our results can be interpreted as a kind of autoduality of the Hitchin fibration. Conversely, some of our results can be derived from a theorem of E. Frenkel and C. Teleman [15] (see Theorem 15). We explore this relation in more details in Section 7.
(iii) Recall that the curve C is assumed to be integral with planar singularities. We assume integrality of C to avoid working with stability conditions for sheaves on C. It is likely that our argument works without this assumption if one fixes an ample line bundle on C and defines the compactified Jacobian J to be the moduli space of semi-stable torsion-free sheaves of degree zero. Such generalization is natural in view of the previous remark, because some fibers of the Hitchin fibration are compactified Jacobians of non-integral curves.
On the other hand, the assumption that C has planar singularities is more important. There are two reasons why the assumption is natural. First of all, J is irreducible if and only if the singularities of C are planar ( [20]); so if one drops this assumption, J is no longer dense in J. Secondly, only compactified Jacobians of curves with planar singularities appear in the Hitchin fibration.

Main results
Fix a ground field k. For convenience, let us assume that k is algebraically closed. Let C be an integral projective curve over k. Denote by J its Jacobian, that is, J is the moduli space of line bundles on C of degree zero. Denote by J the compactified Jacobian; in other words, J is the moduli space of torsion-free sheaves on C of generic rank one and degree zero. (For a sheaf F of generic rank one, the degree is deg( Let P be the Poincaré bundle; it is a line bundle on J × J. Its fiber over (L, F ) ∈ (J × J) equals More explicitly, we can write L ≃ O( a i x i ) for a divisor a i x i supported by the smooth locus of C, and then From now on, we assume that C has planar singularities; that is, the tangent space to C at any point is at most two-dimensional. Our main result is the computation of the direct image of P : Here O ζ is the structure sheaf of the neutral element ζ = [O C ] ∈ J, and p 1 : J ×J → J is the projection.
Let us now view P as a family of line bundles on J parametrized by J. For fixed L ∈ J, denote the corresponding line bundle on J by P L . In other words, P L is the restriction of P to {L} × J. Applying base change, we can use Theorem A to compute cohomology of P L : Remark. Theorem C answers the question raised in [11]. Following [17], set (4) Pic τ (J ) = {L ∈ Pic(J) : L ⊗n ∈ Pic 0 (J ) for some n > 0}, The main result of [11] is the Autoduality Theorem, which claims that if all singularities of C are double points, then ρ : J → Pic 0 (J ) and Pic 0 (J ) = Pic τ (J). Theorem C generalizes the first statement to curves with planar singularities; as for the second statement, we show in Proposition 12 that Pic 0 (J) = Pic σ (J). We do not know whether Pic τ (J) and Pic σ (J) coincide when char(k) > 0 and C has planar singularities.
Theorem A can be reformulated in terms of the Fourier functor given by P . Recall that D b (J) stands for the (bounded) derived category of quasicoherent sheaves on J. The functor F admits a left adjoint given by . This formula relies on the computation of the dualizing sheaf on J: see Corollary 9.
Proof. The first statement follows from Theorem A by base change. (This is completely analogous to the original argument of [22,Theorem 2.2].) This implies the second statement, because the functors F ∨ and F are adjoint.
Remark. For simplicity, we considered a single curve C in this section. However, all our results hold for families of curves. Actually, we prove Theorem A for the universal family of curves (Theorem 10); base change then implies that the statement holds for any family, and, in particular, for any single curve.

Line bundles on a compactified Jacobian
Proof. Let T → J be the G m -torsor corresponding to (P L )| J . One easily sees that T is naturally an abelian group that is an extension of J by G m . The action of J on J lifts to an action of T on P L , therefore, T also acts on H i (J , P L ). Note that G m ⊂ T acts via the tautological character.
Let V ⊂ H i (J , P L ) be an irreducible T -submodule. Since T is commutative, dim(V ) = 1. The action of T on V is given by a character χ : T → G m . Since χ| Gm = id, we see that χ gives a splitting T ≃ G m ×J. This implies the statement.
Remark. If C is smooth, Proposition 1 is equivalent to observation (vii) in [23, Section II.8]; however, our proof uses a slightly different idea, which is better adapted to the singular case.
Let C 0 ⊂ C be the smooth locus of C.
Proof. Fix a degree minus one line bundle ℓ on C. It defines an Abel-Jacobi map Here ℓ(c) can be defined as the sheaf of homomorphisms from the ideal sheaf of c ∈ C to ℓ. Notice that α * (P L ) ≃ L and α(C 0 ) ⊂ J. Now Proposition 1 completes the proof.
Clearly, N ⊂ J is closed (by the Semicontinuity Theorem), and N = supp(Rp 1, * P ), where p 1 : J × J → J is the projection (by base change).
Corollary 3. Let g be the (arithmetic) genus of C andg be its geometric genus, that is, the genus of its normalization. Then dim(N ) ≤ (g −g).
Proof. Let ν :C → C be the normalization, and letJ be the Jacobian ofC. The map ν * : J →J is smooth and surjective; its fibers have dimension (g −g). Denote byÑ ⊂J the set of line bundles onC that are trivial on ν −1 (C 0 ) ⊂C. By Corollary 2, ν * (N ) ⊂Ñ . Now it suffices to note thatÑ is a countable set.

Moduli of curves
Let M = M g be the moduli stack of integral projective curves C of genus g with planar singularities. The following properties of M are well known: Remark. Denote by C the universal curve over M; that is, C is the moduli stack of pairs (C ∈ M, c ∈ C). One easily checks that C is a smooth stack of dimension 3g − 2. This is similar to the statement (ii') after Theorem 8.
Consider the normalizationC of a curve C ∈ M, and letg be the genus ofC (that is, the geometric genus of C). We need some results on the stratification of M by geometric genus due to Tessier ( [26]), Diaz and Harris ( [10]), and Laumon ([21]). Since our settings are somewhat different, we provide the proofs.
Denote by M (g) ⊂ M the locus of curves C ∈ M of geometric genusg. Note that we view M (g) simply as a subset of the set of points of M, rather than a substack. Therefore, it suffices to show that π is projective. Let S ′′ be the stack of collections (C, F, s), where C ∈ M, F is a torsion-free sheaf on C of generic rank one and degree g −g, s ∈ H 0 (C, F ). Also, let S ′ be the stack of collections (C, F, s, µ), where (C, F, s) ∈ S ′′ and µ : where µ is the product on the sheaf of algebras ν * (OC ). This identifies S and S ′ . The forgetful map S ′ → S ′′ : (C, F, s, µ) → (C, F, s) is a closed embedding (essentially because µ is uniquely determined by µ(s⊗s) = s). Finally, the map S ′′ → M : (C, F, s) → C is projective.
T νπ −1 (C) is isomorphic to the space of first-order deformations of O C viewed as a sheaf of subalgebras of ν * OC . This yields an isomorphism Now it suffices to notice that the fibers of the cotangent sheaf Ω C are at most two-dimensional, and that the length of the sky-scraper sheaf ν * OC /O C equals g −g.
Remark. By looking at nodal curves, one sees that codim(M (g) ) = g −g.

Universal Jacobian
Let J (resp. J ⊂ J) be the relative compactified Jacobian (resp. relative Jacobian) of C over M. Here is the precise definition: Definition 7. For a scheme S, letĴ S be the following groupoid: • Objects ofĴ S are pairs (C, F ), where C → S is a flat family of integral projective curves with planar singularities (that is, C ∈ M S ), and F is a S-flat coherent sheaf on C whose restriction to the fibers of C → S is torsion free of generic rank one and degree zero; where φ is a morphism of S-schemes, and ℓ is an invertible sheaf on S.
As S varies, groupoidsĴ S form a pre-stack; let J be the stack associated to it. Also, consider pairs (C, F ) where C ∈ M S and F is a line bundle on C (of degree zero along the fibers of S → C); such pairs form a sub-prestack ofĴ; let J ⊂ J be the associated stack. Remark. The key step in the proof of (i) is Iarrobino's calculation (see [19]): For other proofs of (5), see [7], [24,Theorem 1.13] and [5]. Also, J. Rego gives an alternative inductive proof of (i) in [25].
Denote by j the rank g vector bundle on M whose fiber over C ∈ M is H 1 (C, O C ). Alternatively, j can be viewed as the bundle of (commutative) Lie algebras corresponding to the group scheme p : J → M. The relative dualizing sheaf for p then equals Ω g J/M = p * (det(j) −1 ). It is easy to find the dualizing sheaf for p : J → M: The relative dualizing sheaf ω p of p equals p * (det(j) −1 ).
Proof. By Theorem 8(ii), p is Gorenstein, so ω p is a line bundle. Since ω p | J = Ω g J/M , it suffices to check that codim(J − J) ≥ 2. But this is clear because a generic curve C ∈ M is smooth (see Proposition 6).

Proof of Theorem A
Consider the Poincaré bundle on J × M J. We still denote it by P .
By (6), we see that both Rp 1, * P and R Hom(Rp 1, * P, O J )[−g] are concentrated in cohomological degrees from zero to g. It is now easy to see that Rp 1, * P is concentrated in cohomological degree g, and that R g p 1, * P is a coherent Cohen-Macaulay sheaf of codimension g. Next, notice that the restriction of P to ζ(M) × M J is trivial. This provides a map ζ * (R g p 1, * P ) → R g p * (O J ). By Serre's duality, R g p * O J = det(j). Now by adjunction, we obtain a morphism (7) R g p 1, * P → ζ * det j.
It remains to verify that (7) is an isomorphism. Since (7) is an isomorphism over ζ(M) by construction, we need to verify that supp(R g p 1, * P ) = ζ(M) ⊂ J. Let us check that supp(R g p 1, * P ) equals ζ(M) as a set. As a set, supp(R g p 1, * P ) consists of pairs (C, L) ∈ J such that the line bundle L on C satisfies H g (J, P L ) = 0. In this case, H 0 (J , P −1 L ) = 0 by Serre's duality. Since J is irreducible, we see that the line bundle P −1 L has a subsheaf isomorphic to O J . On the other hand, the line bundles P −1 L = P L −1 and O J = P O are algebraically equivalent, and therefore their Hilbert polynomials coincide. Hence P L ≃ O J . Finally, we can restrict P L to the image of the Abel-Jacobi map (see the proof of Corollary 2) to obtain L ≃ O C .
To complete the proof, let us verify that supp(R g 1, * P ) = ζ(M) as a scheme. Since R g 1, * P is Cohen-Macaulay of codimension g, it suffices to check the claim generically on ζ(M). We can thus restrict ourselves to the open substack of smooth curves in M, and the claim reduces to (1).
Remark. The proof is similar to an argument of S. Lysenko (see proof of Theorem 4 in [4]), see also D. Mumford's proof of the theorem in [23, Section III.13].
Using base change, one easily derives Theorem A from Theorem 10.

Autoduality
Recall that the morphism ρ : J → Pic J is given by L → P L . Since the tangent space to J at Let us fix a line bundle ℓ on C of degree minus one. It defines an Abel-Jacobi map α : C → J, as in the proof of Corollary 2. We then obtain a morphism α * : Pic(J) → Pic(C) : L → α * L.
Remark. Injectivity of dρ follows from the existence of the left inverse. Once injectivity is known, bijectivity follows from the equality Proof of Theorem C. Pic(J) is a group scheme of locally finite type (see [16, is an isomorphism. Bijectivity of dρ (Proposition 11) implies that K is a disjoint union of points. Therefore, the connected component of identity of Pic(J) is contained in ρ(J). Now it remains to notice that J is connected. isétale for all n coprime to char k. Therefore, K σ isétale over M.
Finally, the morphism K σ → M is separated, and over the locus of smooth curves C ∈ M, we have Pic 0 (J C ) = Pic σ (J C ) by [23,Corollary IV.19.2]. Therefore, K σ is the zero group scheme, and Pic σ (J/M) = J, as required.

Fibers of the Hitchin fibration
Recall the construction of the Hitchin fibration [18] (for GL(n)). Fix a smooth curve X and an integer n.

Definition 13.
A Higgs bundle is a rank n vector bundle E on X together with a Higgs field A : Given a Higgs bundle (E, A), consider the characteristic polynomial of A: (8) det(λI − A) = λ n + a 1 λ n−1 + · · · + a n ; a i ∈ H 0 (X, Ω ⊗i X ). The zero locus of (8) is a curve C ⊂ T * X: the spectral curve of A. Higgs bundle (E, A) gives rise to a coherent sheaf F on C; informally, F is the 'sheaf of coeigenspaces': its fiber over a point (x, µ) ∈ T * X is the co-eigenspace Here x ∈ X, µ ∈ Ω X,x .

Proposition 14.
(i) F is a torsion-free sheaf on C whose fiber at a generic point of C has length equal to the multiplicity of the corresponding component of C. In particular, if C is reduced, F is a torsion-free sheaf of generic rank one.
(ii) Fix a spectral curve C (that is, fix a polynomial (8)). Then (E, A) → F is a one-to-one correspondence between Higgs bundles with spectral curve C and sheaves F as in (i).
Given F , E is reconstructed as the push-forward of F with respect to C → X. Therefore, F and E have equal Euler characteristics. We have χ(O C ) = n 2 χ(O X ) = n 2 (1 − g), where g is the genus of X. Hence deg(F ) = 0 if and only if deg(E) = n(n − 1)(1 − g), where g is the genus of X. (Recall that deg(F ) = χ(F ) − χ(O C ).) Also, note that (E, A) is (semi)stable if and only if F is (semi)stable. If C is integral, F has generic rank one and stability is automatic.
Let Higgs be the moduli space of semi-stable Higgs bundles (E, A) with rk(E) = n and deg(E) = n(n − 1)(1 − g). Also, let SCurves be the space of spectral curves C ⊂ T * X; explicitly, SCurves is the space of coefficients (a 1 , . . . , a n ) of (8): Finally, let SCurves ′ ⊂ SCurves be the locus of integral spectral curves C ⊂ T * X.
The correspondence (E, A) → C gives a map h : Higgs → SCurves (the Hitchin fibration). For C ∈ SCurves, the fiber h −1 (C) is the space of Higgs bundles with spectral curve C; Proposition 14 identifies h −1 (C) with the moduli space of semistable coherent sheaves F on C that satisfy Proposition 14(i) and have degree zero. In other words, the fiber is the compactified Jacobian of C.
The results of this paper can be applied to integral spectral curves C ∈ SCurves ′ . For instance, Theorem B(ii) implies that Actually, applying the relative version of Theorem B(ii) to the universal family of spectral curves, we obtain an isomorphism (9) (R i h * O Higgs )| SCurves ′ = Ω i SCurves ′ , where we used the symplectic form on T * X to identify H 1 (C, O C ) with the cotangent space to C ∈ SCurves ′ . Recently, E. Frenkel and C. Teleman proved that the isomorphism (9) can be extended to the space of all spectral curves: Theorem 15. There is an isomorphism SCurves .
Remarks. (i) In [18], N. Hitchin works with the Hitchin fibration for the group SL(2), but his argument can be used to compute R i h * O Higgs for arbitrary n (still assuming i = 0, 1). Actually, essentially the same argument computes R i p * O J for i = 0, 1. (Recall that p : J → M is the universal compactified Jacobian over the moduli stack of curves M.) (ii) In [15], Theorem 15 is stated for the Hitchin fibration for arbitrary group, not just GL(n).
(iii) One can derive some of our results from Theorem 15, at least for integral curves C that appear as spectral curves of the Hitchin fibration. Indeed, for such C ∈ SCurves ′ , Theorem 15 implies Theorem B(ii). In turn, this implies Theorem 8. Also, one can easily derive from Theorem B(ii) that the isomorphism of Theorem A exists on some neighborhood U of ζ ∈ J, so Theorem B(i) holds for L ∈ U . Similarly, we see that P defines a fully faithful Fourier-Mukai transform from D b (U ) to D b (J ).