DEFORMATIONS OF THE PICARD BUNDLE

I. BISWAS, L. BRAMBILA-PAZ, AND P. E. NEWSTEAD

Abstract. Let X be a nonsingular algebraic curve of genus g ≥ 3, and M_{ξ} the moduli space of stable vector bundles of rank n ≥ 2 and degree d with fixed determinant ξ over X such that n and d coprime and d > n(2g − 2). We assume that if g = 3 then n ≥ 4 and if g = 4 then n ≥ 3. Let W_{ξ}(L) denote the vector bundle over M_{ξ} defined as the direct image π_{∗}(U_{ξ} ⊗ p_{1}^{∗}L) where U_{ξ} is a universal vector bundle over X × M_{ξ} and L a line bundle over X of degree zero. The space of infinitesimal deformations of W_{ξ}(L) is proved to be isomorphic to H^{1}(X, O_{X}). This construction gives a complete family of vector bundles over M_{ξ} parametrized by Pic^{0}(X) such that W_{ξ}(L) is the vector bundle corresponding to L ∈ Pic^{0}(X). The connected component of the moduli space of stable sheaves with the same Hilbert polynomial as W_{ξ}(O) over M_{ξ} containing W_{ξ}(O) is in fact isomorphic to Pic^{0}(X) as a polarised variety.

1. Introduction

Let X be a connected nonsingular projective algebraic curve of genus g ≥ 3. Consider a universal line bundle L over X ×Pic^{d}(X), where d ≥ 2g −1. For X non-hyperelliptic, G. Kempf computed the space of infinitesimal deformations of the Picard bundle π_{∗}L over Pic^{d}(X), where π denotes the projection to Pic^{d}(X). In particular,

dim H^{i}(Pic^{d}(X), End(π_{∗}L)) = \{ \begin{matrix}1 & if i = 0 \\ 2g & if i = 1. \end{matrix}

Moreover, the deformations of π_{∗}L arise from translations in Pic^{d}(X) and deformations of the universal line bundle L [8]. Note that the space of universal bundles is an affine space for Pic(Pic^{d}(X)), that is for any L ∈ Pic(Pic^{d}(X)), the line bundle L ⨂ π^{∗}L is also universal.

We prove an analogue of the result of Kempf for the moduli spaces of higher rank vector bundles over X with fixed determinant. In particular, we compute the space of infinitesimal deformations of the Picard bundle.

Fix a holomorphic line bundle ξ over X of degree d. Let M_{ξ} := M_{ξ}(n, d) be the moduli space of stable vector bundles E over X with rank(E) = n ≥ 2, deg(E) = d and ⋀^{n} E = ξ. Assume that n and d are coprime, ensuring the smoothness and completeness of M_{ξ}. We will assume that if g = 3 then n ≥ 4 and if g = 4 then n ≥ 3. The three

Mathematics Subject Classification: 14H60. All authors are members of the research group VBAC (Vector Bundles on Algebraic Curves), which is partially supported by EAGER (EC FP5 Contract no. HPRN-CT-2000-00099) and by EDGE (EC FP5 Contract no. HPRN-CT-2000-00101). The second author acknowledges the support of CONACYT grant 28492-E.

2 I. BISWAS, L. BRAMBILA-PAZ, AND P. E. NEWSTEAD

special cases, namely g = 3 with n = 2, 3 and g = 4 with n = 2 are omitted in our main result since the method of proof does not cover these cases.

It is known that there is a universal vector bundle over X ×M_{ξ}. Two such universal vector bundles differ by tensoring with the pullback of a line bundle on M_{ξ}. However, since Pic(M_{ξ}) = ℤ, it is possible to choose canonically a universal bundle. Let l be the smallest positive number such that ld ≡ 1 mod n. There is a unique universal vector bundle U_{ξ} over X × M_{ξ} such that ⋀^{n}U_{ξ}|_{z×M_{ξ}} = Θ^{⊗l} [15], where z ∈ X and Θ is the ample generator of Pic(M_{ξ}). Henceforth, by a universal bundle we will always mean this canonical one. We denote by

π : X × M_{ξ} ⟶ M_{ξ}

the natural projection and by p_{1} the projection of X × M_{ξ} to X.

Assume that d > n(2g − 2). For any L in Pic^{0}(X), let

W_{ξ}(L) := π_{∗}(U_{ξ} ⊗ p_{1}^{∗}L)

be the direct image. The assumption on d ensures that W_{ξ}(L) is a locally free sheaf on M_{ξ} and all the higher direct images of U_{ξ} ⊗ p_{1}^{∗}L vanish. The rank of W_{ξ}(L) is d + n(1 − g) and H^{i}(M_{ξ}, W_{ξ}(L)) ≅ H^{i}(X × M_{ξ}, U_{ξ} ⊗ p_{1}^{∗}L). When L is the trivial line bundle, W_{ξ} := W_{ξ}(O) is called the Picard bundle. These vector bundles have been studied from various points of view (see [10], [11]).

Here we consider the infinitesimal deformations of W_{ξ}(L). In Theorem 2.9 and Corollary 2.8 we prove that

H^{1}(M_{ξ}, End(W_{ξ}(L))) ≅ H^{1}(X, O_{X})

and

H^{0}(M_{ξ}, End(W_{ξ}(L))) ≅ ℂ .

In the special case where n = 2 with L as the trivial bundle and g ≥ 3, this was proved by V. Balaji and P. Vishwanath in [1] using a construction of M. Thaddeus [16]. It is known that W_{ξ} is simple [5] and indeed that it is stable [4].

We prove that the family of vector bundles {W_{ξ}(L)} over M_{ξ}, parametrized by Pic^{0}(X), is complete (Corollary 5.2). In particular, the infinitesimal deformation map

H^{1}(X, O_{X}) ⟶ H^{1}(M_{ξ}, End(W_{ξ}(L)))

is an isomorphism. As we noted, W_{ξ} is stable. Thus, if we denote by M_{X}^{0}(W_{ξ}) the connected component of the moduli space of stable sheaves with the same Hilbert polynomial as W_{ξ} on M_{ξ} containing W_{ξ}, then in Theorem 5.1 we prove that M_{X}^{0}(W_{ξ}) is isomorphic to Pic^{0}(X) as a polarised variety. Moreover, for two smooth projective curves X and X′ and line bundles ξ, ξ′ of degree d > n(2g −2) over X, X′ respectively, if M_{X}^{0}(W_{ξ}) ≅ M_{X′}^{0}(W_{ξ}) as polarised varieties, then X ≅ X′ (Corollary 5.3).

DEFORMATIONS OF THE PICARD BUNDLE 3

The vector bundle W_{ξ}(L) can be considered as the Fourier-Mukai transform of L constructed using U_{ξ}. So Theorem 5.1 can be interpreted as an assertion that the Fourier-Mukai transformation map is an isomorphism of Pic^{0}(X) with a component of the moduli space of stable sheaves on M_{ξ}.

Acknowledgements. Part of this work was done during a visit of the authors to ICTP, Italy. The authors thank ICTP for its hospitality.

2. Cohomology of W_{ξ}(L_{1}) ⊗ W_{ξ}(L_{2})^{∗}

We continue with the notation set up in the introduction. To compute the cohomology groups H^{i}(M_{ξ}, W_{ξ}(L_{1}) ⊗ W_{ξ}(L_{2})^{∗}) we need the following propositions.

For i, j ∈ {1, 2, 3} and i < j, let p_{ij} denote the projection of X × M_{ξ} × X onto the Cartesian product of the i-th and the j-th factor. The projection of X × M_{ξ} × X onto the j-th factor will be denoted by p_{j}.

Proposition 2.1. If L_{1} and L_{2} are two line bundles in Pic^{0}(X) then

H^{i}(M_{ξ}, W_{ξ}(L_{1}) ⊗ W_{ξ}(L_{2})^{∗}) ≅ H^{i}(X × M_{ξ}, U_{ξ} ⊗ p_{1}^{∗}L_{1} ⊗ π^{∗}W_{ξ}(L_{2})^{∗})

for any i ≥ 0. Moreover,

H^{i}(M_{ξ}, W_{ξ}(L_{1})⊗W_{ξ}(L_{2})^{∗}) ≅ H^{i+1}(X ×M_{ξ} ×X, p_{12}^{∗}U_{ξ} ⊗p_{1}^{∗}L_{1}⊗p_{23}^{∗}U_{ξ}^{∗}⊗p_{3}^{∗}L_{2}^{∗}⊗p_{3}^{∗}K_{X})

for i ≥ 0, where K_{X} is the canonical line bundle over X.

Proof: By the assumption that d > n(2g − 2) we have that H^{1}(X, U_{ξ}|_{X×v} ⊗ L_{1}) = 0 for all v ∈ M_{ξ}. Using the projection formula and the Leray spectral sequence we have

H^{i}(X × M_{ξ}, U_{ξ} ⊗ p_{1}^{∗}L_{1} ⊗ π^{∗}W_{ξ}(L_{2})^{∗}) ≅ H^{i}(M_{ξ}, W_{ξ}(L_{1}) ⊗ W_{ξ}(L_{2})^{∗}) .

This proves the first part.

For every stable vector bundle E of rank n and degree d,

H^{0}(X, E^{∗} ⊗ L_{2}^{∗} ⊗ K_{X}) ≅ H^{1}(X, E ⊗ L_{2})^{∗} = 0 .

Consequently, the projection formula gives

R^{i}p_{12∗}(p_{23}^{∗}(U_{ξ}^{∗}) ⊗ p_{3}^{∗}L_{2}^{∗} ⊗ p_{3}^{∗}K_{X}) = 0

for i ≠ 1, and since H^{1}(X, E^{∗} ⨂ L_{2}^{∗} ⨂ K_{X}) ≅ H^{0}(X, E ⊗ L_{2})^{∗}, we have

R^{1}p_{12∗}(p_{23}^{∗}(U_{ξ}^{∗}) ⊗ p_{3}^{∗}L_{2}^{∗} ⊗ p_{3}^{∗}K_{X}) ≅ π^{∗}W_{ξ}(L_{2})^{∗} .

Finally, using the projection formula and the Leray spectral sequence, it follows that

H^{i}(X × M_{ξ}, U_{ξ} ⊗ p_{1}^{∗}L_{1} ⊗ π^{∗}W_{ξ}(L_{2})^{∗})
≅ H^{i+1}(X × M_{ξ} × X, p_{12}^{∗}U_{ξ} ⊗ p_{1}^{∗}L_{1} ⊗ p_{23}^{∗}U_{ξ}^{∗} ⊗ p_{3}^{∗}L_{2}^{∗} ⊗ p_{3}^{∗}K_{X})

4 I. BISWAS, L. BRAMBILA-PAZ, AND P. E. NEWSTEAD

for i ≥ 0. Thus,

H^{i+1}(X ×M_{ξ} ×X, p_{12}^{∗}U_{ξ} ⊗p_{1}^{∗}L_{1}⊗p_{23}^{∗}U_{ξ}^{∗}⊗p_{3}^{∗}L_{2}^{∗}⊗p_{3}^{∗}K_{X}) ≅ H^{i}(M_{ξ}, W_{ξ}(L_{1})⊗W_{ξ}(L_{2})^{∗})

as asserted in the proposition. □

Remark 2.2. Proposition 2.1 can be formulated in a more general context. Let V be a flat family of vector bundles over X parametrized by a complete irreducible variety Y such that for each y ∈ Y we have H^{1}(X, V |_{X×y}) = 0. Under this assumption

H^{i}(Y, End(π_{∗}V)) ≅ H^{i+1}(X × Y × X, p_{12}^{∗}V ⊗ p_{23}^{∗}V^{∗} ⊗ p_{3}^{∗}K_{X}) .

The proof is the same as for Proposition 2.1. □

Denote by R^{i} the i-direct image of p_{12}^{∗}U_{ξ} ⊗ p_{1}^{∗}L_{1} ⊗ p_{23}^{∗}U_{ξ}^{∗} for the projection p_{3}, that is

R^{i} := R^{i}p_{3∗}(p_{12}^{∗}U_{ξ} ⊗ p_{1}^{∗}L_{1} ⊗ p_{23}^{∗}U_{ξ}^{∗}).

From the Leray spectral sequence the following two exact sequences

0 ⟶ H^{1}(X, R^{0}⊗L_{2}^{∗}⊗K_{X}) ⟶ H^{1}(X ×M_{ξ}×X, p_{12}^{∗}U_{ξ}⊗p_{1}^{∗}L_{1}⊗p_{23}^{∗}U_{ξ}^{∗}⊗p_{3}^{∗}L_{2}^{∗}⊗p_{3}^{∗}K_{X})

⟶ H^{0}(X, R^{1} ⊗ L_{2}^{∗} ⊗ K_{X}) ⟶ 0  (1)

and

0 ⟶ H^{1}(X, R^{1}⊗L_{2}^{∗}⊗K_{X}) ⟶ H^{2}(X ×M_{ξ} ×X, p_{12}^{∗}U_{ξ} ⊗p_{1}^{∗}L_{1}⊗p_{23}^{∗}U_{ξ}^{∗}⊗p_{3}^{∗}L_{2}^{∗}⊗p_{3}^{∗}K_{X})

⟶ H^{0}(X, R^{2} ⊗ L_{2}^{∗} ⊗ K_{X}) ⟶ 0  (2)

are obtained.

Proposition 2.3. R^{0} = 0.

Proof: Note that for any x ∈ X

H^{0}(X × M_{ξ}, U_{ξ} ⊗ p_{1}^{∗}L_{1} ⊗ π^{∗}U_{x}^{∗}) ≅ H^{0}(X, L_{1} ⊗ p_{1∗}(U_{ξ} ⊗ π^{∗}U_{x}^{∗})).  (3)

From [3] and [13] we know that, for generic y ∈ X, the two vector bundles U_{y} and U_{x} are non-isomorphic and stable. Hence H^{0}(M_{ξ}, U_{y} ⊗ U_{x}^{∗}) = 0. This implies that

p_{1∗}(U_{ξ} ⊗ π^{∗}U_{x}^{∗}) = 0  (4)

So (3) gives

R^{0} = R^{0}p_{3∗}(p_{12}^{∗}U_{ξ} ⊗ p_{1}^{∗}L_{1} ⊗ p_{23}^{∗}U_{ξ}^{∗}) = 0

and the proof is complete. □

In the following sections we will use Hecke transformations to compute the fibre of R^{i} for i = 1, 2, and will prove the following propositions.

Proposition 2.4. R^{2} = 0.

Proposition 2.5. R^{1} is a line bundle. Moreover, R^{1} ≅ L_{1} ⊗ T X.

DEFORMATIONS OF THE PICARD BUNDLE 5

Assume for the moment Propositions 2.4 and 2.5.

From the exact sequence (1) and the previous propositions we have the following theorem.

Theorem 2.6. H^{0}(M_{ξ}, W_{ξ}(L_{1}) ⊗ W_{ξ}(L_{2})^{∗}) = H^{0}(X, L_{1} ⊗ L_{2}^{∗}).

Proof: Combining (1), Proposition 2.3 and the second part of Proposition 2.1 it follows that

H^{0}(M_{ξ}, W_{ξ}(L_{1}) ⊗ W_{ξ}(L_{2})^{∗}) ≅ H^{0}(X, R^{1} ⊗ L_{2}^{∗} ⊗ K_{X}) .

From Proposition 2.5 it follows immediately that

H^{0}(X, R^{1} ⊗ L_{2}^{∗} ⊗ K_{X}) ≅ H^{0}(X, L_{1} ⊗ L_{2}^{∗})

and hence the proof is complete. □

Corollary 2.7. If L_{1} ≅ L̸_{2}, then W_{ξ}(L_{1}) ≅ W̸_{ξ}(L_{2}).

Proof: Since L_{1} ≅ L̸_{2}, we have H^{0}(M_{ξ}, W_{ξ}(L_{1}) ⊗ W_{ξ}(L_{2})^{∗}) = 0. Consequently, W_{ξ}(L_{1}) ≅ W̸_{ξ}(L_{2}). □

The following is also a corollary of Theorem 2.6.

Corollary 2.8. The vector bundle W_{ξ}(L) is simple for any L ∈ Pic^{0}(X). In other words, H^{0}(M_{ξ}, W_{ξ}(L) ⊗ W_{ξ}(L)^{∗}) = ℂ.

The following theorem gives the infinitesimal deformations of W_{ξ}(L).

Theorem 2.9. For any line bundle L ∈ Pic^{0}(X), the space of infinitesimal deformations of the vector bundle W_{ξ}(L), namely H^{1}(M_{ξ}, End(W_{ξ}(L))), is canonically isomorphic to H^{1}(X, O_{X}). In particular,

dim H^{1}(M_{ξ}, End(W_{ξ}(L))) = g .

Proof: From (2) and Proposition 2.4 we obtain an isomorphism

H^{1}(X, R^{1} ⊗ L^{∗} ⊗ K_{X}) ≅ H^{2}(X × M_{ξ} × X, p_{12}^{∗}U_{ξ} ⊗ p_{1}^{∗}L ⊗ p_{23}^{∗}U_{ξ}^{∗} ⊗ p_{3}^{∗}L^{∗} ⊗ p_{3}^{∗}K_{X}). (5)

From Proposition 2.5 we have

\begin{aligned}H^{1}(X, R^{1} ⊗ L^{∗} ⊗ K_{X}) & ≅ & H^{1}(X, L ⊗ T X ⊗ L^{∗} ⊗ K_{X}) \\ & ≅ & H^{1}(X, O). \end{aligned}

Combining this observation with Proposition 2.1 and (5) we get

H^{1}(M_{ξ}, End(W_{ξ}(L))) ≅ H^{1}(X, O_{X}).
□

Remark 2.10. From the proof of Theorem 2.9 and Proposition 2.1 we see that if L_{1} and L_{2} are not isomorphic then

H^{1}(M_{ξ}, W_{ξ}(L_{1}) ⊗ W_{ξ}(L_{2})^{∗}) ≅ H^{1}(X, L_{1} ⊗ L_{2}^{∗}) .

Moreover dimH^{1}(M_{ξ}, W_{ξ}(L_{1}) ⊗ W_{ξ}(L_{2})^{∗}) = g − 1 .

3. The Hecke transformation

In this section we will use Hecke transformations to compute the cohomology groups H^{i}(X × M_{ξ}, U_{ξ} ⊗ p_{1}^{∗}L_{1} ⊗ π^{∗}U_{x}^{∗}) for any x ∈ X and prove Proposition 2.4. The details of the Hecke transformation and its properties can be found in [13, 14]. We will briefly describe it and note those properties that will be needed here.

Fix a point x ∈ X. Let ℙ(U_{x}) denote the projective bundle over M_{ξ} consisting of lines in U_{x}. If f denotes the natural projection of ℙ(U_{x}) to M_{ξ} and O_{ℙ(U_{x})}(−1) the tautological line bundle then

f_{∗}O_{ℙ(U_{x})}(1) ≅ U_{x}^{∗} ,

and R^{j}f_{∗}O_{ℙ(U_{x})}(1) = 0 for all j > 0. From the commutative diagram

X × ℙ(U_{x}) ⟶^{Id_{X}×f} X × M_{ξ}
q↓  ↓π
ℙ(U_{x})  ⟶^{f}  M_{ξ}

(where q is the natural projection) and the base change theorem, we deduce that

H^{i}(X × M_{ξ}, U_{ξ} ⊗ p_{1}^{∗}L_{1} ⊗ π^{∗}U_{x}^{∗}) ≅ H^{i}(X × ℙ(U_{x}), (Id_{X} × f)^{∗}(U_{ξ} ⊗ p_{1}^{∗}L_{1}) ⊗ q^{∗}O_{ℙ(U_{x})}(1))

(6)

for all i.

Moreover, since q_{∗}(Id_{X} × f)^{∗}(U_{ξ} ⊗ p_{1}^{∗}L_{1}) ≅ f^{∗}W_{ξ}(L_{1}), there is a canonical isomorphism

H^{i}(X×ℙ(U_{x}), (Id_{X} ×f)^{∗}(U_{ξ}⊗p_{1}^{∗}L_{1})⊗q^{∗}O_{ℙ(U_{x})}(1)) ≅ H^{i}(ℙ(U_{x}), f^{∗}W_{ξ}(L_{1})⊗O_{ℙ(U_{x})}(1))

(7)

for every i ≥ 0.

To compute the cohomology groups H^{i}(ℙ(U_{x}), f^{∗}W_{ξ}(L_{1}) ⊗ O_{ℙ(U_{x})}(1)) we use Hecke transformations.

A point in ℙ(U_{x}) represents a vector bundle E and a line l in the fiber E_{x} at x, or equivalently a non-trivial exact sequence

0 ⟶ E ⟶ F ⟶ ℂ_{x} ⟶ 0  (8)

DEFORMATIONS OF THE PICARD BUNDLE 7

determined up to a scalar multiple; here ℂ_{x} denotes the torsion sheaf supported at x with stalk ℂ. The sequences (8) fit together to form a universal sequence

0 ⟶ (Id_{X} × f)^{∗}U_{ξ} ⊗ q^{∗}O_{ℙ(U_{x})}(1) ⟶ F ⟶ p_{1}^{∗}ℂ_{x} ⟶ 0  (9)

on X × ℙ(U_{x}). If η denotes the line bundle ξ ⊗ O_{X}(x) over X and M_{η} the moduli space of stable bundles M_{η}(n, d + 1) then from (9) we get a rational map

γ : ℙ(U_{x}) − − → M_{η}

which sends any pair (E, l) to F. This map is not everywhere defined since the bundle F in (8) need not be stable.

Our next object is to find a Zariski open subset Z of M_{η}, over which γ is defined and is a projective fibration, such that the complement of Z in M_{η} has codimension at least 4. The construction and calculations are similar to those of [13, Proposition 6.8], but our results do not seem to follow directly from that proposition.

As in [13, §8] or [14, §5], we define a bundle F to be (0, 1)-stable if, for every proper subbundle G of F,

\frac{deg G}{rkG} < \frac{deg F − 1}{rkF}.

Clearly every (0, 1)-stable bundle is stable. We denote by Z the subset of M_{η} consisting of (0, 1)-stable bundles.

Lemma 3.1. (i) Z is a Zariski-open subset of M_{η} whose complement has codimension at least 4.

(ii) γ is a projective fibration over Z and γ^{−1}(Z) is a Zariski-open subset of ℙ(U_{x}) whose complement has codimension at least 4.

Proof: (i) The fact that Z is Zariski-open is standard (see [14, Proposition 5.3]). The bundle F ∈ M_{η} of rank n and degree d + 1 fails to be (0, 1)-stable if and only

if it has a subbundle G of rank r and degree e such that ne ≥ r((d + 1) − 1), i.e.

rd ≤ ne.  (10)

By considering the extensions

0 ⟶ G ⟶ F ⟶ H ⟶ 0,

we can estimate the codimension of M_{η} − Z and show that it is at least

δ = r(n − r)(g − 1) + (ne − r(d + 1))  (11)

(compare the proof of [14, Proposition 5.4]). Note that, since (n, d) = 1, (10) implies that rd ≤ ne − 1. Given that g ≥ 3, we see that δ < 4 only if g = 3, n = 2, 3 or g = 4, n = 2. These are exactly the cases that were excluded in the introduction.

8 I. BISWAS, L. BRAMBILA-PAZ, AND P. E. NEWSTEAD

(ii) γ^{−1}(Z) consists of the pairs (E, l) for which the bundle F in (8) is (0, 1)-stable. As in (i), this is a Zariski-open subset. It follows at once from (10) that, if F is (0, 1)- stable, then E is stable. So, if F ∈ Z, it follows from (8) that γ^{−1}(F) can be identified with the projective space ℙ(F_{x}^{∗}). Using the universal projective bundle on X × M_{η}, we see that γ^{−1}(Z) is a projective fibration over Z (not necessarily locally trivial).

Suppose now that (E, l) belongs to the complement of γ^{−1}(Z) in ℙ(U_{x}). This means that the bundle F in (8) is not (0, 1)-stable and therefore possesses a subbundle G satisfying (10). If G ⊂ E, this contradicts the stability of E. So there exists an exact sequence

0 ⟶ G′ ⟶ G ⟶ ℂ_{x} ⟶ 0

with G′ a subbundle of E of rank r and degree e − 1. Moreover, since G is a subbundle of F, G′_{x} must contain the line l. For fixed r, e, these conditions determine a subvariety of ℙ(U_{x}) of dimension at most

(r^{2}(g −1)+1)+((n−r)^{2}(g −1)+1)−g +(r −1)+((g −1)r(n−r)+(rd−n(e−1))−1).

Since dim ℙ(U_{x}) = n^{2}(g − 1) − g + n, a simple calculation shows that the codimension is at least the number δ given by (11). As in (i), this gives the required result. □

By Lemma 3.1(ii) and a suitable Hartogs-type theorem (see [7, Theorem 3.8 and Proposition 1.11]) we have an isomorphism

H^{i}(ℙ(U_{x}), f^{∗}W_{ξ}(L_{1}) ⊗ O_{ℙ(U_{x})}(1)) ≅ H^{i}(γ^{−1}(Z), f^{∗}W_{ξ}(L_{1}) ⊗ O_{ℙ(U_{x})}(1)|_{γ^{−1}(Z)}) (12)

for i ≤ 2.

Now let F ∈ Z. As in the proof of Lemma 3.1, we identify γ^{−1}(F) with ℙ(F_{x}^{∗}) and denote it by ℙ. On X × ℙ there is a universal exact sequence

0 ⟶ E ⟶ p_{X}^{∗}F ⟶ p_{ℙ}O_{ℙ}(1) ⊗ p_{X}^{∗}ℂ_{x} ⟶ 0  (13)

where p_{X}, p_{ℙ} are the canonical projections. The restriction of (13) to any point of ℙ is isomorphic to the corresponding sequence (8).

Proposition 3.2. Let F be defined by the universal sequence (9). Then

F|_{X×ℙ} ≅ p_{X}^{∗}F ⊗ p_{ℙ}^{∗}O_{ℙ}(−1).

Proof: Restricting (9) to X × ℙ gives

0 ⟶ (Id_{X} × f)^{∗}U_{ξ} ⊗ q^{∗}O_{ℙ(U_{x})}(1)|_{X×ℙ} ⟶ F|_{X×ℙ} ⟶ p_{X}^{∗}ℂ_{x} ⟶ 0.

This must coincide with the universal sequence (13) up to tensoring by some line bundle lifted from ℙ. The result follows. □

Next we tensor (9) by p_{1}^{∗}L_{1}, restrict it to X × γ^{−1}(Z) and take the direct image on γ^{−1}(Z). This gives

0 ⟶ f^{∗}W_{ξ}(L_{1}) ⊗ O_{ℙ(U_{x})}(1)|_{γ^{−1}(Z)} ⟶ q_{∗}(F ⊗ p_{1}^{∗}L_{1})|_{γ^{−1}(Z)} ⟶ O_{γ^{−1}(Z)} ⟶ 0. (14)

Proposition 3.3. R_{γ∗}^{i}(q_{∗}(F ⊗ p_{1}^{∗}L_{1})|_{γ^{−1}(Z)}) = 0 for all i.

Proof: It is sufficient to show that q_{∗}(F ⊗p_{1}^{∗}L_{1})|_{ℙ} has trivial cohomology. By Proposition 3.2

q_{∗}(F ⊗ p_{1}^{∗}L_{1})|_{ℙ} ≅ p_{ℙ∗}(p_{X}^{∗}F ⊗ p_{X}^{∗}L_{1} ⊗ p_{ℙ}^{∗}O_{ℙ}(−1))
≅ H^{0}(F ⊗ L_{1}) ⊗ O_{ℙ}(−1)

and the result follows. □

Corollary 3.4. R_{γ_{∗}}^{i}((f^{∗}W_{ξ}(L_{1}) ⊗ O_{ℙ(U_{x})}(1))|_{γ^{−1}(Z)}) = 0 for i ≠ 1. Moreover

R_{γ_{∗}}^{1}((f^{∗}W_{ξ}(L_{1}) ⊗ O_{ℙ(U_{x})}(1))|_{γ^{−1}(Z)}) ≅ O.

Proof: This follows at once from (14) and Proposition 3.3. □ Now we are in a position to compute the cohomology of H^{i}(X×M_{ξ}, U_{ξ}⊗p_{1}^{∗}L_{1}⊗π^{∗}U_{x}^{∗})

for i = 1, 2.

Proposition 3.5. H^{2}(X × M_{ξ}, U_{ξ} ⊗ p_{1}^{∗}L_{1} ⊗ π^{∗}U_{x}^{∗}) = 0 for any x ∈ X.

Proof: The combination of (6), (7) and (12) yields

H^{2}(X × M_{ξ}, U_{ξ} ⊗ p_{1}^{∗}L_{1} ⊗ π^{∗}U_{x}^{∗}) ≅ H^{2}(γ^{−1}(Z), f^{∗}W_{ξ}(L_{1}) ⊗ O_{ℙ(U_{x})}(1)|_{γ^{−1}(Z)}) .

Using Corollary 3.4 and Lemma 3.1(i), the Leray spectral sequence for the map γ gives

\begin{aligned}H^{2}(γ^{−1}(Z), f^{∗}W_{ξ}(L_{1}) ⊗ O_{ℙ(U_{x})}(1)|_{γ^{−1}(Z)}) & ≅ & H^{1}(Z, O) \\ & ≅ & H^{1}(M_{η}, O). \end{aligned}

It is known that H^{1}(M_{η}, O) = 0 [6]. Therefore,

H^{2}(X × M_{ξ}, U_{ξ} ⊗ p_{1}^{∗}L_{1} ⊗ π^{∗}U_{x}^{∗}) = 0.
□

Proof of Proposition 2.4. Proposition 2.4 is an immediate consequence of Proposition 3.5. □

Proposition 3.6. For any point x ∈ X, dim H^{1}(X × M_{ξ}, U_{ξ} ⊗ p_{1}^{∗}L_{1} ⊗ π^{∗}U_{x}^{∗}) = 1.

Proof: As in the proof of Proposition 3.5 we conclude that

H^{1}(X × M_{ξ}, U_{ξ} ⊗ p_{1}^{∗}L_{1} ⊗ π^{∗}U_{x}^{∗}) ≅ H^{0}(M_{η}, O) .

Now M_{η} is just the non-singular part of the moduli space of semistable bundles of rank n and determinant η, and the latter space is normal. So dim H^{0}(M_{η}, O) = 1.□

4. Proof of Proposition 2.5

Let Δ be the diagonal divisor in X × X. Pull back the exact sequence

0 ⟶ O(−Δ) ⟶ O ⟶ O_{Δ} ⟶ 0

to X ×M_{ξ} ×X and tensor it with p_{12}^{∗}U_{ξ} ⊗p_{1}^{∗}L_{1} ⊗p_{23}^{∗}U_{ξ}^{∗}. Now the direct image sequence for the projection p_{3} gives the following exact sequence over X

⟶ R^{i}p_{3∗}(p_{12}^{∗}U_{ξ} ⊗ p_{1}^{∗}L_{1} ⊗ p_{23}^{∗}U_{ξ}^{∗}(−Δ)) ⟶ R^{i} ⟶ R^{i}p_{3∗}(p_{12}^{∗}U_{ξ} ⊗ p_{1}^{∗}L_{1} ⊗ p_{23}^{∗}U_{ξ}^{∗}|_{Δ×M_{ξ}})

⟶ R^{i+1}p_{3∗}(p_{12}^{∗}U_{ξ} ⊗ p_{1}^{∗}L_{1} ⊗ p_{23}^{∗}U_{ξ}^{∗}(−Δ)) ⟶ .  (15)

The following propositions will be used in computing the direct images R^{i}.

Proposition 4.1. For any L_{1} ∈ Pic^{0}(X), the direct images of

p_{12}^{∗}U_{ξ} ⊗ p_{1}^{∗}L_{1} ⊗ p_{23}^{∗}U_{ξ}^{∗}|_{Δ×M_{ξ}}

have the following description:

1. R^{0}p_{3∗}(p_{12}^{∗}U_{ξ} ⊗ p_{1}^{∗}L_{1} ⊗ p_{23}^{∗}U_{ξ}^{∗}|_{Δ×M_{ξ}}) ≅ L_{1}

2. R^{1}p_{3∗}(p_{12}^{∗}U_{ξ} ⊗ p_{1}^{∗}L_{1} ⊗ p_{23}^{∗}U_{ξ}^{∗}|_{Δ×M_{ξ}}) ≅ L_{1} ⊗ T X

3. R^{2}p_{3∗}(p_{12}^{∗}U_{ξ} ⊗ p_{1}^{∗}L_{1} ⊗ p_{23}^{∗}U_{ξ}^{∗}|_{Δ×M_{ξ}}) = 0

where T X is the tangent bundle of X.

Proof: Identifying Δ with X we have

R^{i}p_{3∗}(p_{12}^{∗}U_{ξ} ⊗ p_{1}^{∗}L_{1} ⊗ p_{23}^{∗}U_{ξ}^{∗}|_{Δ×M_{ξ}}) ≅ R^{i}p_{1∗}(U_{ξ} ⊗ p_{1}^{∗}L_{1} ⊗ U_{ξ}^{∗})

The proposition follows from a result of Narasimhan and Ramanan [13, Theorem 2] that says

H^{i}(M_{ξ}, U_{x} ⊗ U_{x}^{∗}) ≅ \{ \begin{matrix}ℂ & if i = 0, 1 \\ 0 & if i = 2 \end{matrix}  (16)

where U_{x} is the restriction of U_{ξ} to x × M_{ξ}. For i = 0 the isomorphism is given by the obvious inclusion of O in U_{x} ⊗ U_{x}^{∗} and therefore globalises to give R^{0}p_{1∗}(U_{ξ} ⊗ U_{ξ}^{∗}) ≅ O. Similarly for i = 1 the isomorphism is given by the infinitesimal deformation map of U_{ξ} regarded as a family of bundles over M_{ξ} parametrised by X; this globalises to R^{1}p_{1∗}(U_{ξ} ⊗ U_{ξ}^{∗}) ≅ T X. □

Propositions 4.1, 2.3 and the exact sequence (15) together give the following exact sequence of direct images

0 ⟶ L_{1} ⟶ R^{1}p_{3∗}(p_{12}^{∗}U_{ξ} ⊗ p_{1}^{∗}L_{1} ⊗ p_{23}^{∗}U_{ξ}^{∗}(−Δ)) ⟶ R^{1} ⟶^{α} L_{1} ⊗ T X

⟶ R^{2}p_{3∗}(p_{12}^{∗}U_{ξ} ⊗ p_{1}^{∗}L_{1} ⊗ p_{23}^{∗}U_{ξ}^{∗}(−Δ)) ⟶ .  (17)

For any x ∈ X we have the cohomology exact sequence

⟶ H^{i}(X × M_{ξ}, U_{ξ} ⊗ p_{1}^{∗}L_{1} ⊗ p_{2}^{∗}U_{x}^{∗}(−x)) ⟶ H^{i}(X × M_{ξ}, U_{ξ} ⊗ p_{1}^{∗}L_{1} ⊗ p_{2}^{∗}U_{x}^{∗})

DEFORMATIONS OF THE PICARD BUNDLE 11

⟶ H^{i}(M_{ξ}, U_{x} ⊗ U_{x}^{∗}) ⊗ (L_{1})_{x} ⟶ H^{i+1}(X × M_{ξ}, U_{ξ} ⊗ p_{1}^{∗}L_{1} ⊗ p_{2}^{∗}U_{x}^{∗}(−x)) , (18)

where (L_{1})_{x} is the fiber of L_{1} at x.

By (4), p_{1∗}(U_{ξ} ⊗ π^{∗}U_{x}^{∗}) = 0. So the Leray spectral sequence for p_{1} gives

H^{i}(X × M_{ξ}, U_{ξ} ⊗ p_{1}^{∗}L_{1} ⊗ π^{∗}U_{x}^{∗}) ≅ H^{i−1}(X, R^{1}p_{1∗}(U_{ξ} ⊗ p_{1}^{∗}L_{1} ⊗ π^{∗}U_{x}^{∗})) for i = 0, 1, and also

H^{i}(X × M_{ξ}, U_{ξ} ⊗ p_{1}^{∗}L_{1} ⊗ π^{∗}U_{x}^{∗}(−x)) ≅ H^{i−1}(X, R^{1}p_{1∗}(U_{ξ} ⊗ p_{1}^{∗}L_{1} ⊗ π^{∗}U_{x}^{∗})(−x)) for i = 0, 1. Since U_{x} is simple [13, Theorem 2], (18) gives the exact sequence

0 ⟶ (L_{1})_{x} ⟶ H^{0}(X, R^{1}p_{1∗}(U_{ξ} ⊗ p_{1}^{∗}L_{1} ⊗ π^{∗}U_{x}^{∗})(−x))

⟶^{β} H^{0}(X, R^{1}p_{1∗}(U_{ξ} ⊗ p_{1}^{∗}L_{1} ⊗ π^{∗}U_{x}^{∗})) ⟶ .

This implies that R^{1}p_{1∗}(U_{ξ} ⊗p_{1}^{∗}L_{1} ⊗π^{∗}U_{x}^{∗}) has torsion at x. Now from (16) we conclude that it is a torsion sheaf with x in the support. Therefore,

H^{1}(X, R^{1}p_{1∗}(U_{ξ} ⊗ p_{1}^{∗}L_{1} ⊗ π^{∗}U_{x}^{∗})(−x)) ≅ H^{1}(X, R^{1}p_{1∗}(U_{ξ} ⊗ p_{1}^{∗}L_{1} ⊗ π^{∗}U_{x}^{∗})) = 0 . Since R^{1}p_{1∗}(U_{ξ} ⊗ π^{∗}U_{x}^{∗}) is a torsion sheaf and p_{1∗}(U_{ξ} ⊗ π^{∗}U_{x}^{∗}) = 0 (noted earlier),

using the Leray spectral sequence for p_{1} yields

H^{2}(X × M_{ξ}, U_{ξ} ⊗ p_{1}^{∗}L_{1} ⊗ π^{∗}U_{x}^{∗}) ≅ H^{0}(X, R^{2}p_{1∗}(U_{ξ} ⊗ p_{1}^{∗}L_{1} ⊗ π^{∗}U_{x}^{∗})) . But Proposition 3.5 says that H^{2}(X × M_{ξ}, U_{ξ} ⊗ p_{1}^{∗}L_{1} ⊗ π^{∗}U_{x}^{∗}) = 0.

Similarly, we have

H^{2}(X × M_{ξ}, U_{ξ} ⊗ p_{1}^{∗}L_{1} ⊗ π^{∗}U_{x}^{∗}(−x)) ≅ H^{0}(X, R^{2}p_{1∗}(U_{ξ} ⊗ p_{1}^{∗}L_{1} ⊗ π^{∗}U_{x}^{∗})(−x)) and from (16) it follows that R^{2}p_{1∗}(U_{ξ} ⊗ p_{1}^{∗}L_{1} ⊗ π^{∗}U_{x}^{∗}) is a torsion sheaf and the point x ∈ X is not in the support. Consequently,

H^{0}(X, R^{2}p_{1∗}(U_{ξ} ⊗ p_{1}^{∗}L_{1} ⊗ π^{∗}U_{x}^{∗})(−x)) ≅ H^{0}(X, R^{2}p_{1∗}(U_{ξ} ⊗ p_{1}^{∗}L_{1} ⊗ π^{∗}U_{x}^{∗})) = 0 . Therefore,

H^{2}(X × M_{ξ}, U_{ξ} ⊗ p_{1}^{∗}L_{1} ⊗ π^{∗}U_{x}^{∗}(−x)) ≅ H^{2}(X × M_{ξ}, U_{ξ} ⊗ p_{1}^{∗}L_{1} ⊗ π^{∗}U_{x}^{∗}) = 0 . So, we have the following proposition:

Proposition 4.2. R^{2}p_{3∗}(p_{12}^{∗}U_{ξ} ⊗ p_{1}^{∗}L_{1} ⊗ p_{23}^{∗}U_{ξ}^{∗}(−Δ)) = 0.

Now Proposition 2.5 is easy to derive.

Proof of Proposition 2.5. By Proposition 3.6

H^{1}(X × M_{ξ}, U_{ξ} ⊗ p_{1}^{∗}L_{1} ⊗ π^{∗}U_{x}^{∗}) ≅ ℂ .

Therefore, R^{1} = R^{1}p_{3∗}(p_{12}^{∗}U_{ξ}⊗p_{1}^{∗}L_{1}⊗p_{23}^{∗}U_{ξ}^{∗}) is a line bundle. Moreover, by Proposition 4.2,

R^{2}p_{3∗}(p_{12}^{∗}U_{ξ} ⊗ p_{1}^{∗}L_{1} ⊗ p_{23}^{∗}U_{ξ}^{∗}(−Δ)) = 0.

12 I. BISWAS, L. BRAMBILA-PAZ, AND P. E. NEWSTEAD

Consequently, the map α : R^{1} ⟶ L_{1} ⊗ T X in the exact sequence (17) is surjective. Therefore α must be an isomorphism, and the proof is complete. □

Earlier Theorems 2.6 and 2.9 were proved assuming Propositions 2.4 and 2.5. Therefore Theorems 2.6 and 2.9 are now established.

5. Family of Deformations

Fix a point x ∈ X. Let L be the Poincaré line bundle over X × Pic^{0}(X) which is trivial on x × Pic^{0}(X). It was proven in [2] that the family Ũ_{ξ} := p_{12}^{∗}L ⊗ p_{13}^{∗}U_{ξ} over X ×Pic^{0}(X)×M_{ξ} is a complete family of deformations of U_{ξ} parametrized by Pic^{0}(X), where p_{ij} are the projections of X × Pic^{0}(X) × M_{ξ} on the ij-factor. In this section we will prove that the direct image

W̃ := p_{23∗}(Ũ_{ξ})

is a complete family of deformations of W_{ξ}(L) for any L ∈ Pic^{0}(X).

First note that W̃ is locally free. Indeed, R_{p_{23∗}}^{i}(Ũ_{ξ}) = 0 for i ≠ 0. Moreover, for each L ∈ Pic^{0}(X),

W̃|_{L×M_{ξ}} ≅ W_{ξ}(L).

In [4] it was proved that W_{ξ} is stable. From the proof of the Theorem in [4] we can see that for any line bundle L ∈ Pic^{0}(X), W_{ξ}(L) is also stable. Since Pic^{0}(M_{ξ}) is discrete and Pic^{0}(X) irreducible , ⋀^{top}W_{ξ}(L) is independent of the choice of L.

Denote by M_{X}^{0}(W_{ξ}) the connected component of the moduli space of stable sheaves on M_{ξ}, with the same Hilbert polynomial as W_{ξ}, containing W_{ξ}. As in [3] the determinant line bundle M on M_{X}^{0}(W_{ξ}) defines a polarisation on M_{X}^{0}(W_{ξ}).

Define the morphism

ϕ : Pic^{0}(X) → M_{X}^{0}(W_{ξ})

by L ↦ W_{ξ}(L).

Theorem 5.1. The morphism

ϕ : Pic^{0}(X) ⟶ M_{X}^{0}(W_{ξ})

is an isomorphism of polarised varieties.

Proof: By Corollary 2.7 ϕ is injective, so its image has dimension g. On the other hand, by Theorem 2.9, the Zariski tangent space of M_{X}^{0}(W_{ξ}) also has dimension g at every point of the image of ϕ. It follows that M_{X}^{0}(W_{ξ}) is smooth of dimension g at every point of the image of ϕ. Hence, by Zariski’s Main Theorem, ϕ is an isomorphism onto an open subset of M_{X}^{0}(W_{ξ}). Finally Pic^{0}(X) is complete and M_{X}^{0}(W_{ξ}) is connected, so ϕ is an isomorphism.

DEFORMATIONS OF THE PICARD BUNDLE 13

Let ζ be the polarisation on M_{X}^{0}(W_{ξ}) given by the determinant line bundle [3, Section 4]. Let Θ denote the principal polarisation on Pic^{0}(X) defined by a theta divisor. We wish to show that the isomorphism ϕ takes ζ to a nonzero constant scalar multiple (independent of the curve X) of Θ.

Take any family of pairs (X, ξ), where X is a connected non-singular projective curve of genus g and ξ is a line bundle on X of degree d > n(2g − 2), parametrized by a connected space T. Consider the corresponding family of moduli spaces M_{X}^{0}(W_{ξ}) (respectively, Picard groups Pic^{0}(X)) over T, where (X, ξ) runs over the family. Using the map ϕ an isomorphism between these two families is obtained. The polarisation ζ (respectively, Θ) defines a constant section of the second direct image over T of the constant sheaf ℤ over the family. It is known that for the general curve X of genus g, the Neron-Severi group of Pic^{0}(X) is ℤ. Therefore, for such a curve, ϕ takes ζ to a nonzero constant scalar multiple of Θ. Since T is connected, if T contains a curve with NS(Pic^{0}(X)) = ℤ, then ϕ takes ζ to the same nonzero constant scalar multiple of Θ for every curve in the family. Since the moduli space of smooth curves of genus g is connected, the proof is complete. □

Corollary 5.2. The family W̃ parametrised by Pic^{0}(X) is complete. Moreover, the infinitesimal deformation map is an isomorphism.

From the Torelli Theorem we have finally

Corollary 5.3. Let X and X′ be two non-singular algebraic curves of genus g ≥ 3 and let ξ (respectively ξ′) be a line bundle of degree d > n(2g − 2) on X (respectively X′). If M_{X}^{0}(W_{ξ}) ≅ M_{X′}^{0}(W_{ξ′}) as polarised varieties then X ≅ X′.

References

[1] V. Balaji and P. A. Vishwanath, Deformations of Picard sheaves and moduli of pairs, Duke Math. Journal. 76 (1994), 773–792.

[2] V. Balaji, L. Brambila-Paz and P. E. Newstead, Stability of the Poincaré bundle, Math. Nachr. 188 (1997), 5–15.

[3] I. Biswas and L. Brambila-Paz, Restrictions of the Poincaré bundle to Calabi-Yau hypersurface J. reine angew. Math. 513 (1999), 1–16.

[4] I. Biswas, L. Brambila-Paz, T. L. Gómez and P. E. Newstead, Stability of the Picard bundle, preprint.

[5] L. Brambila-Paz, E. Gómez-Gonzáles and F. Pioli, On Picard bundles over Prym varieties, preprint CIMAT, No I-99-21.

[6] J.-M. Drezet and M. S. Narasimhan, Groupe de Picard des variétés de modules de fibrés semistables sur les courbes algébriques, Inv. Math. 97 (1989), 53–94.

[7] R. Hartshorne, Local cohomology, A seminar given by A. Grothendieck, Harvard University, Fall, 1961. Lecture Notes in Mathematics, No. 41, Springer-Verlag, Berlin-New York 1967.

[8] G. Kempf, Toward the inversion of abelian integrals: I, Ann. of Math. 110 (1979), 243–273.

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[10] A. Mattuck, Picard bundles, Illinois J. Math. 5 (1961), 550–564.

[11] S. Mukai, Duality between D(X) and { \atop D(X)} with applications to Picard sheaves, Nagoya Math. Jour. 81 (1981), 153–175.

[12] S. Mukai, Symplectic structure of the moduli space of sheaves on an abelian or K3 surface, Inv. Math. 77 (1984), 101–116.

[13] M. S. Narasimhan and S. Ramanan, Deformations of the moduli space of vector bundles over an algebraic curve, Ann. of Math. 101 (1975), 391–417.

[14] M. S. Narasimhan and S. Ramanan, Geometry of Hecke cycles I, in C. P. Ramanujan - A Tribute, Studies in Math. No. 8, T.I.F.R., Bombay, 1978.

[15] S. Ramanan, The moduli spaces of vector bundles over an algebraic curve, Math. Ann. 200 (1973), 69–84.

[16] M. Thaddeus, Stable pairs, linear systems and the Verlinde formula, Invent. Math 117 (1994), 317–353.

[17] A. N. Tyurin, The geometry of moduli of vector bundles, Russ. Math. Surveys 29:6 (1974), 57–88.

School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India

  E-mail address: indranil@math.tifr.res.in

CIMAT, Apdo. Postal 402, C.P. 36240. Guanajuato, Gto, México

  E-mail address: lebp@fractal.cimat.mx

Department of Mathematical Sciences, The University of Liverpool, Liverpool, L69 3BX England

  E-mail address: newstead@liverpool.ac.uk