3 Generic Vanishing , Gaussian Maps , and Fourier - Mukai Transform

A quite basic fact about abelian varieties (and complex tori) is that the only line bundle in Pic 0 with non-trivial cohomology is the structure sheaf. Mumford, in his treatment of the dual abelian variety, made the far-reaching remark that this yields the following sheaf-theoretic formulation. Let A be an abelian variety (over an algebraically closed field) of dimension g, let us denote A = Pic 0 A and let P be a Poincaré line bundle on A × A. Then ([M] §13, see also and [K], Th. 3.15 for the complex-analityc setting) R i p A * P = 0 for i < g k(0) for i = g (0.1) where k(0) denotes the one-dimensional skyscraper sheaf at the identity point of A. It is worth to remark that the Fourier-Mukai equivalence between the derived categories of A and A ([Mu1]) is a direct consequence of (0.1). The theme of generic vanishing can be seen as a vast generalization of the above to varieties mapping to abelian varieties (or complex tori). Given a morphism a : X → A from a compact Kähler manifold to a complex torus (e.g. the Albanese map), works of Green-Lazarsfeld and Simpson ([GE1],[GE2],[S], see also [EL]) provide a fairly complete description of the loci V i a = {ξ ∈ A | h i (X, a * P ξ) > 0 } (here P ξ denotes the line bundle parametrised by the point ξ ∈ A ∨ via the choice of the Poincaré line bundle P). The main result of this paper is instead a generalization of (0.1), conjectured by Green and Lazars-feld themselves. The methods of proof are completely algebraic and very different from the Hodge-theoretic ones of Green-Lazarsfeld. As a byproduct we get an algebraic proof of Green-Lazarsfeld's Generic Vanishing Theorem working on any algebraically closed field as well, under the separability assumption. Theorem 1. (compare [GL2], Problem 6.2) On an algebraically closed field, let a : X → A be a separable morphism from an irreducible, Gorenstein variety X to an abelian variety A. Then R i p A * ((a, id A) * P) = 0 for i < dim a(X). Corollary 2. On an algebraically closed field, let a : X → A be a separable morphism from an irreducible, Gorenstein variety X to an abelian variety A. Then: (a) (compare [GL1]) codim A V i a ≥ max{0, …