Generic vanishing, gaussian maps, and Fourier-Mukai transform

TL;DR

This paper establishes a vanishing criterion for higher direct images of line bundles on Cohen-Macaulay varieties using Gaussian maps, and applies it via Fourier-Mukai transform to prove a stronger algebraic version of the Generic Vanishing Theorem.

Contribution

It introduces a new vanishing criterion based on global co-gaussian maps and applies it to prove an algebraic version of the Generic Vanishing Theorem using Fourier-Mukai transform.

Findings

Proved a vanishing criterion involving global co-gaussian maps.

Established an algebraic version of Green-Lazarsfeld's Generic Vanishing Theorem.

Extended results to higher direct images of Poincaré line bundles.

Abstract

In the first part of this paper we prove a vanishing criterion for higher direct images of projective families of line bundles on a Cohen-Macaulay variety X. The result involves certain first-order deformations of certain curves on X, and makes essential use of the notion of global co-gaussian maps, a generalization of Wahl's gaussian maps. In the second part we apply the criterion above, combined with Fourier-Mukai transform on abelian varieties, to prove an algebraic version of Green-Lazarsfeld's Generic Vanishing Theorem. In fact we prove a stronger result concerning higher direct images of Poincar\'e line bundles, which -- in the compact K\"ahler setting -- was conjectured by Green and Lazarsfeld and was recently proved, by completely different methods, by Hacon (math.AG/0308198)