GV-sheaves, Fourier-Mukai transform, and Generic Vanishing

TL;DR

This paper develops a formal criterion for Generic Vanishing using homological methods within Fourier-Mukai frameworks, leading to new vanishing theorems for line bundles and applications to higher rank moduli spaces.

Contribution

It introduces a general homological criterion for Generic Vanishing applicable to arbitrary Fourier-Mukai correspondences, extending classical results to broader contexts.

Findings

Kodaira-type generic vanishing theorem for $K_X + L$ with nef $L$

A generic Nadel-type vanishing theorem for multiplier ideal sheaves

Applications to higher rank moduli spaces on curves and Calabi-Yau threefolds

Abstract

We use homological methods to establish a formal criterion for Generic Vanishing, in the sense originated by Green and Lazarsfeld and pursued further by Hacon and the first author, but in the context of an arbitrary Fourier-Mukai correspondence. For smooth projective varieties we apply this to deduce a Kodaira-type generic vanishing theorem for adjoint bundles of the form $K_X + L$ with $L$ a nef line bundle, and in fact a more general generic Nadel-type vanishing theorem for multiplier ideal sheaves. Still in the context of the Picard variety, the same method generates various other generic vanishing results, by reduction to standard vanishing theorems. We further use the formal criterion in order to address examples related to generic vanishing on higher rank moduli spaces (on curves and on some threefold Calabi-Yau fiber spaces).