Generic vanishing theory via mixed Hodge modules

TL;DR

This paper generalizes generic vanishing theory to broader classes of sheaves on irregular varieties using advanced Hodge module techniques, revealing new categorical structures on moduli spaces.

Contribution

It extends generic vanishing results to holomorphic forms and local systems via mixed Hodge modules, introducing new categories of perverse coherent sheaves on moduli spaces.

Findings

Dimension and linearity results are extended to new sheaf classes.

Identification of two natural categories of perverse coherent sheaves.

Application of mixed Hodge modules and Fourier-Mukai transform in this context.

Abstract

We extend the dimension and strong linearity results of generic vanishing theory to bundles of holomorphic forms and rank one local systems, and more generally to certain coherent sheaves of Hodge-theoretic origin associated to irregular varieties. Our main tools are Saito's mixed Hodge modules, the Fourier-Mukai transform for D-modules on abelian varieties introduced by Laumon and Rothstein, and Simpson's harmonic theory for flat bundles. In the process, we discover two natural categories of perverse coherent sheaves, one on the Picard variety, and the other on the moduli space of Higgs line bundles or that of line bundles with flat connection.