On generic vanishing for pluricanonical bundles

TL;DR

This paper investigates the structure of cohomology support loci for pluricanonical bundles on smooth projective varieties, revealing their torsion translate structure and providing a counterexample related to higher direct images and GV-sheaves.

Contribution

It proves that 0-th cohomology support loci of log pluricanonical bundles are finite unions of torsion translates of subtori and constructs a counterexample concerning higher direct images.

Findings

0-th cohomology support loci are torsion translate unions

Counterexample for higher direct images not being GV-sheaves

Insights into the structure of cohomology support loci

Abstract

We study cohomology support loci and higher direct images of (log) pluricanonical bundles of smooth projective varieties or log canonical pairs. We prove that the 0-th cohomology support loci of log pluricanonical bundles are finite unions of torsion translates of subtori. We also construct an example of morphism from a smooth projective variety to an abelian variety such that a higher direct image of a pluricanonical bundle to the abelian variety is not a GV-sheaf.