Cohomological support loci and Pluricanonical systems on irregular varieties

TL;DR

This paper investigates the relationship between cohomological support loci and pluricanonical systems on irregular varieties, establishing conditions under which these loci generate the Picard variety and imply birationality of pluricanonical maps.

Contribution

It introduces new conditions linking cohomological support loci to the birationality of pluricanonical maps on irregular varieties.

Findings

Cohomological support loci can generate the Picard variety under certain conditions.

These conditions can be used to prove birationality of pluricanonical maps.

The results connect Hodge theory with birational geometry.

Abstract

For an irregular variety $X$ of general type, we show that if a general fiber $F$ of the Albanese morphism of $X$ satisfies certain Hodge theoretic condition, the $0$-th cohomological support loci of $K_X$ generates the Picard variety of $X$ . We then show that the condition that the $0$-th cohomological support loci of $K_X$ generates the Picard variety of $X$ can often be applied to prove the birationality of certain pluricanonical maps of $X$.