Regularity on abelian varieties III: relationship with Generic Vanishing and applications

TL;DR

This paper explores the relationship between M-regular and GV-sheaves on abelian varieties, providing algebraic criteria, new properties, and applications to various geometric concepts like Seshadri constants and pluricanonical maps.

Contribution

It establishes a precise algebraic criterion distinguishing M-regular sheaves from GV-sheaves and applies this to derive new properties and results in the geometry of abelian varieties.

Findings

Characterization of M-regular sheaves among GV-sheaves

New properties of M-regular and GV-sheaves

Applications to Seshadri constants, Picard bundles, and pluricanonical maps

Abstract

We describe the relationship between the notions of $M$-regular sheaf and $GV$-sheaf in the case of abelian varieties. The former is a natural strengthening of the latter, and we provide an algebraic criterion characterizing it among the larger class. Based on this we deduce new basic properties of both $M$-regular and $GV$-sheaves. In the second part we give a number of applications of generation criteria for $M$-regular sheaves to the study of Seshadri constants, Picard bundles, pluricanonical maps on irregular varieties, and semihomogeneous vector bundles. This second part of the paper is based on our earlier preprint math.AG/0306103, with some improved statements and shortened arguments.