Irreducible Theta Divisors of PPAV's are Strongly F-regular

TL;DR

This paper proves that irreducible Theta divisors of PPAVs are strongly F-regular in positive characteristic, extending classical results and utilizing recent generic vanishing theorems to analyze singularities and birational geometry.

Contribution

It establishes the strong F-regularity of irreducible Theta divisors of PPAVs in positive characteristic, extending prior characteristic-zero results using new techniques.

Findings

Irreducible Theta divisors of PPAVs are strongly F-regular in positive characteristic.

Develops a positive characteristic analogue of a key result on Albanese images.

Extends classical results to fields of positive characteristic.

Abstract

We study the birational geometry of irregular varieties and the singularities of Theta divisors of PPAV's in positive characteristic by applying recent generic vanishing results of Hacon and Patakfalvi. In particular, we prove that irreducible Theta divisors of principally polarized abelian varieties are strongly F-regular, which extends an old result of Ein and Lazarsfeld to fields of positive characteristic. In order to prove this, we formulate a positive characteristic analogue of another result of Ein and Lazarsfeld, to the effect that the Albanese image of a smooth projective variety of maximal Albanese dimension with vanishing holomorphic Euler characteristic is fibered by abelian subvarieties.