Singularities of Theta Divisors, and the Birational Geometry of Irregular Varieties

TL;DR

This paper uses generic vanishing theorems to address key questions in the geometry of irregular varieties, including characterizations of theta divisors, properties of varieties of general type, and Albanese mappings.

Contribution

It proves a conjecture on singularities of theta divisors, verifies a conjecture of Kollár for certain subvarieties, and offers a new proof of Kawamata's theorem on Albanese mappings.

Findings

Characterization of theta divisors with singularities in codimension one

Verification of Kollár's conjecture for subvarieties of abelian varieties

Simplified proof of Kawamata's theorem on Albanese mappings

Abstract

The purpose of this paper is to show how the generic vanishing theorems of M. Green and the second author can be used to settle several questions and conjectures concerning the geometry of irregular complex projective varieties. First, we prove a conjecture of Arbarello and DeConcini characterizing principally polarized abelian varieties whose theta divisors are singular in codimension one. Secondly, we study the holomorphic Euler characteristics of varieties of general type having maximal Albanese dimension: we verify a conjecture of Kollar for subvarieties of abelian varieties, but show that it fails in general. Finally, we give a surprisingly simple new proof of a fundamental theorem of Kawamata concerning the Albanese mapping of projective varieties of Kodaira dimension zero.