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

1996/03/21 by Lawrence Ein, Robert Lazarsfeld, Ein, Lawrence +1 · 4 citations
Mathematics · #Algebraic Geometry and Number Theory #Advanced Algebra and Geometry #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.alg-geom/9603017

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.

Cited by

• Singularities of base loci on abelian varieties
• An effective version of a theorem of Kawamata on the Albanese map
• On the Euler characteristic of weakly ordinary varieties of maximal Albanese dimension
• Generic vanishing theory in positive characteristic

Related

• Irreducible Theta Divisors of PPAV's are Strongly F-regular
• Singularities of the theta divisor at points of order two
• On the bicanonical map of irregular varieties
• Geometry of theta divisors --- a survey
• Varieties with vanishing holomorphic Euler characteristic