  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.
