  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.
