Generic vanishing theory in positive characteristic

Abstract

We simplify and improve the main fundamental theorems of positive characteristic generic vanishing theory. As a quick corollary of the theory, we prove that a normal variety $X$ of maximal Albanese dimension satisfies $H^0(X,\ddot{I}{‰}_X)\ne 0$ and that if $\mathrm{Alb}(X)$ is ordinary, then $S^0(X,\ddot{I}{‰}_X)\ne 0$.

Final version. To appear in L'Enseignement Mathématique