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 proper variety $X$ of maximal Albanese dimension satisfies $H^0(X,{\omega }_X)\ne 0$. If $\mathrm{Alb}(X)$ is in addition ordinary, then $S^0(X,{\omega }_X)\ne 0$.