Generic vanishing theory in positive characteristic

TL;DR

This paper advances the understanding of generic vanishing theory in positive characteristic by simplifying key theorems and deriving new results about the cohomology of varieties with maximal Albanese dimension.

Contribution

It simplifies and strengthens the main theorems of positive characteristic generic vanishing theory, leading to new cohomological results for varieties.

Findings

Normal varieties of maximal Albanese dimension have non-zero global sections of the canonical bundle.

If the Albanese variety is ordinary, then the space of sections of the canonical bundle is non-zero.

The paper provides a streamlined proof of fundamental theorems in positive characteristic generic vanishing theory.

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 $H^0(X, \omega_X) \neq 0$ and that if $\mathrm{Alb}(X)$ is ordinary, then $S^0(X, \omega_X) \neq 0$.