Mumford vanishing for threefolds in positive characteristic

TL;DR

This paper proves vanishing theorems for certain threefolds in positive characteristic, extending Mumford vanishing results under specific conditions on the canonical divisor.

Contribution

It establishes new vanishing results for projective klt threefolds in characteristic p>5, broadening Mumford vanishing applicability in positive characteristic.

Findings

H^1(X, -L)=0 when K_X is not big and L is big

H^1(X, -L)=0 when -K_X is nef and L has numerical dimension two

Results hold for characteristic p>5

Abstract

Let $X$ be a projective klt threefold in characteristic $p>5$ and let $L$ be a nef Cartier divisor on $X$. We show that $H^1(X, -L)=0$ for the following two cases: (1) $K_X$ is not big and $L$ is big; (2) $-K_X$ is nef and $L$ is of numerical dimension two.