Higher direct images of the structure sheaf in positive characteristic

TL;DR

This paper proves vanishing theorems for higher direct images of structure sheaves in positive characteristic and establishes the birational invariance of the structure sheaf's cohomology for smooth projective varieties.

Contribution

It extends known characteristic zero results to positive characteristic, providing new vanishing theorems and invariance properties.

Findings

Vanishing of higher direct images in positive characteristic.

Cohomology of the structure sheaf is a birational invariant in positive characteristic.

Results generalize characteristic zero theorems to positive characteristic.

Abstract

We prove vanishing of the higher direct images of the structure (and the canonical) sheaf for a proper birational morphism with source a smooth variety and target the quotient of a smooth variety by a finite group of order prime to the characteristic of the ground field. We also show that for smooth projective varieties the cohomology of the structure sheaf is a birational invariant. These results are well-known in characteristic zero.