Local vanishing for F-pure threefolds

TL;DR

This paper proves vanishing theorems for F-pure threefolds in characteristic p>5, leading to new results in the extension of differential forms and singularity theory.

Contribution

It establishes Grauert--Riemenschneider and Steenbrink vanishing theorems for F-pure threefolds, advancing the understanding of singularities in positive characteristic.

Findings

Proves Grauert--Riemenschneider vanishing for F-pure threefolds

Establishes Steenbrink vanishing for sharply F-pure pairs in characteristic p>5

Derives logarithmic extension for one-forms in this setting

Abstract

We establish Grauert--Riemenschneider vanishing for $F$-pure threefolds over a perfect field $k$ of characteristic $p>5$. We apply this to prove Steenbrink vanishing for three-dimensional sharply $F$-pure pairs in characteristic $p>5$. As a consequence, we obtain the logarithmic extension for one-forms in this setting.