# Differential forms on log canonical spaces in positive characteristic

## TL;DR

This paper proves extension properties of differential forms on log canonical surfaces in positive characteristic, establishing new residue and restriction sequences, and providing a new proof of the characteristic zero theorem.

## Contribution

It extends the understanding of differential forms on log canonical spaces in positive characteristic, including new residue and restriction sequences for tamely dlt pairs.

## Key findings

- Differential forms extend with logarithmic poles on log canonical surfaces in characteristic p ≥ 7.
- Residue and restriction sequences are established for tamely dlt pairs.
- Results are sharp in surfaces and demonstrate failure in higher dimensions.

## Abstract

Given a logarithmic $1$-form on the snc locus of a log canonical surface pair $(X, D)$ over a perfect field of characteristic $p \ge 7$, we show that it extends with at worst logarithmic poles to any resolution of singularities. We also prove the analogous statement for regular differential forms, under an additional tameness hypothesis. In addition, residue and restriction sequences for tamely dlt pairs are established. We give a number of examples showing that our results are sharp in the surface case, and that they fail in higher dimensions. On the other hand, our techniques yield a new proof of the characteristic zero Logarithmic Extension Theorem in any dimension.