Differential Forms on Log Canonical Spaces

TL;DR

This paper proves that differential forms on log canonical varieties extend across singularities, develops a theory for dlt pairs, and applies these results to key problems like the Lipman-Zariski conjecture.

Contribution

It establishes extension theorems for differential forms on log canonical spaces and develops a differential forms theory for dlt pairs, extending known results to singular settings.

Findings

Differential forms extend regularly from the smooth locus to resolutions.

A theory of differential forms on dlt pairs is developed.

Positive results on the Lipman-Zariski conjecture for klt spaces.

Abstract

The present paper is concerned with differential forms on log canonical varieties. It is shown that any p-form defined on the smooth locus of a variety with canonical or klt singularities extends regularly to any resolution of singularities. In fact, a much more general theorem for log canonical pairs is established. The proof relies on vanishing theorems for log canonical varieties and on methods of the minimal model program. In addition, a theory of differential forms on dlt pairs is developed. It is shown that many of the fundamental theorems and techniques known for sheaves of logarithmic differentials on smooth varieties also hold in the dlt setting. Immediate applications include the existence of a pull-back map for reflexive differentials, generalisations of Bogomolov-Sommese type vanishing results, and a positive answer to the Lipman-Zariski conjecture for klt spaces.