Log canonical singularities are Du Bois

TL;DR

This paper proves that log canonical singularities are Du Bois, extending understanding of their properties and implications for moduli spaces in algebraic geometry.

Contribution

It establishes that all log canonical singularities are Du Bois, a significant step in understanding their structure and behavior in the Minimal Model Program.

Findings

Log canonical singularities are Du Bois.

Flatness of cohomology sheaves in families with Du Bois fibers.

Moduli space components parametrize Cohen-Macaulay or non-Cohen-Macaulay objects.

Abstract

A recurring difficulty in the Minimal Model Program is that while log terminal singularities are quite well behaved (for instance, they are rational), log canonical singularities are much more complicated; they need not even be Cohen-Macaulay. The aim of this paper is to prove that log canonical singularities are Du Bois. The concept of Du Bois singularities, introduced by Steenbrink, is a weakening of rationality. We also prove flatness of the cohomology sheaves of the relative dualizing complex of a projective family with Du Bois fibers. This implies that each connected component of the moduli space of stable log varieties parametrizes either only Cohen-Macaulay or only non-Cohen-Macaulay objects.