Potentially Du Bois spaces

Patrick Graf and Sándor J. Kovács

Journal of Singularities
volume 8 (2014), 117-134

Received: 18 February 2014. Received in revised form: 14 October 2014.

DOI: 10.5427/jsing.2014.8i

Add a reference to this article to your citeulike library.

Abstract:

We investigate properties of potentially Du Bois singularities, that is, those that occur on the underlying space of a Du Bois pair. We show that a normal variety X with potentially Du Bois singularities and Cartier canonical divisor K_X is necessarily log canonical, and hence Du Bois. As an immediate corollary, we obtain the Lipman-Zariski conjecture for varieties with potentially Du Bois singularities. We also show that for a normal surface singularity, the notions of Du Bois and potentially Du Bois singularities coincide. In contrast, we give an example showing that in dimension at least three, a normal potentially Du Bois singularity x in X need not be Du Bois even if one assumes the canonical divisor K_X to be Q-Cartier.

Keywords:

Singularities of the minimal model program, Du Bois pairs, differential forms, Lipman-Zariski conjecture

Mathematical Subject Classification:

14B05, 32S05

Author(s) information:

Patrick Graf	Sándor Kovács
Lehrstuhl für Mathematik I	Department of Mathematics, Box 354350
Universität Bayreuth	University of Washington
95440 Bayreuth, Germany	Seattle, WA 98195-4350, USA
email: patrick.graf@uni-bayreuth.de	email: skovacs@uw.edu