Asian Journal of Mathematics

Volume 24 (2020)

Number 2

A class of singularity of arbitrary pairs and log canonicalizations

Pages: 207 – 238

DOI: https://dx.doi.org/10.4310/AJM.2020.v24.n2.a2

Author

Kenta Hashizume (Graduate School of Mathematical Sciences, University of Tokyo, Meguro-ku, Tokyo, Japan; and Department of Mathematics, Graduate School of Science, Kyoto University, Kyoto, Japan)

Abstract

We define a class of singularity on arbitrary pairs of a normal variety and an effective $\mathbb{R}$‑divisor on it, which we call pseudo‑$\operatorname{lc}$ in this paper. This is a generalization of the usual $\operatorname{lc}$ singularity of pairs and log canonical singularity of normal varieties introduced by de Fernex and Hacon. By giving examples of pseudo‑$\operatorname{lc}$ pairs which are not $\operatorname{lc}$ or log canonical in the sense of de Fernex–Hacon’s paper, we show that pseudo‑$\operatorname{lc}$ singularity is a strictly extended notion of those singularities. We prove that pseudo‑$\operatorname{lc}$ pairs admit a small $\operatorname{lc}$ modification. We also discuss a criterion of log canonicity.

Keywords

singularity of pairs, log canonicalization, log canonical criterion

2010 Mathematics Subject Classification

14E30, 14J17

Full Text (PDF format)

Received 29 September 2018

Accepted 17 May 2019

Published 8 September 2020