Asian Journal of Mathematics

Volume 24 (2020)

Number 2

A class of singularity of arbitrary pairs and log canonicalizations

Pages: 207 – 238



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)


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.


singularity of pairs, log canonicalization, log canonical criterion

2010 Mathematics Subject Classification

14E30, 14J17

Received 29 September 2018

Accepted 17 May 2019

Published 8 September 2020