A class of singularity of arbitrary pairs and log canonicalizations

TL;DR

This paper introduces pseudo-lc singularities, a broader class of singularities for pairs of varieties and divisors, extending the concept of log canonical singularities, and proves their key properties including the existence of small lc modifications.

Contribution

It defines pseudo-lc singularities as a new, more general class of singularities for pairs, extending the existing notions of lc and log canonical singularities, and establishes foundational properties.

Findings

Pseudo-lc pairs are strictly more general than lc or log canonical pairs.

Pseudo-lc pairs admit small lc modifications.

A criterion for log canonicity of pseudo-lc pairs is provided.

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-lc in this paper. This is a generalization of the usual lc singularity of pairs and log canonical singularity of normal varieties introduced by de Fernex and Hacon. By giving examples of pseudo-lc pairs which are not lc or log canonical in the sense of de Fernex--Hacon's paper, we show that pseudo-lc singularity is a strictly extended notion of those singularities. We prove that pseudo-lc pairs admit a small lc modification. We also discuss a criterion of log canonicity.