Some remarks on the volume of log varieties

TL;DR

This paper investigates the behavior of volumes of log varieties, showing that under certain conditions, these volumes form a DCC set and are discrete when boundary coefficients are finite, extending understanding of their accumulation points.

Contribution

It establishes that the set of volumes of log varieties with DCC boundary coefficients is itself DCC and closed under limits, and proves discreteness of volumes for finite boundary coefficient sets in epsilon-log canonical cases.

Findings

Volumes form a DCC set under specified conditions.

Volumes are discrete when boundary coefficients are finite.

Set of volumes is closed under limits and accumulation points.

Abstract

In this note, using methods introduced by Hacon, McKernan and Xu, we study the accumulation points of volumes of varieties of log general type. First, we show that, if the set of boundary coefficients $\Lambda$ is DCC, closed under limits and contains 1, then also the corresponding set of volumes is DCC and closed under limits. Then, we consider the case of $\epsilon$-log canonical varieties, for $0 < \epsilon < 1$. In this situation, we prove that, if $\Lambda$ is finite, then the corresponding set of volumes is discrete.