Some remarks on the volume of log varieties

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.