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.

arXiv commentFixed some typos (including an inequality in the statement of Theorem 1.3). arXiv admin note: text overlap with arXiv:1706.09913

References in corpus (3)

• The minimal volume of log surfaces of general type with positive geometric genus
• Open surfaces of small volume
• Boundedness of varieties of log general type

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• On infinite accumulation points of log canonical volumes
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