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Some remarks on the volume of log varieties

Part of: Surfaces and higher-dimensional varieties Birational geometry

Published online by Cambridge University Press:  18 December 2019

Stefano Filipazzi
Stefano Filipazzi*
Affiliation:
UCLA Mathematics Department, Box 951555, Los Angeles, CA90095-1555, USA (filipazzi@math.ucla.edu)
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Abstract

In this note, using methods introduced by Hacon et al. [‘Boundedness of varieties of log general type’, Proceedings of Symposia in Pure Mathematics, Volume 97 (American Mathematical Society, Providence, RI, 2018) 309–348], we study the accumulation points of volumes of varieties of log general type. First, we show that if the set of boundary coefficients Λ satisfies the descending chain condition (DCC), is closed under limits and contains 1, then the corresponding set of volumes satisfies the DCC and is closed under limits. Then, we consider the case of ε-log canonical varieties, for 0 < ε < 1. In this situation, we prove that if Λ is finite, then the corresponding set of volumes is discrete.

Keywords

log canonical volumeaccumulation pointsε-log canonical

MSC classification

Primary: 14E99: None of the above, but in this section
Secondary: 14J40: $n$-folds ($n>4$)
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 63 , Issue 2 , May 2020 , pp. 314 - 322
Copyright
Copyright © Edinburgh Mathematical Society 2019

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References

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