Taiwanese Journal of Mathematics

Boundedness of Log Canonical Surface Generalized Polarized Pairs

Stefano Filipazzi

Abstract

In this paper, we study the behavior of the sets of volumes of the form $\operatorname{vol}(X,K_X+B+M)$, where $(X,B)$ is a log canonical pair, and $M$ is a nef $\mathbb{R}$-divisor. After a first analysis of some general properties, we focus on the case when $M$ is $\mathbb{Q}$-Cartier with given Cartier index, and $B$ has coefficients in a given DCC set. First, we show that such sets of volumes satisfy the DCC property in the case of surfaces. Once this is established, we show that surface pairs with given volume and for which $K_X+B+M$ is ample form a log bounded family. These generalize results due to Alexeev [1].

Article information

Source
Taiwanese J. Math., Volume 22, Number 4 (2018), 813-850.

Dates
Revised: 8 October 2017
Accepted: 19 December 2017
First available in Project Euclid: 4 January 2018

https://projecteuclid.org/euclid.twjm/1515034830

Digital Object Identifier
doi:10.11650/tjm/171204

Mathematical Reviews number (MathSciNet)
MR3830822

Zentralblatt MATH identifier
06965398

Citation

Filipazzi, Stefano. Boundedness of Log Canonical Surface Generalized Polarized Pairs. Taiwanese J. Math. 22 (2018), no. 4, 813--850. doi:10.11650/tjm/171204. https://projecteuclid.org/euclid.twjm/1515034830

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