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August, 2018 Boundedness of Log Canonical Surface Generalized Polarized Pairs
Stefano Filipazzi
Taiwanese J. Math. 22(4): 813-850 (August, 2018). DOI: 10.11650/tjm/171204


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].


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Stefano Filipazzi. "Boundedness of Log Canonical Surface Generalized Polarized Pairs." Taiwanese J. Math. 22 (4) 813 - 850, August, 2018.


Received: 29 August 2017; Revised: 8 October 2017; Accepted: 19 December 2017; Published: August, 2018
First available in Project Euclid: 4 January 2018

zbMATH: 06965398
MathSciNet: MR3830822
Digital Object Identifier: 10.11650/tjm/171204

Primary: 14E30 , 14J10 , 14J29

Keywords: boundedness , DCC , generalized polarized pair , Volume

Rights: Copyright © 2018 The Mathematical Society of the Republic of China


Vol.22 • No. 4 • August, 2018
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