Vol. 13, No. 6, 2019

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Positivity functions for curves on algebraic varieties

Brian Lehmann and Jian Xiao

Vol. 13 (2019), No. 6, 1243–1279
DOI: 10.2140/ant.2019.13.1243
Abstract

This is the second part of our work on Zariski decomposition structures, where we compare two different volume type functions for curve classes. The first function is the polar transform of the volume for divisor classes. The second function captures the asymptotic geometry of curves analogously to the volume function for divisors. We prove that the two functions coincide, generalizing Zariski’s classical result for surfaces to all varieties. Our result confirms the log concavity conjecture of the first named author for weighted mobility of curve classes in an unexpected way, via Legendre–Fenchel type transforms. During the course of the proof, we obtain a refined structure theorem for the movable cone of curves.

Keywords
algebraic varieties, positivity of curves, mobility of cycles, volume-type function, Zariski decomposition
Mathematical Subject Classification 2010
Primary: 14C25
Secondary: 14C20, 32J25
Milestones
Received: 2 March 2018
Revised: 20 February 2019
Accepted: 8 April 2019
Published: 18 August 2019
Authors
Brian Lehmann
Department of Mathematics
Boston College
Chestnut Hill, MA
United States
Jian Xiao
Department of Mathematical Sciences and Yau Mathematical Sciences Center
Tsinghua University
Beijing
China