Positivity functions for curves on algebraic varieties

Lehmann, Brian; Xiao, Jian

doi:10.2140/ant.2019.13.1243

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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

Vol. 13, No. 6, 2019