Convexity and Zariski decomposition structure

TL;DR

This paper introduces a Zariski decomposition framework for curve classes using Legendre-Fenchel transforms, leading to new positivity results and geometric insights for varieties with special structures.

Contribution

It develops a novel Zariski decomposition for curve classes via Legendre-Fenchel transforms, expanding the theory of volume functions and positivity in algebraic geometry.

Findings

Defined a Zariski decomposition for curve classes

Established fundamental positivity results for curve classes

Provided geometric interpretations for special varieties

Abstract

This is the first part of our work on Zariski decomposition structures, where we study Zariski decompositions using Legendre-Fenchel type transforms. In this way we define a Zariski decomposition for curve classes. This decomposition enables us to develop the theory of the volume function for curves defined by the second named author, yielding some fundamental positivity results for curve classes. For varieties with special structures, the Zariski decomposition for curve classes admits an interesting geometric interpretation.