Correspondences between convex geometry and complex geometry

TL;DR

This paper explores deep analogies between convex geometry and complex algebraic geometry, establishing new correspondences and inequalities involving divisor classes and geometric structures.

Contribution

It introduces a Blaschke addition for divisor classes and derives novel geometric inequalities, bridging convex and complex geometry theories.

Findings

Established analogies between convex and complex geometry.

Defined a Blaschke addition for divisor classes.

Proved new geometric inequalities for divisor classes.

Abstract

We present several analogies between convex geometry and the theory of holomorphic line bundles on smooth projective varieties or K\"ahler manifolds. We study the relation between positive products and mixed volumes. We define and study a Blaschke addition for divisor classes and mixed divisor classes, and prove new geometric inequalities for divisor classes. We also reinterpret several classical convex geometry results in the context of algebraic geometry: the Alexandrov body construction is the convex geometry version of divisorial Zariski decomposition; Minkowski's existence theorem is the convex geometry version of the duality between the pseudo-effective cone of divisors and the movable cone of curves.