Extremal potentials and equilibrium measures associated to collections of K\"ahler classes

TL;DR

This paper introduces new notions of extremal potentials and equilibrium measures for collections of K"ahler classes on complex manifolds, establishing regularity results and a variational framework, with applications to Fekete points.

Contribution

It develops a generalized framework for extremal potentials and equilibrium measures for multiple K"ahler classes, extending classical concepts and providing foundational regularity results.

Findings

Defined new extremal potentials and equilibrium measures for collections of K"ahler classes.

Proved regularity results for these new notions.

Established a variational framework for further analysis.

Abstract

Given a collection of K\"ahler forms and a continuous weight on a compact complex manifold we show that it is possible to define natural new notions of extremal potentials and equilibrium measures which coincide with classical notions when the collection is a singleton. We prove two regularity results and set up a variational framework. Applications to Fekete points are treated elsewhere.