Measures of finite energy in pluripotential theory: a synthetic approach

TL;DR

This paper develops a synthetic framework for global pluripotential theory, defining and analyzing measures of finite energy on complex and non-Archimedean spaces, with implications for energy functionals and their properties.

Contribution

It introduces a unified synthetic approach to measures of finite energy in pluripotential theory, extending to Kähler and Berkovich spaces, and studies properties of associated energy functionals.

Findings

Defined measures of finite energy in a synthetic setting

Introduced twisted and free energy functionals

Proved coercivity of energy functionals is an open condition

Abstract

We introduce a synthetic approach to global pluripotential theory, covering in particular the case of a compact K\"ahler manifold and that of a projective Berkovich space over a non-Archimedean field. We define and study the space of measures of finite energy, introduce twisted energy and free energy functionals thereon, and show that coercivity of these functionals is an open condition with respect to the polarization.