Fekete points and convergence towards equilibrium measures on complex manifolds

TL;DR

This paper establishes a general criterion for the convergence of Bergman measures to equilibrium measures on complex manifolds, confirming a key conjecture in pluripotential theory and advancing understanding of Fekete points.

Contribution

It introduces a new criterion based on growth properties and L-functionals for convergence, proving a conjecture about Fekete points and extending results to Bernstein-Markov measures.

Findings

Proves convergence of Bergman measures to equilibrium measures.

Confirms a well-known conjecture in pluripotential theory.

Shows convergence for Bernstein-Markov measures.

Abstract

Building on the first two authors' previous results, we prove a general criterion for convergence of (possibly singular) Bergman measures towards equilibrium measures on complex manifolds. The criterion may be formulated in terms of growth properties of balls of holomorphic sections, or equivalently as an asymptotic minimization of generalized Donaldson L-functionals. Our result yields in particular the proof of a well-known conjecture in pluripotential theory concerning the equidistribution of Fekete points, and it also gives the convergence of Bergman measures towards equilibrium for Bernstein-Markov measures. Applications to interpolation of holomorphic sections are also discussed.