Equidistribution speed for Fekete points associated with an ample line bundle

TL;DR

This paper provides explicit estimates on how quickly Fekete points become evenly distributed according to the equilibrium measure, extending results to line bundles over projective manifolds.

Contribution

It introduces a new estimate on Bergman kernels and applies quantitative pluripotential theory to analyze Fekete points' distribution speed.

Findings

Explicit convergence speed estimate for Fekete points

Extension to ample line bundles over projective manifolds

New bounds on Bergman kernels

Abstract

Let K be the closure of a bounded open set with smooth boundary in C^n. A Fekete configuration of order p for K is a finite subset of K maximizing the Vandermonde determinant associated with polynomials of degree at most p. A recent theorem by Berman, Boucksom and Witt Nystrom implies that Fekete configurations for K are asymptotically equidistributed with respect to a canonical equilibrium measure, as p tends to infinite. We give here an explicit estimate for the speed of convergence. The result also holds in a general setting of Fekete points associated with an ample line bundle over a projective manifold. Our approach requires a new estimate on Bergman kernels for line bundles and quantitative results in pluripotential theory which are of independent interest.