Equidistribution estimates for Fekete points on complex manifolds

TL;DR

This paper establishes quantitative estimates for the equidistribution of Fekete points on complex manifolds by linking them to sampling and interpolation problems on line bundles, providing new geometric characterizations.

Contribution

It introduces a novel approach connecting Fekete points to sampling and interpolation, enabling quantitative equidistribution estimates and density conditions on complex manifolds.

Findings

Quantitative bounds on the Kantorovich-Wasserstein distance of Fekete points

Complete geometric characterization of sampling and interpolation arrays in one dimension

No simultaneous sampling and interpolation arrays in semipositive line bundles

Abstract

We study the equidistribution of Fekete points in a compact complex manifold. These are extremal point configurations defined through sections of powers of a positive line bundle. Their equidistribution is a known result. The novelty of our approach is that we relate them to the problem of sampling and interpolation on line bundles, which allows us to estimate the equidistribution of the Fekete points quantitatively. In particular we estimate the Kantorovich-Wasserstein distance of the Fekete points to its limiting measure. The sampling and interpolation arrays on line bundles are a subject of independent interest, and we provide necessary density conditions through the classical approach of Landau, that in this context measures the local dimension of the space of sections of the line bundle. We obtain a complete geometric characterization of sampling and interpolation arrays in the case of compact manifolds of dimension one, and we prove that there are no arrays of both sampling and interpolation in the more general setting of semipositive line bundles.