Mutual Asymptotic Fekete Sequences

TL;DR

This paper characterizes the existence of asymptotically Fekete point sequences on complex manifolds with multiple line bundles, providing conditions, construction methods, and new equidistribution results.

Contribution

It establishes necessary and sufficient conditions for the existence of asymptotically Fekete sequences across collections of toric Hermitian ample line bundles and introduces a construction method.

Findings

Necessary and sufficient conditions for existence of asymptotically Fekete sequences.

Construction method for such sequences when conditions are met.

New equidistribution property for maximizers of Vandermonde determinants.

Abstract

A sequence of point configurations on a compact complex manifold is asymptotically Fekete if it is close to maximizing a sequence of Vandermonde determinants. These Vandermonde determinants are defined by tensor powers of a Hermitian ample line bundle and the point configurations in the sequence possess good sampling properties with respect to sections of the line bundle. In this paper, given a collection of toric Hermitian ample line bundles, we give necessary and sufficient condition for existence of a sequence of point configurations which is asymptotically Fekete (and hence possess good sampling properties) with respect to each one of the line bundles. When they exist, we also present a way of constructing such sequences. As a byproduct we get a new equidistribution property for maximizers of products of Vandermonde determinants.