Spaces of norms, determinant of cohomology and Fekete points in non-Archimedean geometry

TL;DR

This paper investigates the asymptotic behavior of the determinant of cohomology for line bundles over non-Archimedean varieties, establishing results on Fekete points and transfinite diameters using Berkovich geometry.

Contribution

It introduces a non-Archimedean analogue of complex asymptotic expansions for cohomology determinants, extending equidistribution and Fekete point results to this setting.

Findings

Asymptotic description of cohomology determinants in non-Archimedean geometry.

Existence of transfinite diameters and Fekete points in this setting.

Development of a systematic study of spaces of norms and Fubini-Study metrics.

Abstract

Let L be an ample line bundle on a (geometrically reduced) projective variety X over any complete valued field. Our main result describes the leading asymptotics of the determinant of cohomology of large powers of L, with respect to the supnorm of a continuous metric on the Berkovich analytification of L. As a consequence, we establish in this setting the existence of transfinite diameters and equidistribution of Fekete points, following a strategy going back Berman, Witt Nystr\"om and the first author for complex manifolds. In the non-Archimedean case, our approach relies on a version of the Knudsen-Mumford expansion for the determinant of cohomology on models over the (possibly non-Noetherian) valuation ring, as a replacement for the asymptotic expansion of Bergman kernels in the complex case, and on the reduced fiber theorem, as a replacement for the Bernstein-Markov inequalities. Along the way, a systematic study of spaces of norms and the associated Fubini-Study type metrics is undertaken.