Distribution of logarithmic spectra of the equilibrium energy

TL;DR

This paper studies the asymptotic distribution of eigenvalues of transition matrices between two metrics on sections of line bundles over complex varieties, extending known results to more general settings and including a p-adic analogue.

Contribution

It generalizes the concept of energy at equilibrium and extends Berndtsson's results to graded linear series and broader line bundle classes.

Findings

Eigenvalue distributions converge to a Borel probability measure

Extension of energy at equilibrium to new contexts

p-adic analogue of the main result obtained

Abstract

Let $L$ be a big invertible sheaf on a complex projective variety, equipped with two continuous metrics. We prove that the distribution of the eigenvalues of the transition matrix between the $L^2$ norms on $H^0(X,nL)$ with respect to the two metriques converges (in law) as $n$ goes to infinity to a Borel probability measure on $\mathbb R$. This result can be thought of as a generalization of the existence of the energy at the equilibrium as a limit, or an extension of Berndtsson's results to the more general context of graded linear series and a more general class of line bundles. Our approach also enables us to obtain a $p$-adic analogue of our main result.