Growth rate of balls of holomorphic sections on compact Riemann surfaces

TL;DR

This paper studies the asymptotic growth of the volume of unit balls in spaces of holomorphic sections of line bundles over compact Riemann surfaces, linking it to equilibrium measures.

Contribution

It provides new quantitative estimates on the growth rate of these volumes and the convergence speed of Fekete measures to equilibrium measures.

Findings

Derived explicit growth speed of unit ball volumes in holomorphic sections.

Established convergence rates of Fekete measures to equilibrium measures.

Connected geometric growth with potential-theoretic equilibrium concepts.

Abstract

Let $X$ be a compact Riemann surface and let $L$ be a positive line bundle on $X$. We obtain the growth speed of unit ball volume in $H^0(X,L^n)$ towards the energy at equilibrium. As an application, we also obtain the speed of Fekete measures converging to the equilibrium measure.