Growth of balls of holomorphic sections on projective toric varieties

TL;DR

This paper investigates the asymptotic growth of holomorphic sections' balls on projective toric varieties, linking it to an energy concept derived from metrics on line bundles.

Contribution

It introduces an energy at equilibrium for toric metrics and connects it to the asymptotic volume growth of holomorphic sections.

Findings

Energy at equilibrium describes asymptotic volume behavior.

Asymptotic growth of section spaces is characterized by the energy.

Provides a new perspective on metrics and section growth in toric geometry.

Abstract

Let $\mathcal{O}(D)$ be an equivariant line bundle which is big and nef on a complex projective nonsingular toric variety $X$. Given a continuous toric metric $\|\cdot\|$ on $\mathcal{O}(D)$, we define the energy at equilibrium of $(X,\phi_{\bar{D}})$ where $\phi_{\bar{D}}$ is the weight of the metrized toric divisor $\bar{D}=(D,\|\cdot\|)$. We show that this energy describes the asymptotic behaviour as $k\rightarrow \infty$ of the volume of the $L^2$-norm unit ball induced by $(X,k\phi_{\bar{D}})$ on the space of global holomorphic sections $H^0(X,\mathcal{O}(kD))$.