Growth of balls of holomorphic sections on projective toric varieties

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 $\Vert \cdot \Vert$ on $\mathcal{O}(D)$, we define the energy at equilibrium of $(X,{\varphi }_{\bar{D}})$ where ${\varphi }_{\bar{D}}$ is the weight of the metrized toric divisor $\bar{D}=(D,\Vert \cdot \Vert )$. We show that this energy describes the asymptotic behaviour as $k\to \infty$ of the volume of the $L^2$-norm unit ball induced by $(X,k{\varphi }_{\bar{D}})$ on the space of global holomorphic sections $H^0(X,\mathcal{O}(kD))$.

Accepted for publication in IJM