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 $\|\cdot\|$ 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 $H^0(X,\mathcal{O}(kD))$.