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Growth of balls of holomorphic sections on projective toric varieties (1603.02118v1)
Published 7 Mar 2016 in math.AG and math.DG
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 $L2$-norm unit ball induced by $(X,k\phi_{\bar{D}})$ on the space of global holomorphic sections $H0(X,\mathcal{O}(kD))$.