Existence of the Sectional Capacity

Let $K$ be a global field, and let $X/K$ be an equidimensional, geometrically reduced projective variety. For an ample line bundle $\overline{\mathcal L}$ on $X$ with norms $\|\ \|_v$ on the spaces of sections $K_v \otimes_K \Gamma(X,\mathcal{L}^{\otimes n})$, we prove the existence of the sectional capacity $S_\gamma(\overline{\mathcal L})$, giving content to a theory proposed by Chinburg. In the language of Arakelov Theory, the quantity $-\log(S_\gamma(\overline{\mathcal L}))$ generalizes the top arithmetic self-intersection number of a metrized line bundle, and the existence of the sectional capacity is equivalent to an arithmetic Hilbert-Samuel Theorem for line bundles with singular metrics. In the case where the norms are induced by metrics on the fibres of ${\mathcal L}$, we establish the functoriality of the sectional capacity under base change, pullbacks by finite surjective morphisms, and products. We study the continuity of $S_\gamma(\overline{\mathcal L})$ under variation of the metric and line bundle, and we apply this to show that the notion of $v$-adic sets in $X(\mathbb C_v)$ of capacity $0$ is well-defined. Finally, we show that sectional capacities for arbitrary norms can be well-approximated using objects of finite type.