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 $overlinemathcal L$ on $X$ with norms $ v$ on the spaces of sections $Kv otimesK Gamma(X,Lotimes n)$, we prove the existence of the sectional capacity $SGamma(overlinemathcal L)$, giving content to a theory proposed by Chinburg. In the language of Arakelov Theory, the quantity $-log(SGamma(overlinemathcal L))$ 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 $SGamma(overlinemathcal L)$ under variation of the metric and line bundle, and we apply this to show that the notion of $v$-adic sets in $X(mathbb Cv)$ of capacity $0$ is well-defined. Finally, we show that sectional capacities for arbitrary norms can be well-approximated using objects of finite type.