Existence of the sectional capacity

Let K be a global field, and let X/K be an equidimensional, geometrically r educed projective variety. For an ample line bundle (L) over bar on X with norms \\ \\ v on the spaces of sections K-v x(K) Gamma(X, L-xn), we prove t he existence of the sectional capacity S gamma ((L) over bar), giving conte nt to a theory proposed by Chinburg. In the language of Arakelov Theory, th e quantity - log(S gamma ((L) over bar)) generalizes the top arithmetic sel f-intersection number of a metrized line bundle, and the existence of the s ectional 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 L, we e stablish the functoriality of the sectional capacity under base change, pul l-backs by finite surjective morphisms, and products. We study the continui ty of S-gamma ((L) over bar) under variation of the metric and line bundle, and we apply this to show that the notion of v-adic sets in X(Cv) of capac ity 0 is well-defined. Finally, we show that sectional capacities for arbit rary norms can be well-approximated using objects of finite type.