Existence of the sectional capacity
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Bibliographic Information
Existence of the sectional capacity
（Memoirs of the American Mathematical Society, no. 690）
American Mathematical Society, 2000
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"May 2000, volume 145, number 690 (third of 4 numbers)"
Bibliography: p. 130
Description and Table of Contents
Description
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 $K_v \otimes_K \Gamma(X,\L^{\otimes n})$, 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.
Table of Contents
Introduction The standard hypothesis The definition of the sectional capacity Reductions Existence of the monic basis for very ample line bundles Zaharjuta's construction Local capacities Existence of the global sectional capacity A positivity criterion Base change Pullbacks Products Continuity, Part I Continuity, Part II Local capacities of sets Approximation theorems Appendix A. Ample divisors and cohomology Appendix B. A lifting lemma Appendix C. Bounds for volumes of convex bodies Bibliography.
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