Memoirs of the American Mathematical Society 2000; 130 pp; softcover Volume: 145 ISBN10: 0821820583 ISBN13: 9780821820582 List Price: US$53 Individual Members: US$31.80 Institutional Members: US$42.40 Order Code: MEMO/145/690
 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 selfintersection number of a metrized line bundle, and the existence of the sectional capacity is equivalent to an arithmetic HilbertSamuel 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 welldefined. Finally, we show that sectional capacities for arbitrary norms can be wellapproximated using objects of finite type. Readership Graduate students and research mathematicians interested in algebraic geometry. 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
