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Existence of the Sectional Capacity
Robert Rumely, University of Georgia, Athens, GA, Chi Fong Lau, DKB Financial Products, Ltd., Hong Kong, Hong Kong, and Robert Varley, University of Georgia, Athens, GA
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Memoirs of the American Mathematical Society
2000; 130 pp; softcover
Volume: 145
ISBN-10: 0-8218-2058-3
ISBN-13: 978-0-8218-2058-2
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 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.

Graduate students and research mathematicians interested in algebraic geometry.

• 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