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.

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