# Existence of the Sectional Capacity

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*Robert Rumely; Chi Fong Lau; Robert Varley*

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.

#### Table of Contents

# Table of Contents

## Existence of the Sectional Capacity

- Contents vii8 free
- Introduction 110 free
- §1. The Standard Hypotheses 1019 free
- §2. The Definition of the Sectional Capacity 2332
- §3. Reductions 2736
- §4. Existence of the Monic Basis for Very Ample Line Bundles 4049
- §5. Zaharjuta's Construction 5766
- §6. Local Capacities 6473
- §7. Existence of the Global Sectional Capacity 7382
- §8. A Positivity Criterion 7988
- §9. Base Change 8392
- §10. Pullbacks 8493
- §11. Products 9099
- §12. Continuity, Part I 94103
- §13. Continuity, Part II 104113
- §14. Local Capacities of Sets 109118
- §15. Approximation Theorems 115124
- Appendix A. Ample Divisors and Cohomology 124133
- Appendix B. A Lifting Lemma 127136
- Appendix C. Bounds for Volumes of Convex Bodies 128137
- Bibliography 130139