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Existence of the sectional capacity. (English) Zbl 0987.14018
Mem. Am. Math. Soc. 690, 130 p. (2000).
Let $$X$$ be an equidimensional, geometrically reduced projective variety over a global field $$K$$. Let $$\overline{\mathcal L}$$ be an ample line bundle on $$X$$ with adelically normed sections; i.e., norms $$\|\;\|_v$$ on $$\Gamma(X,\mathcal L^{\otimes n})\otimes_K K_v$$ for all places $$v$$ of $$K$$ and all $$n\geq 0$$, subject to certain compatibility conditions. For integers $$n\geq 0$$ let $\mathcal F(\overline{\mathcal L}^{\otimes n}) =\{f\in\Gamma(X,\mathcal L^{\otimes n})\otimes_K\mathbb A_K :\|f_v\|_v\leq 1 \text{ for all }v\},$ where $$\mathbb A_K$$ is the adele ring of $$K$$ and $$d=\dim X$$. Fix a Haar measure $$\text{vol}(\cdot)$$ on $$\Gamma(X,\mathcal L^{\otimes n})\otimes_K\mathbb A_K$$, and let $$\text{covol}(\Gamma(X,\mathcal L^{\otimes n}))$$ be the volume of a fundamental domain for $$\Gamma(X,\mathcal L^{\otimes n})$$ in $$\Gamma(X,\mathcal L^{\otimes n})\otimes_K\mathbb A_K$$. This paper proves, under fairly general conditions, that the limit $-\log S_\gamma(\overline{\mathcal L}) = \lim_{n\to\infty}\frac{(d+1)!}{n^{d+1}} \log\bigl(\text{vol}(\mathcal F(\overline{\mathcal L}^{\otimes n})) /\text{covol}(\Gamma(X,\mathcal L^{\otimes n}))\bigr)$ exists. Here $$S_\gamma(\overline{\mathcal L})$$ is the sectional capacity. More generally, for certain sets $$\mathbb E\subseteq X(\mathbb A_K)$$, the logarithmic capacity $$S_\gamma(\mathbb E,D)$$ is shown to exist, where $$D$$ is an ample Cartier divisor on $$X$$. This was originally proposed by T. Chinburg [Compos. Math. 80, No. 1, 75-84 (1991; Zbl 0761.11028)].
In addition to the application to Chinburg’s theory, limits of the above sort also occur in Arakelov theory, leading to a generalized arithmetic Hilbert-Samuel theorem in which the line bundle may have adelically normed sections. Specifically, this theorem asserts that a limit as in the above definition of $$-\log S_\gamma(\overline{\mathcal L})$$ is equal to the self-intersection number $$(\overline{\mathcal L}^{d+1})$$. This generalizes work in the Arithmetic Amplitude paper of H. Gillet and C. Soulé [C. R. Acad. Sci., Paris, Sér. I 307, No. 17, 887-890 (1988; Zbl 0676.14007)]. It also extends work of S. Zhang [J. Am. Math. Soc. 8, No. 1, 187-221 (1995; Zbl 0861.14018)], in which Zhang proved the arithmetic Hilbert-Samuel theorem for singular varieties using adelic metrized line bundles. In this paper, the sectional capacity is shown to exist for a general type of adelic metriced line bundle which generalizes the sectional capacities mentioned earlier.
The existence of $$S_\gamma(\overline{\mathcal L})$$ is proved by further developing the method of V. P. Zakharyuta [Math. USSR, Sb. 25 (1975), 350-364 (1976); translation from Mat. Sb., Nov. Ser. 96(138), 374-389 (1975; Zbl 0324.32009)]. The limit $$S_\gamma(\overline{\mathcal L})$$ is decomposed into a product of local sectional capacities $$S_\gamma(\overline{\mathcal L})_v$$, which are in turn shown to exist because they are equal to certain “local Chebyshev constants,” the existence of which is clear because they are integrals of well-behaved functions.
The main ingredient in the proof is the construction of an ordered basis for the graded algebra $$R=\bigoplus\Gamma(X,\mathcal L^{\otimes n})$$ with remarkable multiplicative properties. The proof in this paper uses an argument based on a theorem of Mumford; this is the paper’s main technical improvement.
The paper also shows that, in the case where the norms are induced by metrics on the fibers of $$\mathcal L$$, the sectional capacity is compatible with base change, pull-backs by finite surjective morphisms, and products. The continuity of $$S_\gamma(\overline{\mathcal L})$$ under variation of metric and line bundle is shown to hold; as a consequence it is shown that the notion of $$v$$-adic sets in $$X(\mathbb C_v)$$ of capacity $$0$$ is well defined.

##### MSC:
 14G40 Arithmetic varieties and schemes; Arakelov theory; heights 31C15 Potentials and capacities on other spaces 11G35 Varieties over global fields
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