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.

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.

Reviewer: Paul Vojta (Berkeley)

##### MSC:

14G40 | Arithmetic varieties and schemes; Arakelov theory; heights |

31C15 | Potentials and capacities on other spaces |

11G35 | Varieties over global fields |