Capacity theory and arithmetic intersection theory
type: Item
instance of: scholarly article
title: Capacity theory and arithmetic intersection theory
zbMATH Open document ID: 1026.11056
DOI: 10.1215/S0012-7094-03-11722-6
author: Chi Fong Lau
author: Ted Chinburg
author: Robert S. Rumely
published in: Duke Mathematical Journal
publication date: 2003-12-12T00:00:00Z
review text: In the paper under review, the authors establish a formula for the sectional capacity of an adelic set in terms of arithmetic intersection theory. An adelic set is a subset of the adelic points of a projective variety over a number field. The sectional capacity of such a set is an arithmetic measure of its size relative to an ample divisor, and it generalizes the classical logarithmic capacity to an adelic setting on varieties of arbitrary dimension. The main result of the paper asserts that the sectional capacity is a limit of top self-intersection numbers of metrized fractional line bundles in Gillet-Soulé theory [\textit{H. Gillet} and \textit{C. Soulé}, Amplitude arithmétique, C. R. Acad. Sci., Paris, Sér. I 307, 887-890 (1988; Zbl 0676.14007); An arithmetic Riemann-Roch theorem, Invent. Math. 110, 473-543 (1992; Zbl 0777.14008]. These line bundles are supported on a canonical sequence of models determined by the non-Archimedean components of the set, with metrics given by smoothings of extremal plurisubharmonic Green's functions attached to the Archimedean components of the set. This result gives the first explicit relation between capacity theory and arithmetic intersection theory in dimension \(d\geq 2\). The connection is made via extremal plurisubharmonic Green's functions and the Gillet-Soulé arithmetic intersection theorem. The main theorem: Let \(K\) be a number field, and let \(X/K\) be a smooth, connected, projective variety of dimension \(d\). Let an adelic set \({\mathbf{E}}=\prod_v E_v\) and an effective, ample \(K\)-rational divisor \(D\) on \(X\) satisfying the standard hypothesis be given. Assume that the sectional capacity \(S_{\gamma}({\mathbf{E}},D)>0\), and assume that each Archimedean \(E_v\) is compact and not pluripolar. Put \(L={\mathcal{O}}_X(D)\). Then there is a canonical sequence of models \(\{{\mathcal{X}}_n\}_{1\leq n\leq \infty}\) determined by the non-Archimedean part of \({\mathbf{E}}\) and a natural sequence of ample metrized fr…
Mathematics Subject Classification ID: 11G35
Mathematics Subject Classification ID: 32U35
Mathematics Subject Classification ID: 32U20
Mathematics Subject Classification ID: 31C10
Mathematics Subject Classification ID: 31C12
Mathematics Subject Classification ID: 14G40
zbMATH DE Number: 1933165
zbMATH Keywords: capacity theory
zbMATH Keywords: arithmetic intersection theory
zbMATH Keywords: projective variety over a number field
zbMATH Keywords: Green's function
reviewed by: Noriko Yui
MaRDI profile type: Publication
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full work available at URL: s0012-7094-03-11722-6
OpenAlex ID: W2017819413
description: scientific article; zbMATH DE number 1933165