Capacity theory and arithmetic intersection theory
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 fractional line bundles \(({\mathcal{X}}_n,{\mathcal{L}}_n,\{|\quad|\}_n)\) inducing \(L\) on \(X\), where \(\{|\quad|\}_n\) are smooth positive metrics given by smoothings of Green's functions \(G(z;E_v,D)\) of the Archimedean components of \({\mathbf{E}}\), such that \[ -\log(S_{\gamma}({\mathbf{E}},D)) =\lim_{n\to\infty}({\mathcal{X}}_n,{\mathcal{L}}_n,\{|\;|\}_n)^{d+1}. \] A more precise formulation of the theorem is given as follows. For each \(n\geq 1\), let \[ S_n=\{f\in\Gamma(nD): \sup_{x\in E_v}(|f(x)|_v)\leq 1 \text{ for each non-Archimedean }v\}, \] and let \(O_K[S_n]\) denote the graded \(O_K\)-algebra generated in degree \(1\) by \(S_n\). Then \({\mathcal{X}}_n=\text{Proj}(O_K[S_n])\), and \({\mathcal{L}}_n\) is a fractional line bundle induced by \(L\) on \({\mathcal{X}}_n\) for which \(L^{\otimes n}\) extends to \(\mathcal O_{\mathcal X_n}(1)\). The metric \(\{|\;|\}_n\) is obtained by smoothing the Green's function of an appropriate enlargement of the Archimedean component of \({\mathbf{E}}\) constructed in the theorem. The enlargements have the property that their Green's functions are continuous and their capacities approach the capacity of the original set. Such a sequence of enlargements and smoothings is not unique, but any such sequence given by the construction in the theorem yields metrics \(\{|\;|\}_n\) for which the formula in the theorem holds. For each place \(v\) of \(K\), let \(\mathbb{C}_v\) denote the completion of the algebraic closure of \(K_v\), and let \(\text{Gal}^c(\mathbb{C}_v/K_v)\) denote the Galois group of continuous automorphisms of \(\mathbb{C}_v/K_v\). Here the following conditions are referred to as the standard hypothesis: \(S_{\gamma}({\mathbf{E}}, D)\) exists and is finite (possibly zero): (1) each \(E_v\) is nonempty and stable under the Galois action \(\text{Gal}^c(\mathbb{C}_v/K_v)\); (2) each \(E_v\) is bounded away from \(\text{supp}(D)(\mathbb{C}_v)\) under the \(v\)-adic metric induced by the given projective embedding of \(X\); and for all but finitely many \(v\), \(E_v\) and \(\text{supp}(D)(\mathbb{C}_v)\) reduce to disjoint sets (mod \(v\)). The above result is the strongest one which can be obtained within Gillet-Soulé theory. Beyond Gillet-Soulé theory, a conjecture is formulated that the sectional capacity should be given exactly by a top self-intersection number of an adelic metrized line bundle with singular metrics, in an appropriate adelic intersection theory.
scientific article; zbMATH DE number 1933165
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