The arithmetic Hodge index theorem for adelic line bundles
type: Item
instance of: scholarly article
title: The arithmetic Hodge index theorem for adelic line bundles
zbMATH Open document ID: 1372.14017
author: Xinyi Yuan
author: Shou-Wu Zhang
published in: Mathematische Annalen
publication date: 2017-04-18T00:00:00Z
full work available at URL: https://arxiv.org/abs/1304.3538
review text: The main theorem (Theorem 3.2) of this paper is the arithmetic Hodge index theorem for integrable adelic line bundles. This generalizes the index theorem for line bundles [\textit{A. Moriwaki}, Math. Res. Lett. 3, No. 2, 173--183 (1996; Zbl 0873.14005)], which in turn is a higher-dimensional generalization of Faltings-Hriljac. More precisely, let \(X\) be a normal and geometrically integral projective variety over \(\overline{\mathbb Q}\) of dimension \(n\geq 1\). Let \(\overline M\) be an integral adelic \(\mathbb Q\)-line bundle on \(X\) as in [\textit{S. Zhang}, J. Algebr. Geom. 4, No. 2, 281--300 (1995; Zbl 0861.14019)], and let \(\overline L_1, \ldots, \overline L_{n-1}\) be nef adelic \(\mathbb Q\)-line bundles. Suppose also that each \(L_i\) is big and \(M\cdot L_1\cdots L_{n-1} = 0\). Then \(\overline M^2\cdot \overline L_1 \cdots \overline L_{n-1} \leq 0\), and if in addition each \(\overline L_i\) is arithmetically positive and \(\overline M\) is \(\overline L_i\)-bounded for each \(i\), the equality holds if and only if \(\overline M\in \pi^*\widehat{\mathrm{Pic}(\overline{\mathbb Q})}_{\mathbb Q}\), where \(\pi\) is the structure morphism. The authors first prove the local index theorem for integrable metrized line bundles, using the induction formula for integration on the Berkovich analytic space \(X^{\mathrm{an}}\) [\textit{A. Chambert-Loir} and \textit{A. Thuillier}, Ann. Inst. Fourier 59, No. 3, 977--1014 (2009; Zbl 1192.14020)]. Then the authors use the vanishing result of curvatures of flat metrics [\textit{W. Gubler}, Invent. Math. 169, No. 2, 321--376 (2007; Zbl 1153.14036)] and a Lefschetz-type theorem to obtain Theorem 3.2. An appendix discusses many of these ingredients in detail, including local and arithmetic intersections, arithmetic positivity, flat metrics, and the Lefschetz theorems. As a consequence of the local index theorem, the authors derive a non-archimedean analog of [\textit{E. Calabi}, The space of Kähler Metrics, Proc. Interna…
reviewed by: Yu Yasufuku
Mathematics Subject Classification ID: 14G40
Mathematics Subject Classification ID: 37P55
Mathematics Subject Classification ID: 14G22
zbMATH DE Number: 6706027
zbMATH Keywords: Hodge index theorem
zbMATH Keywords: integrable adelic line bundles
zbMATH Keywords: Calabi theorem
zbMATH Keywords: preperiodic points
MaRDI profile type: Publication
OpenAlex ID: W1507561980
arXiv ID: 1304.3538
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DOI: 10.1007/S00208-016-1414-1
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description: scientific article