Metric distortion in the geometric Schottky problem
type: Item
instance of: scholarly article
title: Metric distortion in the geometric Schottky problem
zbMATH Open document ID: 1427.14063
author: Lizhen Ji
published in: Science China. Mathematics
publication date: 2019-11-27T00:00:00Z
review text: The Schottky problem is the problem of characterizing Jacobian varieties among all abelian varieties. Let \(\mathcal{H}_g=\{Z\in M_g(\mathbb{C}) : Z=Z^\top,\operatorname{Im} Z>0\}\), be the Siegel upper half-space and \(\mathcal{A}_g=\mathcal{H}_g/\mathrm{Sp}(2g,\mathbb{Z})\) the moduli space of principally polarized abelian varieties of dimension \(g\), \(J_g\subset\mathcal{A}_g\) the locus of Jacobians. The problem is to find explicit equations for \(J_g\) (or rather its closure \(\overline{J}_g\)) in \(\mathcal{A}_g\). The application (periods) \(\mathcal{M}_g \longrightarrow \mathcal{A}_g\), which associates with each curve its jacobian is well defined, it is an injective morphism and according to Torelli's theorem this application is an immersion. Riemann surfaces of genus \(g\) are classified by \(3g-3\) parameters for \(g\geq 2\). We have \(\dim\mathcal{M}_g=3g-3\), and \(\dim\mathcal{A}_g=\dim\mathcal{H}_g=\frac{1}{2}g(g+1)\). For \(g=2,3\), we have \(\dim\mathcal{M}_g=\dim\mathcal{A}_g\), and for \(g\geq 4\), we have \(\dim\mathcal{M}_g<\dim\mathcal{A}_g\). This means that for \(g\geq 4\), there are non-trivial relations satisfied by period matrix of Riemann surfaces. The Schottky problem consists to make explicit these relations. Roughly speaking, it is a question of finding criteria so that a period matrix belonging to the set \(\mathcal{H}_g \) is the period matrix of a surface of Riemann. An analytical approach to studying Schottky problem is to try to write the equations of the subvariety \(\mathcal{J}_g\) in \(\mathcal{A}_g\) using modular forms on \(\mathcal{H}_g\) and geometrically, the Schottky problem consists in characterizing Jacobian varieties among all polarized abelian varieties or in explicitly writing the subvariety \(\mathcal{J}_g\) or its adherence \(\overline{\mathcal{J}_g}\) in \(\mathcal{A}_g\) (see \textit{A. Beauville}'s beautiful 1987 Seminaire Bourbaki talk for an overview on the same subject, Exposé No. 675, [Astérisque 152, 101--…
reviewed by: Ahmed Lesfari
Mathematics Subject Classification ID: 14H42
Mathematics Subject Classification ID: 14H10
Mathematics Subject Classification ID: 14K10
Mathematics Subject Classification ID: 30F60
zbMATH DE Number: 7137529
zbMATH Keywords: metric distortion
zbMATH Keywords: Schottky problem
zbMATH Keywords: noncompact metric space
MaRDI profile type: Publication
full work available at URL: s11425-019-1598-y
OpenAlex ID: W2980572265
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Wikidata QID: Q127147132
DOI: 10.1007/S11425-019-1598-Y
description: scientific article; zbMATH DE number 7137529