Metric distortion in the geometric Schottky problem
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--112 (1987; Zbl 0637.14021)]). By viewing \(\mathcal{A}_g\) as a noncompact metric space coming from its structure as a locally symmetric space and \(\mathcal{J}_g\) as a metric subspace, the author compares in this paper the subspace metric \(d\) and the induced length metric \(l\) on \(\mathcal{J}_g\). Consequently, he clarifies the nature of the metric distortion of the subspace \(\mathcal{J}_g\) and hence settle a problem posed by \textit{B. Farb} [Proc. Symp. Pure Math. 74, 11--55 (2006; Zbl 1191.57015)] on the metric distortion of \(\mathcal{J}_g\) inside \(\mathcal{A}_g\) in a certain sense.
scientific article; zbMATH DE number 7137529
scientific article