Metric distortion in the geometric Schottky problem

Abstract

The classical Schottky problem is concerned with characterization of Jacobian varieties of compact Riemann surfaces among all abelian varieties, or the identification of the Jacobian locus $J(\mathcal{M}_g)$ in the moduli space $\mathcal{A}_g$ of principally polarized abelian varieties as an algebraic subvariety. By viewing $\mathcal{A}_g$ as a noncompact metric space coming from its structure as a locally symmetric space and $J(\mathcal{M}_g)$ as a metric subspace, we compare the subspace metric d and the induced length metric ℓ on $J(\mathcal{M}_g)$. Consequently, we clarify the nature of the metric distortion of the subspace $J(\mathcal{M}_g)$ and hence settle a problem posed by Farb (2006) on the metric distortion of $J(\mathcal{M}_g)$ inside $\mathcal{A}_g$ in a certain sense (see Theorem 1.5 and Corollary 1.6).