The asymptotic Schottky problem

Abstract

Let $\mathcal M_g$ denote the moduli space of compact Riemann surfaces of genus $g$ and let $\mathcal A_g$ be the space of principally polarized abelian varieties of (complex) dimension $g$. Let $J:\mathcal M_g\longrightarrow \mathcal A_g$ be the map which associates to a Riemann surface its Jacobian. The map $J$ is injective, and the image $J(\mathcal M_g)$ is contained in a proper subvariety of $\mathcal A_g$ when $g\geq 4$. The classical and long-studied Schottky problem is to characterize the Jacobian locus $\mathcal J_g:=J(\mathcal M_g)$ in $\mathcal A_g$. In this paper we adress a large scale version of this problem posed by Farb and called the {\em coarse Schottky problem}: How does $\mathcal J_g$ look "from far away", or how "dense" is $\mathcal J_g$ in the sense of coarse geometry? The coarse geometry of the Siegel modular variety $\mathcal A_g$ is encoded in its asymptotic cone $\textup{Cone}_\infty(\mathcal A_g)$, which is a Euclidean simplicial cone of (real) dimension $g$. Our main result asserts that the Jacobian locus $\mathcal J_g$ is "asymptotically large", or "coarsely dense" in $\mathcal A_g$. More precisely, the subset of $\textup{Cone}_\infty(\mathcal A_g)$ determinded by $\mathcal J_g$ actually coincides with this cone. The proof also shows that the Jacobian locus of hyperelliptic curves is coarsely dense in $\mathcal A_g$ as well. We also study the boundary points of the Jacobian locus $\mathcal J_g$ in $\mathcal A_g$ and in the Baily-Borel and the Borel-Serre compactification. We show that for large genus $g$ the set of boundary points of $\mathcal J_g$ in these compactifications is "small".