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 moduli space of principally polarized abelian varieties of dimension g. Let $${J : \mathcal{M}_g \rightarrow \mathcal{A}_g}$$ be the map which associates to a Riemann surface its Jacobian. The map J is injective, and the image $${\mathcal{J}_g := J(\mathcal{M}_g)}$$ is contained in a proper subvariety of $${\mathcal{A}_g}$$ when g ≥ 4. The classical and long-studied Schottky problem is to characterize the Jacobian locus $${\mathcal{J}_g}$$ in $${\mathcal{A}_g}$$ . In this paper we address a large scale version of this problem posed by Farb and called the coarse Schottky problem: What does $${\mathcal{J}_g}$$ look like “from far away”, or how “dense” is $${\mathcal{J}_g}$$ in the sense of coarse geometry? The large scale geometry of $${\mathcal{A}_g}$$ is encoded in its asymptotic cone, $${{\rm Cone}_\infty(\mathcal{A}_g)}$$ , which is a Euclidean simplicial cone of real dimension g. Our main result asserts that the Jacobian locus $${\mathcal{J}_g}$$ is “coarsely dense” in $${\mathcal{A}_g}$$ , which implies that the subset of $${{\rm Cone}_\infty(\mathcal{A}_g)}$$ determined by $${\mathcal{J}_g}$$ actually coincides with this cone. The proof 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”.