The Asymptotic Schottky Problem

Let ${\mathcal{M}_g}Let M_g{\mathcal{M}_g} denote the moduli space of compact Riemann surfaces of genus g and let A_g{\mathcal{A}_g} be the moduli space of principally polarized abelian varieties of dimension g. Let J : M_g ? A_g{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 J_g : = J(M_g){\mathcal{J}_g := J(\mathcal{M}_g)} is contained in a proper subvariety of A_g{\mathcal{A}_g} when g ≥  4. The classical and long-studied Schottky problem is to characterize the Jacobian locus J_g{\mathcal{J}_g} in A_g{\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 J_g{\mathcal{J}_g} look like “from far away”, or how “dense” is J_g{\mathcal{J}_g} in the sense of coarse geometry? The large scale geometry of A_g{\mathcal{A}_g} is encoded in its asymptotic cone, Cone_￥(A_g){{\rm Cone}_\infty(\mathcal{A}_g)}, which is a Euclidean simplicial cone of real dimension g. Our main result asserts that the Jacobian locus J_g{\mathcal{J}_g} is “coarsely dense” in A_g{\mathcal{A}_g}, which implies that the subset of Cone_￥(A_g){{\rm Cone}_\infty(\mathcal{A}_g)} determined by J_g{\mathcal{J}_g} actually coincides with this cone. The proof shows that the Jacobian locus of hyperelliptic curves is coarsely dense in A_g{\mathcal{A}_g} as well. We also study the boundary points of the Jacobian locus J_g{\mathcal{J}_g} in A_g{\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 J_g{\mathcal{J}_g} in these compactifications is “small”.