The asymptotic Schottky problem
| dc.creator | Ji, Lizhen | |
| dc.creator | Leuzinger, Enrico | |
| dc.date | 2008-11-25 | |
| dc.date.accessioned | 2026-07-07T10:36:31Z | |
| dc.date.available | 2026-07-07T10:36:31Z | |
| dc.description | 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". | |
| dc.identifier | https://arxiv.org/abs/0811.4059 | |
| dc.identifier | http://arxiv.org/abs/0811.4059 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/180080 | |
| dc.subject | Geometric Topology | |
| dc.subject | Differential Geometry | |
| dc.subject | 14H42, 32G15 | |
| dc.title | The asymptotic Schottky problem | |
| dc.type | text |