渐近肖特基问题
The asymptotic Schottky problem
摘要：
设$\mathcal M_g$表示亏格为$g$的紧致黎曼曲面的模空间，$\mathcal A_g$表示复维数为$g$的主极化阿贝尔簇的空间.设$J:\mathcal M_g\longrightarrow \mathcal A_g$是将黎曼曲面与其雅可比矩阵相关联的映射。映射$J$是内射的，当$g\geq 4$时，像$J(\mathcal M_g)$包含在$\mathcal A_g$的一个真子类中。经典的和长期研究的肖特基问题是描述$\mathcal A_g$中的雅可比轨迹$\mathcal J_g:=J(\mathcal M_g)$。在本文中，我们解决了一个大规模的版本，这个问题所提出的Farb和所谓的{ \em粗糙肖特基问题}：如何$\mathcal J_g$看“从远处”，或如何“密集”是$\mathcal J_g$在这个意义上的粗糙几何？西格尔模簇$\mathcal A_g$的粗糙几何被编码在它的渐近锥$\textup{Cone}_\infty(\mathcal A_g)$中，这是一个（实）维$g$的欧几里得单纯锥。我们的主要结果断言，雅可比轨迹$\mathcal J_g$是“渐近大”，或“粗糙密集”在$\mathcal A_g$。更准确地说，由$\mathcal J_g$确定的$\textup{Cone}_\infty(\mathcal A_g)$的子集实际上与该锥重合。证明了超椭圆曲线的雅可比轨迹在$\mathcal A_g$中也是粗稠密的。我们还研究了$\mathcal A_g$和Baily-Borel紧化及Borel-Serre紧化中Jacobian轨迹$\mathcal J_g$的边界点。我们证明了对于大亏格$g$，在这些紧化中$\mathcal J_g$的边界点集是“小的”。
英文摘要：
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".
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