The Geometry on Smooth Toroidal Compactifications of Siegel varieties

Abstract

This is a part of our joint program. The purpose of this paper is to study smooth toroidal compactifications of Siegel varieties and their applications, we also try to understand the Kähler-Einstein metrics on Siegel varieties through the compactifications. Let $A_{g,\Gamma }:=H_g/\Gamma$ be a Siegel variety, where $H_g$ is the genus-$g$ Siegel space and $\Gamma$ is an arithmetic subgroup in $Aut(H_g)$. There are four aspects of this paper : 1.There is a correspondence between the category of degenerations of Abelian varieties and the category of limits of weight one Hodge structures. We show that any cusp of Siegel space ${\mathfrak{H}}_g$ can be identified with the set of certain weight one polarized mixed Hodge structures. 2.In general, the boundary of a smooth toroidal compactification ${\bar{A}}_{g,\Gamma }$ of $A_{g,\Gamma }$ has self-intersections.For most geometric applications, we would like to have a nice toroidal compactification such that the added infinity boundary $D_{\infty }={\bar{A}}_{g,\Gamma }-A_{g,\Gamma }$ is a normal crossing divisor, We actually obtain a sufficient and necessary combinatorial condition for toroidal compactifications. 3. A toroidal compactification ${\bar{A}}_{g,\Gamma }$ of is totally determined by a combinatorial condition : an admissible family of polyhedral decompositions of certain positive cones. We show that the unique Kähler-Einstein metric on $A_{g,\Gamma }$ endows some restraint combinatorial conditions for all toroidal smooth compactifications of $A_{g,\Gamma }.$ 4.We study the asymptotic behaviour of logarithmical canonical line bundles on smooth toroidal compactifications of $A_{g,\Gamma }$ and get an integral formula for intersection numbers.

To appear in American Journal of Mathematics 2014,81 pages

Cited by in corpus (1)

• The fundamental group of a toroidal compactification of a Hermitian locally symmetric space