The Geometry on Smooth Toroidal Compactifications of Siegel varieties

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 K\"ahler-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 $\frak{H}_g$ 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 K\"ahler-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.