The geometry on smooth toroidal compactifications of Siegel varieties

We study smooth toroidal compactifications of Siegel varieties thoroughly from the viewpoints of Hodge theory and K\"ahler-Einstein metric. We observe that any cusp of a Siegel space can be identified as a set of certain weight one polarized mixed Hodge structures. We then study the infinity boundary divisors of toroidal compactifications, and obtain a global volume form formula of an arbitrary smooth Siegel variety $\scr{A}_{g,\Gamma} (g>1)$ with a smooth toroidal compactification $\overline{\scr{A}}_{g,\Gamma}$ such that $D_\infty:=\overline{\scr{A}}_{g,\Gamma}\setminus\scr{A}_{g,\Gamma}$ is normal crossing. We use this volume form formula to show that the unique group-invariant K\"ahler-Einstein metric on $\scr{A}_{g,\Gamma}$ endows some restraint combinatorial conditions for all smooth toroidal compactifications of $\scr{A}_{g,\Gamma}$. Again using the volume form formula, we study the asymptotic behavior of logarithmical canonical line bundle on any smooth toroidal compactification of $\scr{A}_{g,\Gamma}$ carefully and we obtain that the logarithmical canonical bundle degenerate sharply even though it is big and numerically effective.