THE GEOMETRY ON SMOOTH TOROIDAL COMPACTIFICATIONS OF SIEGEL VARIETIES

摘要

We study smooth toroidal compactifications of Siegel varieties thoroughly from the viewpoints of Hodge theory and K?hler-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 A_(g,Γ)(g >1) with a smooth toroidal compactification A_(g,Γ) such that D∞:= A_(g,Γ) A_(g,Γ) is normal crossing. We use this volume form formula to show that the unique group-invariant K?hler-Einstein metric on A_(g,Γ) endows some restraint combinatorial conditions for all smooth toroidal compactifications of A_(g,Γ). Again using the volume form formula, we study the asymptotic behavior of logarithmical canonical line bundle on any smooth toroidal compactification of A_(g,Γ) carefully and we obtain that the logarithmical canonical bundle degenerate sharply even though it is big and numerically effective.