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The geometry on smooth toroidal compactifications of Siegel varieties
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 136, Number 4, August 2014
- pp. 859-941
- 10.1353/ajm.2014.0024
- Article
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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.