The Geometry on Smooth Toroidal Compactifications of Siegel varieties

Shing-Tung Yau Harvard University Yi Zhang Fudan University

Arithmetic Geometry and Commutative Algebra mathscidoc:1705.01004

Distinguished Paper Award in 2017

American Journal of Mathematics, 136, (4), 859-941, 2014.8
We study smooth toroidal compactifications of Siegel varieties thoroughly from the viewpoints of mixed 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 $\sA_{g,\Gamma}(g>1)$ with a smooth toroidal compactification $\overline{\sA}_{g,\Gamma}$ such that $D_\infty:=\overline{\sA}_{g,\Gamma}\setminus \sA_{g,\Gamma}$ is normal crossing. We use this volume form formula to show that the unique group-invariant K\"ahler-Einstein metric on $\sA_{g,\Gamma}$ endows some restraint combinatorial conditions for all smooth toroidal compactifications of $\sA_{g,\Gamma}.$ Again using the volume form formula, we study the asymptotic behaviour of logarithmical canonical line bundle on any smooth toroidal compactification of $\sA_{g,\Gamma}$ carefully and we obtain that the logarithmical canonical bundle degenerate sharply even though it is big and numerically effective.
Toroidal Compactifications, Moduli of abelian variety, Hodge theory, Kaehler-Einstein metric
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  title={The Geometry on Smooth Toroidal Compactifications of Siegel varieties},
  author={Shing-Tung Yau, and Yi Zhang},
  booktitle={American Journal of Mathematics},
Shing-Tung Yau, and Yi Zhang. The Geometry on Smooth Toroidal Compactifications of Siegel varieties. 2014. Vol. 136. In American Journal of Mathematics. pp.859-941.
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