The geometry on smooth toroidal compactifications of Siegel varieties
The geometry on smooth toroidal compactifications of Siegel varieties
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DOI：
10.1353/ajm.2014.0024
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发表时间：
2012-01
影响因子：
1.7
通讯作者：
S. Yau;Yi Zhang
中科院分区：
文献类型：
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作者：
S. Yau;Yi Zhang
本文从Hodge理论和Kahler-Einstein度规的观点出发，对Siegel簇的光滑环面紧化进行了深入的研究。我们发现Siegel空间的任何一个尖点都可以看作是一组权为1的极化混合Hodge结构。然后研究了环面紧化的内边界因子，得到了任意光滑Siegel簇Ag，Γ（g> 1）的整体体积形式公式，其中光滑环面紧化Ag，Γ使得D∞：= Ag，Γ \ Ag，Γ是法交.利用这个体积形式公式，我们证明了Ag，Γ上唯一的群不变Kahler-Einstein度规赋予了Ag，Γ的所有光滑环面紧化的约束组合条件.再次利用体积形式公式，仔细研究了Ag，Γ的任意光滑环面紧化上的代数正则线丛的渐近行为，得到了代数正则丛即使很大，数值有效，也是急剧退化的.
We study smooth toroidal compactifications of Siegel varieties thoroughly from the view- points of Hodge theory and Kahler-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 in- finity boundary divisors of toroidal compactifications, and obtain a global volume form formula of an arbitrary smooth Siegel variety Ag,Γ( g> 1) with a smooth toroidal compactification Ag,Γ such that D∞ := Ag,Γ \ Ag,Γ is normal crossing. We use this volume form formula to show that the unique group-invariant Kahler-Einstein metric on Ag,Γ endows some restraint combinatorial conditions for all smooth toroidal compactifications of Ag,Γ. Again using the volume form formula, we study the asymptotic behavior of logarithmical canonical line bundle on any smooth toroidal compactification of Ag,Γ carefully and we obtain that the logarithmical canonical bundle degenerate sharply even though it is big and numerically effective.