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Arakelov inequalities and the uniformization of certain rigid shimura varieties

Eckart Viehweg Universit¨at Duisburg-Essen Kang Zuo Universit¨at Mainz

Differential Geometry mathscidoc:1609.10041

Journal of Differential Geometry, 77, (2), 291-352, 2007
Let Y be a non-singular projective manifold with an ample canonical sheaf, and let V be a Q-variation of Hodge structures of weight one on Y with Higgs bundle E1,0 ⊕ E0,1, coming from a family of Abelian varieties. If Y is a curve the Arakelov inequality says that the slopes satisfy μ(E1,0) − μ(E0,1) ≤ μ(­1 Y ). We prove a similar inequality in the higher dimensional case. If the latter is an equality, and if the discriminant of E1,0 or the one of E0,1 is zero, one hopes that Y is a Shimura variety, and V a uniformizing variation of Hodge structures. This is verified, in case the universal covering of Y does not contain factors of rank> 1. Part of the results extend to variations of Hodge structures over quasi-projective manifolds U.
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@inproceedings{eckart2007arakelov,
  title={ARAKELOV INEQUALITIES AND THE UNIFORMIZATION OF CERTAIN RIGID SHIMURA VARIETIES},
  author={Eckart Viehweg, and Kang Zuo},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160909101313836143698},
  booktitle={Journal of Differential Geometry},
  volume={77},
  number={2},
  pages={291-352},
  year={2007},
}
Eckart Viehweg, and Kang Zuo. ARAKELOV INEQUALITIES AND THE UNIFORMIZATION OF CERTAIN RIGID SHIMURA VARIETIES. 2007. Vol. 77. In Journal of Differential Geometry. pp.291-352. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160909101313836143698.
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