Stability of hodge bundles and a numerical characterization of shimura varieties

Martin Moller Goethe Universit¨at Frankfurt Eckart Viehweg Universit¨at Duisburg-Essen Kang Zuo Universit¨at Mainz

Differential Geometry mathscidoc:1609.10281

Journal of Differential Geometry, 92, (1), 71-151, 2012
Let U be a connected non-singular quasi-projective variety and f : A → U a family of abelian varieties of dimension g. Suppose that the induced map U → Ag is generically finite and there is a compactification Y with complement S = Y \U a normal crossing divisor such that Ω1 Y (log S) is nef and ωY (S) is ample with respect to U. We characterize whether U is a Shimura variety by numerical data attached to the variation of Hodge structures, rather than by properties of the map U →Ag or by the existence of CM points. More precisely, we show that f : A → U is a Kuga fibre space, if and only if two conditions hold. First, each irreducible local subsystem V of R1f∗CA is either unitary or satisfies the Arakelov equality. Second, for each factor M in the universal cover of U whose tangent bundle behaves like that of a complex ball, an iterated Kodaira-Spencer map associated with V has minimal possible length in the direction of M. If in addition f : A → U is rigid, it is a connected Shimura subvariety of Ag of Hodge type.
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@inproceedings{martin2012stability,
  title={STABILITY OF HODGE BUNDLES AND A NUMERICAL CHARACTERIZATION OF SHIMURA VARIETIES},
  author={Martin Moller, Eckart Viehweg, and Kang Zuo},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160913231450897694943},
  booktitle={Journal of Differential Geometry},
  volume={92},
  number={1},
  pages={71-151},
  year={2012},
}
Martin Moller, Eckart Viehweg, and Kang Zuo. STABILITY OF HODGE BUNDLES AND A NUMERICAL CHARACTERIZATION OF SHIMURA VARIETIES. 2012. Vol. 92. In Journal of Differential Geometry. pp.71-151. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160913231450897694943.
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