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On the Oort conjecture for Shimura varieties of unitary and orthogonal types

Part of: Arithmetic algebraic geometry Curves Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  02 February 2016

Ke Chen ,
Xin Lu  and
Kang Zuo
Ke Chen
Affiliation:
Department of Mathematics, Nanjing University, Nanjing, Jiangsu 210093, China School of Mathematics, University of Science and Technology of China, Hefei 230026, China email kechen@ustc.edu.cn
Xin Lu
Affiliation:
Department of Mathematics, East China Normal University, Shanghai 200241, China email luxin001@uni-mainz.de Current address: Institut für Mathematik, Universität Mainz, 55099 Mainz, Germany
Kang Zuo
Affiliation:
Institut für Mathematik, Universität Mainz, 55099 Mainz, Germany email zuok@uni-mainz.de
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Abstract

In this paper we study the Oort conjecture concerning the non-existence of Shimura subvarieties contained generically in the Torelli locus in the Siegel modular variety ${\mathcal{A}}_{g}$. Using the poly-stability of Higgs bundles on curves and the slope inequality of Xiao on fibered surfaces, we show that a Shimura curve $C$ is not contained generically in the Torelli locus if its canonical Higgs bundle contains a unitary Higgs subbundle of rank at least $(4g+2)/5$. From this we prove that a Shimura subvariety of $\mathbf{SU}(n,1)$ type is not contained generically in the Torelli locus when a numerical inequality holds, which involves the genus $g$, the dimension $n+1$, the degree $2d$ of CM field of the Hermitian space, and the type of the symplectic representation defining the Shimura subdatum. A similar result holds for Shimura subvarieties of $\mathbf{SO}(n,2)$ type, defined by spin groups associated to quadratic spaces over a totally real number field of degree at least $6$ subject to some natural constraints of signatures.

Keywords

Oort conjectureShimura varietiesTorelli locusslope inequality

MSC classification

Primary: 11G15: Complex multiplication and moduli of abelian varieties 14G35: Modular and Shimura varieties 14H40: Jacobians, Prym varieties

Information

Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 5 , May 2016 , pp. 889 - 917
Copyright
© The Authors 2016 

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