Article contents
On the Oort conjecture for Shimura varieties of unitary and orthogonal types
Published online by Cambridge University Press: 02 February 2016
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.
- Type
- Research Article
- Information
- Copyright
- © The Authors 2016
References
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:50521:20160628043850633-0686:S0010437X15007794_inline10.gif?pub-status=live)
- 12
- Cited by