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

TL;DR

This paper advances the understanding of the Oort conjecture by establishing conditions under which certain Shimura subvarieties are not contained in the Torelli locus, using Higgs bundle stability and slope inequalities.

Contribution

It proves new non-containment results for Shimura subvarieties of unitary and orthogonal types in the Torelli locus based on numerical inequalities involving geometric and arithmetic data.

Findings

Shimura curves with certain Higgs bundle properties are not in the Torelli locus.

Shimura subvarieties of SU(n,1)-type are excluded under specific numerical conditions.

Similar non-containment results are shown for SO(n,2)-type Shimura subvarieties.

Abstract

In this paper we study the Oort conjecture on 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 fibred surfaces, we show that a Shimura curve $C$ is not contained generically in the Torelli locus if its canonical Higgs bundles 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.