On Higgs bundles over Shimura varieties of ball quotient type

TL;DR

This paper demonstrates that specific Shimura varieties of unitary and orthogonal types rarely intersect the Torelli locus, using slope inequalities and geometric arguments to establish their generic exclusion in higher dimensions.

Contribution

It introduces a novel application of slope inequalities to exclude certain Shimura varieties from the Torelli locus, advancing understanding of their geometric properties.

Findings

Shimura varieties of certain types are generically excluded from the Torelli locus

The main result shows these varieties only meet the Torelli locus in dimension zero

The proof uses a slope inequality on surface fibrations by G. Xiao

Abstract

We prove the generic exclusion of certain Shimura varieties of unitary and orthogonal types from the Torelli locus. The proof relies on a slope inequality on surface fibration due to G. Xiao, and the main result implies that certain Shimura varieties only meet the Torelli locus in dimension zero.