The Oort Conjecture for Shimura Curves of Small Unitary Rank

Abstract

We prove that a Shimura curve in the Siegel modular variety is not generically contained in the open Torelli locus as long as the rank of unitary part in its canonical Higgs bundle satisfies a numerical upper bound. As an application we show that the Coleman–Oort conjecture holds for Shimura curves associated with partial corestriction upon a suitable choice of parameters, which generalizes a construction due to Mumford.

Keywords

Coleman-Oort conjecture / Torelli locus / Shimura curves / Higgs bundles / Corestriction