The Oort conjecture for Shimura curves of small unitary rank

TL;DR

This paper proves a new result about Shimura curves in the Siegel modular variety, establishing conditions under which they are not contained in the Torelli locus, and applies this to support the Coleman-Oort conjecture for certain cases.

Contribution

It introduces a numerical criterion involving the unitary rank of the Higgs bundle to determine the non-containment of Shimura curves in the Torelli locus, extending previous work by Mumford.

Findings

Shimura curves are not contained in the Torelli locus under certain unitary rank bounds.

The Coleman-Oort conjecture is verified for Shimura curves related to partial corestriction with specific parameters.

The result generalizes Mumford's construction to broader cases.

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 to partial corestriction upon a suitable choice of parameters, which generalizes a construction due to Mumford.