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

Families of curves with Higgs field of arbitrarily large kernel

“…In a first study [19], Lu and Zuo excluded the existence of certain types of Shimura curves in the Torelli locus using properties of U . In the subsequent works [6,7], Chen, Lu and Zuo proved that if rk U 4g+2 5…”

“…In a first study [19], Lu and Zuo excluded the existence of certain types of Shimura curves in the Torelli locus using properties of . In the subsequent works [6, 7], Chen, Lu and Zuo proved that if or , then is not generically contained in the Torelli locus (that is, intersects the Torelli locus at most in isolated points). Therefore, Shimura curves in the Torelli locus cannot have of too big or too small rank.…”