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

In this paper we study the Oort conjecture concerning the non-existence of Shimura subvarieties contained generically in the Torelli locus in the Siegel modular variety. Using the poly-stability of Higgs bundles on curves and the slope inequality of Xiao on fibered surfaces, we show that a Shimura curve is not contained generically in the Torelli locus if its canonical Higgs bundle contains a unitary Higgs subbundle of rank at least. From this we prove that a Shimura subvariety of type is not contained generically in the Torelli locus when a numerical inequality holds, which involves the genus, the dimension, the degree 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 type, defined by spin groups associated to quadratic spaces over a totally real number field of degree at least subject to some natural constraints of signatures.