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${\mathcal{A}}_{g}$. Using the poly-stability of Higgs bundles on curves and the slope inequality of Xiao on fibered surfaces, we show that a Shimura curve$C$is not contained generically in the Torelli locus if its canonical Higgs bundle contains a unitary Higgs subbundle of rank at least$(4g+2)/5$. From this we prove that a Shimura subvariety of$\mathbf{SU}(n,1)$type is not contained generically in the Torelli locus when a numerical inequality holds, which involves the genus$g$, the dimension$n+1$, the degree$2d$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$\mathbf{SO}(n,2)$type, defined by spin groups associated to quadratic spaces over a totally real number field of degree at least$6$subject to some natural constraints of signatures.

THE GEOMETRY AND ARITHMETIC OF A CALABI–YAU SIEGEL THREEFOLD

In this paper we treat in details a Siegel modular variety [Formula: see text] that has a Calabi–Yau model, [Formula: see text]. We shall describe the structure of the ring of modular forms and its geometry. We shall illustrate two different methods of producing the Hodge numbers. The first uses the definition of [Formula: see text] as the quotient of another known Calabi–Yau variety [Formula: see text]. In this case we will get the Hodge numbers considering the action of the group on a crepant resolution [Formula: see text] of [Formula: see text]. The second, purely algebraic geometric, uses the equations derived from the ring of modular forms and is based on determining explicitly the Calabi–Yau model [Formula: see text] and computing the Picard group and the Euler characteristic.