Cones of orthogonal Shimura subvarieties and equidistribution

Abstract

Let X be an orthogonal Shimura variety, and let \(\mathcal {C}^{\textrm{ort}}_{r}(X)\) be the cone generated by the cohomology classes of orthogonal Shimura subvarieties in X of dimension r. We investigate the asymptotic properties of the generating rays of \(\mathcal {C}^{\textrm{ort}}_{r}(X)\) for large values of r. They accumulate towards rays generated by wedge products of the Kähler class of X and the fundamental class of an orthogonal Shimura subvariety. We also compare \(\mathcal {C}^{\textrm{ort}}_{r}(X)\) with the cone generated by the special cycles of dimension r. The main ingredient to achieve the results above is the equidistribution of orthogonal Shimura subvarieties.