Cones of orthogonal Shimura subvarieties and equidistribution

Let X be an orthogonal Shimura variety, and let Crort(X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}<^>{\textrm{ort}}_{r}(X)$$\end{document} 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 Crort(X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}<^>{\textrm{ort}}_{r}(X)$$\end{document} for large values of r. They accumulate towards rays generated by wedge products of the K & auml;hler class of X and the fundamental class of an orthogonal Shimura subvariety. We also compare Crort(X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}<^>{\textrm{ort}}_{r}(X)$$\end{document} 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.