Cones of orthogonal Shimura subvarieties and equidistribution

TL;DR

This paper studies the asymptotic behavior of cones generated by orthogonal Shimura subvarieties in a Shimura variety, revealing their accumulation towards specific geometric rays and utilizing equidistribution results.

Contribution

It provides new insights into the structure and asymptotics of cones generated by orthogonal Shimura subvarieties, connecting them with equidistribution phenomena.

Findings

Rays of the cone C_r(X) accumulate towards wedge products of the Kähler class and fundamental classes.

Comparison between cones generated by orthogonal Shimura subvarieties and special cycles.

Equidistribution of orthogonal Shimura subvarieties is key to the analysis.

Abstract

Let X be an orthogonal Shimura variety, and let C_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 C_r(X) for large values of r. They accumulate towards rays generated by wedge products of the K\"ahler class of X and the fundamental class of an orthogonal Shimura subvariety. We also compare C_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.