Special Cycles on Toroidal Compactifications of Orthogonal Shimura Varieties

TL;DR

This paper studies the behavior of automorphic Green functions on toroidal compactifications of orthogonal Shimura varieties, defining boundary components of special divisors and proving their generating series is modular.

Contribution

It introduces a detailed analysis of Green functions at the boundary, leading to the modularity of the generating series of special divisors.

Findings

Automorphic Green functions' behavior along boundary components is characterized.

Boundary components of special divisors are explicitly defined.

The generating series of these divisors is proven to be a modular form.

Abstract

We determine the behavior of automorphic Green functions along the boundary components of toroidal compactifications of orthogonal Shimura varieties. We use this analysis to define boundary components of special divisors and prove that the generating series of the resulting special divisors on a toroidal compactification is modular.