Modularity of special cycles on unitary Shimura varieties over CM-fields

TL;DR

This paper investigates the modularity of generating series of special cycles on unitary Shimura varieties over CM-fields, extending known results and providing new proofs for cases where the signature varies, under certain conjectural assumptions.

Contribution

It generalizes Kudla's modularity conjecture to cases with multiple real places, proving modularity of special cycle series assuming the Beilinson-Bloch conjecture.

Findings

Proves modularity of special cycle series for $e=1$ using Bruinier's regularized theta lifts.

Establishes modularity of generating series for $e>1$ assuming Beilinson-Bloch conjecture.

Provides an alternative proof of Liu's result for the $e=1$ case.

Abstract

We study the modularity of the generating series of special cycles on unitary Shimura varieties over CM-fields of degree $2d$ associated with a Hermitian form in $n+1$ variables whose signature is $(n,1)$ at $e$ real places and $(n+1,0)$ at the remaining $d-e$ real places for $1\leq e <d$. For $e=1$, Liu proved the modularity and Xia showed the absolute convergence of the generating series. On the other hand, Bruinier constructed regularized theta lifts on orthogonal groups over totally real fields and proved the modularity of special divisors on orthogonal Shimura varieties. By using Bruinier's result, we work on the problem for $e=1$ and give an another proof of Liu's proof. For $e>1$, we prove that the generating series of special cycles of codimension $er$ in the Chow group is a Hermitian modular form of weight $n+1$ and genus $r$, assuming the Beilinson-Bloch conjecture with respect to orthogonal Shimura varieties. Our result is a generalization of $\textit{Kudla's modularity conjecture}$, solved by Liu unconditionally when $e=1$.