The modularity of special cycles on orthogonal Shimura varieties over totally real fields under the Beilinson-Bloch conjecture

Abstract: We study special cycles on a Shimura variety of orthogonal type over a totally real field of degree $d$ associated with a quadratic form in $n+2$ variables whose signature is $(n,2)$ at $e$ real places and $(n+2,0)$ at the remaining $d-e$ real places for $1\leq e <d$. Recently, these cycles were constructed by Kudla and Rosu-Yott and they proved that the generating series of special cycles in the cohomology group is a Hilbert-Siegel modular form of half integral weight. We prove that, assuming the Beilinson-Bloch conjecture on the injectivity of the higher Abel-Jacobi map, the generating series of special cycles of codimension $er$ in the Chow group is a Hilbert-Siegel modular form of genus $r$ and weight $1+n/2$. Our result is a generalization of \textit{Kudla's modularity conjecture}, solved by Yuan-Zhang-Zhang unconditionally when $e=1$.