Arithmetic modularity of Kudla’s generating function

Establish that the formal generating function Z(g,ϕ)_K of special cycles of codimension n on unitary Shimura varieties associated to H=U(V) converges absolutely and defines an element of the adelized space of holomorphic hermitian modular forms of parallel weight m for G together with values in the Chow group CH^n(X); equivalently, prove the Arithmetic Modularity Conjecture for unitary groups as stated by the author.

Background

The paper introduces Kudla’s generating function Z(g,ϕ)_K of special cycles on unitary Shimura varieties and formulates an arithmetic modularity conjecture asserting that this series should be a modular form valued in Chow groups. In the unitary case, Liu proved the divisor case n=1 and reduced n>1 to a convergence problem; Xia recently proved convergence in a few imaginary quadratic fields, settling the conjecture there. The general case across number fields and higher codimensions remains unresolved.