The Geometric Unitary Kudla Conjecture

Abstract

We prove that, over an arbitrary CM field, every symmetric formal Fourier-Jacobi series converges and equals the Fourier-Jacobi expansion of a genuine Hermitian Hilbert modular form. As an application, we show that the Chow-valued Kudla generating series of special cycles on unitary Shimura varieties for Hermitian lattices over CM fields of signature $(p,1)$ at one infinite place and $(p+1,0)$ at all others is modular of weight $p+1$ for a Weil representation, establishing the geometric unitary Kudla Conjecture in arbitrary codimension. This removes the modularity hypothesis from the arithmetic inner product formula by Li-Liu.