The generating series for special cycles in a unitary Shimura variety $X$ associated to a Hermitian lattice of signature $(1,n)$ is a modular form. Such a Shimura variety has a unique toroidal compactification, and one can consider the closures of the special cycles there. We prove that for codimension up to the middle, the generating series for these closures is quasi-modular, and explain how to make boundary corrections to restore modularity, answering a question of Kudla. This is based on joint work with Salim Ta