Kudla-Millson forms and one-variable degenerations of Hodge structure

We consider arbitrary polarized variations of Hodge structure of weight two and $h^{2,0}=1$ over a non–singular complex algebraic curve $S$ and analyze the boundary behaviour of the associated Kudla–Millson theta series using Schmid’s theorems on degenerations of Hodge structure. This allows us to prove that this theta series is always integrable over $S$ and to describe explicitly the non-holomorphic part of the Kudla–Millson generating series in terms of the mixed Hodge structures at infinity.