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

TL;DR

This paper studies the boundary behavior of Kudla--Millson theta series associated with polarized variations of Hodge structure of weight two over complex curves, revealing their integrability and explicit non-holomorphic components.

Contribution

It provides a detailed analysis of Kudla--Millson theta series degenerations using Schmid's theorems, establishing their integrability and describing their non-holomorphic parts explicitly.

Findings

Kudla--Millson theta series are always integrable over the base curve.

Explicit description of the non-holomorphic part in terms of mixed Hodge structures.

Application of Schmid's theorems to analyze boundary degenerations.

Abstract

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.