Superconnections, theta series, and period domains

TL;DR

This paper introduces a new approach using superconnections to construct and analyze differential forms on period domains, linking them to theta series and known forms in hermitian symmetric cases, with explicit cohomology computations.

Contribution

It develops a superconnection-based framework for differential forms on period domains, connecting them to theta series and extending Kudla-Millson forms to broader contexts.

Findings

Constructed forms depend on chosen vectors and peak on Hodge loci.

Formed theta series sum over lattices to produce forms on arithmetic quotients.

Explicitly computed cohomology classes in terms of Hodge loci classes.

Abstract

We use superconnections to define and study some natural differential forms on period domains $\mathbb{D}$ that parametrize polarized Hodge structures of given type on a rational quadratic vector space $V$. These forms depend on a choice of vectors $v_1,\ldots,v_r \in V$ and have a Gaussian shape that peaks on the locus where $v_1,\ldots,v_r$ become Hodge classes. We show that they can be rescaled so that one can form theta series by summing over a lattice $L^r \subset V^r$. These series define differential forms on arithmetic quotients $\Gamma \backslash \mathbb{D}$. We compute their cohomology class explicitly in terms of the cohomology classes of Hodge loci in $\Gamma \backslash \mathbb{D}$. When the period domain is a hermitian symmetric domain of type IV, we show that the components of our forms of appropriate degree recover the forms introduced by Kudla and Millson. In particular, our results provide another way to establish the main properties of these forms.