  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.
