boelkinm.tex

\subsubsection*{The cocycle of Kudla-Millson}

The key point of the work of Kudla and Millson \cite{KM1,KM2} is
that they found (in greater generality) a family of cocycles
$\varphi^V_{q}$ in $(\mathcal{S}(V) \otimes \wedge^q
\mathfrak{p}^{\ast})^K$ with weight $(p+q)/2$ for $\SL_2$. Moreover,
these cocycles give rise to Poincar\'e dual forms for certain totally
geodesic, ``special'' cycles in $X$. Recently, it has now been
shown, first \cite{HoffmanHe} for $\SO(3,2)$, and then \cite{BMM}
for all $\SO(p,q)$ and $p+q>6$ (with $p \geq q$) in the cocompact
(standard arithmetic) case that the geometric theta correspondence
specialized to $\varphi_q^V$ induces on the adelic level an {\it
isomorphism} from the appropriate space of classical modular forms
to $H^q(X)$. In particular, for any congruence quotient, the dual
homology groups are spanned by special cycles. This gives further
justification to the term geometric theta correspondence and
highlights the significance of these cocycles. In \cite{FMcoeff}
we generalize $\varphi^V_{q}$ to allow suitable non-trivial coefficient
systems (and one has an analogous isomorphism in \cite{BMM}).
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\textbf{2.3.4 The main results} 
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In the present paper, we consider the case when $V$ has signature
$(2,2)$ with $\Q$-rank $1$. Then $D \simeq \h \times \h$, and $X$
is a Hilbert modular surface. We let $\overline{X}$ be the Borel-Serre
compactification of $X$ which is obtained by replacing each isolated
cusp associated to a rational parabolic $P$ with a boundary face
$e'(P)$ which turns out to be a torus bundle over a circle, a
$3$-manifold of type Sol. This makes $\overline{X}$ a $4$-manifold
with boundary.  For simplicity, we assume that $X$ has only one
cusp so that $\partial \overline{X} = e'(P)$, and we write $k:
\partial \overline{X}  \hookrightarrow \overline{X}$ for the
inclusion. The special cycles $C_n$\footnote{We distinguish the
relative cycles $C_n$ in $X$ from the Hirzebruch-Zagier cycles $T_n$
in $\tilde{X}$, see below.} in question are now embedded modular
and Shimura curves, and are parameterized by $n \in \N$. They define
relative homology classes in $H_2(X, \partial X,\Q)$.