remainder.tex

\subsection{The Hirzebruch-Zagier Theorem} 

We now view $[\theta_{\varphi}, \theta_{\phi}]$ as a class in $H^2(\tilde{X})$ via the map $j_{\#}: H_c^2(X) \to H^2(\tilde{X})$. We recover the Hirzebruch-Zagier-Theorem. 

\begin{theorem}\label{HZTheorem}
We have 
\[
j_{\#}[\theta_{\varphi}, \theta_{\phi}](\tau) =  -\frac{1}{2\pi}\delta_{h0} [\omega] + \sum_{n>0} [T^c_n] q^n \in H^2(\tilde{X},\Q) \otimes M_2(\G(N)).
\]
In particular,
\[
 -\frac{1}{2\pi}\delta_{h0} \vol(T_m) + \sum_{n>0} (T_n^c \cdot T_m)_{\tilde{X}} q^n \in M_2(\G(N)).
\]
This is the result Hirzebruch-Zagier proved for certain Hilbert modular surfaces (Example~\ref{HZex}) by explicitly computing the intersection numbers $T_m \cdot T^c_n $.
\end{theorem}

\begin{proof}

This follows from Theorem~\ref{FM-main-th} since $j_{\ast} C_n^c = T_n^c$ (Proposition~\ref{CnTn}), combined with the following general principle.
Suppose $\omega$ is a compactly supported form on $X$ such that the cohomology class of $\omega$ is the Poincar\'e dual of the homology class of a cycle $C$: $[\omega] = \PD(C)$. Then we have $ j_{\#}[\omega] = \PD( j_* C)$.
To see this we have only to replace $\omega$ by a cohomologous  `Thom representative' of $\PD(C)$, namely a closed form $\tilde{\omega}$ supported in a tubular neighborhood $N(C)$ of $C$ in $X$ such that the integral of $\tilde{\omega}$ over any disk of $N(C)$ is one. Then it is a general fact from algebraic topology (extension by zero of a Thom class)  that $\tilde{\omega}$ represents the Poincar\'e dual of $C$ in any manifold $M$ containing $N(C)$, in particular for $M = \tilde{X}$.
\end{proof}


\begin{remark}
If one is only interested in recovering the statement of this theorem, then there is also a different way of deriving this from the Kudla-Millson theory. Namely, the lift $\Lambda$ on $H_2(X)$ (Theorem~\ref{KM90}) factors through the quotient of $H_2(X)$ by $H_2(\partial X)$ since the restriction of $\theta_{\varphi_2}$ is exact (Theorem~\ref{globalexact}). But by Proposition~\ref{intersectionhom} we have $j_{\ast} H_2(X) \simeq H_2(X)/  H_2(\partial X)$, and the Hirzebruch-Zagier result exactly stipulates the modularity of the lift of classes in $j_{\ast} H_2(X)$. However, in that way one misses the  remarkable  extra structure coming from $\partial X$ as we will explain in the next subsection.
\end{remark}


\subsection{The lift of special cycles}\label{special-lift-section}

We now consider the lift of a special cycle $C_y$. By Theorem~\ref{FM-main-th} and Lemma~\ref{integralformula} we see
\begin{align}\label{special-lift}
\La^c(C_y,\tau,\calL_V) &= -\frac{1}{2\pi}\delta_{h0} \vol(C_y) + \sum_{n>0} ( C^c_n \cdot C_y) q^n \\
&= \int_{C_y} \theta_{\varphi_2}(\tau,\calL_V) - \sum_{[P]}\int_{(\partial C_y)_P} \theta^P_{\phi_{0,1}}(\tau,\calL_{W_P}). \notag
\end{align}

The two terms on the right, the integrals over $C_y$ and ${\partial C_y}$, are both non-holomorphic modular forms (see below) whose difference is holomorphic (by Theorem~\ref{La^c-hol}). So the generating series series of $(C^c_n \cdot C_y)$ is the sum of two non-holomorphic modular forms. We now give geometric interpretations for the two individual non-holomorphic forms. 

Following \cite{HZ} we define the {\it interior} intersection number of two special cycles by
\[
( C_n \cdot C_y )_X =  (C_n \cdot C_y )^{tr} + \vol(C_n \cap C_y),
\]
the sum of the transversal intersections and the volume of the $1$-dimensional (complex) intersection of  $C_n$ and $C_y$ which occur if one of the components of $C_n$ is equal to $C_y$. 

\begin{theorem}\label{interior-lift}
We have
\[
 \int_{C_y} \theta_{\varphi_2}(\tau,\calL_V) = -\frac{1}{2\pi}\delta_{h0} \vol(C_y) \:+\; 
 \sum_{n=1}^{\infty} ( C_n \cdot C_y )_X q^n \; + \; \sum_{n \in \Q} \sum_{[P]} \int_{(\partial C_y)_{P}}  {\tilde{\psi}_{0,1}^P}(n)(\tau).
 \]
 So the Fourier coefficients of the holomorphic part of the non-holomorphic modular form $\int_{C_y} \theta_{\varphi_2}$ are the interior intersection numbers of the cycles $C_y$ and $C_n$. 
\end{theorem}

\begin{proof}
This is essentially \cite{FCompo}, section~5, where more generally $\Orth(p,2)$ is considered. 
There the interpretation of the holomorphic Fourier coefficients as interior intersection number is given. (For more details of an analogous calculation see \cite{FMspec}, section~8). A little calculation using the formulas in \cite{FCompo} gives the non-holomorphic contribution. A more conceptual proof would use the relationship between $\varphi_2$ and $\tilde{\psi}_1$ (see Proposition~\ref{schluesselV} and Section~\ref{currents}) and the restriction formula for $\tilde{\psi}_1(n)$ (Theorem~\ref{psitilderes}). 
\end{proof}




By slight abuse of notation we write $\Lk(C_n,C_y) = \sum_{[P]} \Lk((\partial C_n)_P, (\partial C_y)_P)$ for the total linking number of $\partial C_n$ and $\partial C_y$. Then by Theorem~\ref{xi'-integralP} we obtain

\begin{theorem}\label{xi'-integral}
\[
\sum_{[P]}\int_{(\partial C_y)_P} \theta^P_{\phi_{0,1}}(\tau,\calL_{W_P}) = 
  \sum_{n>0}  \Lk(C_n,C_y) q^n
 \; + \; \sum_{n \in \Q} \sum_{[P]} \int_{(\partial C_y)_{P}}  {\tilde{\psi}_{0,1}^P}(n)(\tau).
\]
 So the Fourier coefficients of the holomorphic part of $\int_{(\partial C_y)_P} \theta^P_{\phi}(\tau,\calL_{W_P})$ are the linking numbers of the cycles $\partial C_y$ and $ \partial C_n$ at the boundary component $e'(P)$.  
 \end{theorem}

\begin{remark}
There is also another ``global'' proof for Theorem~\ref{xi'-integral}. The cycle $C_y$ intersects $e'(P)$ transversally (when pushed inside) and hence also the cap $A_n$. From this it is not hard to see that we can split the intersection number $C_n^c \cdot C_y$ as 
\[
C^c_n \cdot C_y  =  (C_n \cdot C_y)_X  -  \Lk(C_n,C_y).
\]
Hence Theorem~\ref{xi'-integral} also follows from combining \eqref{special-lift} and Theorem~\ref{interior-lift}. 
\end{remark}

Hirzebruch-Zagier also obtain the modularity of the functions given in Theorems~\ref{interior-lift} and \ref{xi'-integral}, but by quite different methods. In particular, they explicitly calculate the intersection number $T^c_n \cdot T_m$. They split the intersection number into the interior part $(T_n \cdot T_m)_X$ and a `boundary contribution' $(T_n \cdot T_m)_{\infty}$ given by 
 \[
 (T_n \cdot T_m )_{\infty} = (T_n \cdot T_m)_{\tilde{X}-X}  - ({T}_m-T_m^c) \cdot ({T}_n -T_n^c).
 \]
Now by Theorem~\ref{HZTheorem} and its proof we have
\[
T^c_n \cdot T_m = C^c_n \cdot C_m.
\]
We have (per definition) $(T_n \cdot T_m)_X = (C_n \cdot C_m)_X$, so Theorem~\ref{interior-lift} gives the generating series for $(T_n \cdot T_m)_X$. Note that Theorem~5.4 in \cite{FCompo} also compares the explicit formulas in \cite{HZ} for $(T_n \cdot T_m)_X$ with the ones obtained via  $\int_{C_y} \theta_{\varphi_2}(\tau,\calL_V)$. All this implies
\[
 (T_n \cdot T_m)_{\infty}  =\Lk( C_n \cdot C_m).
 \]
Independently, we also obtain this from comparing the explicit formulas for the boundary contribution in \cite{HZ}, Section~1.4 with our formulas for the linking numbers, Theorem~\ref{LinkCnCm} and Example~\ref{LinkCnCmex}.

 



\section{A current approach for the special cycles}\label{currents}

In this section we prove Theorem~\ref{FM-main-th}, the crucial Fourier coefficient formula for our lift $\Lambda^c$.  As a consequence of our approach we will also obtain Theorem~\ref{linking-dual}, the linking number interpretation for the lift at the boundary. 

\subsection{A differential character for $C_n^c$}

The key step for the entire Kudla-Millson theory is that for $n>0$ the form $\varphi_2(n)$ is a Poincar\'e dual form for the cycle $C_n$, i.e., 

\begin{theorem}[\cite{KM2,KMCan}]
Let $\eta$ be a closed rapidly decreasing $2$-form. Then 
\[
\int_X \eta \wedge  \varphi_2(n)= \left(\int_{C_n} \eta \right) e^{-2\pi n}.
\]
\end{theorem}

To show this they employ at some point a homotopy argument which requires $\eta$ to be rapidly decaying. Since we require $\eta$ to be any closed $2$-form on the compactification $\overline{X}$, their approach is not applicable in our case. Instead, we use a differential character argument for $\varphi_2$ which implicitly already occurred in \cite{BFDuke}, Section~7 for general signature $(p,q)$. Namely, we have

\begin{theorem} (\cite{BFDuke}, Section~7)\label{BrFu}
Let $n>0$. 
The singular form $ \tilde{\psi}_1(n)$ is a differential character in the sense of Cheeger-Simons for the cycle $C_n$. More precisely, $\tilde{\psi}_1(n)$ is a locally integrable $1$-form on $X$, and for any compactly supported $2$-form $\eta$ we have 
\[
\int_{X} \eta \wedge \varphi_2(n)  = \left(\int_{C_n}  \eta \right) e^{-2 \pi n}  - \int_{X}  d \eta\wedge \tilde{\psi}_1(n).
\]
\end{theorem}

\begin{proof}
This is the content of the proofs of Theorem~7.1 and Theorem~7.2 in \cite{BFDuke}. There the analogous properties for a singular theta lift associated to $\psi$ is established. However, the proofs boil down to establish the claims for $\tilde{\psi}_1$. The form $\tilde{\psi}$ there is indeed the form $\tilde{\psi}_1$ of this paper. 
\end{proof}

\begin{remark}\label{Kudla-xi}
The form $\tilde{\psi}_1$ is closely related to Kudla's Green function $\xi$ \cite{KAnn97,KBforms} (more generally for $\Orth(p,2)$) which is given by 
\[
\xi(x) =   \left( \int_1^{\infty} \varphi_0^0(\sqrt{r}x)  \frac{dr}{r} \right) e^{- \pi (x,x) }.
\]
Then $\Xi(n) = \sum_{x\in\calL_n} \xi(x)$ gives rise to a Green's function for the divisor $C_n$ and moreover $dd^c \xi = \varphi_2$. Here $d^c = \tfrac{1}{4\pi i}(\partial - \overline{\partial})$. This suggests $d^c \xi = \tilde{\psi}_1$, which indeed follows from $d^c \varphi_0 = -\psi_1$, see \cite{BFDuke}, Remark~4.5. 
\end{remark}

For $n \in \Q$ we define
\[
 \varphi_2^c(n) :=  \varphi_2(n) - \sum_{[P]} d(f \pi^{\ast} \phi^P_{0,1}(n))
\]
and follow the current approach to show that for $n>0$ the form $\varphi_2^c(n)$
is a Poincar\'e dual form for the cycle $C_n^c$. Here we follow the notation of subsection~\ref{mappingconesection}. That is, $\pi^{\ast} \phi^P_{0,1}(n)$ is the pullback to a product neighborhood $V$ of $\partial \overline{X}$, and $f$ is a smooth function on $V$ of the geodesic flow coordinate $t$ which is $1$ near $t=\infty$ and zero else. Note that  $\varphi_2^c(n)$ is exactly the $n$-th Fourier coefficient of the mapping cone element $[\theta_{\varphi},\theta_{\phi}]$, when realized as a rapidly decreasing form on $X$. We also define 
\[
\tilde{\psi}_1^c(n) = \tilde{\psi}_1(n) - f \pi^{\ast} \phi^P_{0,1}(n). 
\]
We call a differential form $\eta$ on $\overline{X}$ special if in a neighborhood of each boundary component $e'(P)$ it is the pullback of a form $\eta_P$ on $e'(P)$ under the geodesic retraction and if the pullback of the form $\eta_P$ to the universal cover $e(P)$ is $N$-left-invariant. The significance of the forms lies in the fact that the complex of special forms also computes the cohomology of $\overline{X}$. Note that the proof of Theorem~\ref{restriction} shows that $\theta_{\varphi_2}$ is `almost' special; it only differs from a special form by a rapidly decreasing form. 




\begin{theorem}\label{newcurrenteq}
Let $n>0$. The form $ \tilde{\psi}_1^c(n)$ is a differential character for the cycle $C^c_n$. More precisely, $\tilde{\psi}_1^c(n)$ is a locally integrable $1$-form on $X$ and satisfies the following current equation on special $2$ forms on $\overline{X}$:
\[
d[\tilde{\psi}_1^c(n)] + \delta_{C_n}  e^{-2\pi n} = [\varphi_2^c(n)].
\]
That is, for any special $2$-form $\eta$ on $\overline{X}$ we have 
\[
\int_{X} \eta \wedge \varphi^c_{2}(n)  = \left(\int_{C^c_n}  \eta \right) e^{-2 \pi n} - \int_{X}  d\eta \wedge \tilde{\psi}^c_{2}(n).
\]
\end{theorem}

This implies Theorem~\ref{FM-main-th} for the positive Fourier coefficients. For $n\leq 0$, the form $\varphi^c_{2}(n)$ is exact with primitive $\tilde{\psi}^c_{2}(n)$ which by Theorem~\ref{psitilderes} is decaying. So Theorem~\ref{newcurrenteq} holds also for $n \leq 0$ with $C_n^c = \emptyset$. Hence for the these coefficients only the term $x=0$ contributes, which gives the integral of $\eta$ against the K\"ahler form. 

\begin{remark}\label{Kudla-modification}
In view of Remark~\ref{Kudla-xi} it is very natural question to ask how one can modify Kudla's Green's function $\Xi(n)$ to obtain a Green's function for the cycle $T_n^c$ in $\tilde{X}$. Extensive discussions with K\"uhn suggest that (if $X$ has only one cusp) 
\[
\Xi(n) - t \sum_{\substack{x \in \calL_W\\ (x,x)=2n}} f \pi^{\ast}(B(x)+B'(x))
\]
is such a Green's function, but we have not checked all details.
\end{remark}


\subsection{Proof of Theorem~\ref{newcurrenteq} }\label{8.1}

For simplicity assume that $X$ has only one cusp and continue the drop the superscript $P$. We let $\rho_{T}$ be a family of smooth functions on a standard fundamental domain $\calF$ of $\G$ in $D$ only depending on $t$ which is $1$ for $t\leq T$ and $0$ for $T+1$. We then have 
\begin{align*}
\int_{X} \eta \wedge \varphi^c_{2}(n)  &= \lim_{T\to \infty} \int_{X} \rho_T \eta \wedge
\left(\varphi_2(n) -  d(f \pi^{\ast} \phi_{0,1}(n)) \right). 
\end{align*}
We apply Theorem~\ref{BrFu} for the compactly supported form $\rho_T\eta$ and obtain
\begin{align}\label{eq1}
\int_{X} \eta \wedge \varphi^c_{2}(n&)=  \lim_{T\to \infty} \Biggl[ \left(\int_{C_n}  \rho_T \eta \right) e^{- 2\pi n} - \int_{X}  d (\rho_T \eta) \wedge \tilde{\psi}_1(n) \\
& \quad - \int_X  d\left( \rho_T \eta \wedge (f \pi^{\ast} \phi_{0,1}(n)) \right) - d(\rho_T \eta) \wedge f \pi^{\ast} \phi_{0,1}(n) \Biggr] \notag
\end{align}
The first term on the right hand side of \eqref{eq1} goes to $\left(\int_{C_n} \eta\right)e^{-2\pi n}$ as $T \to \infty$, while the third vanishes for any $T$ by Stokes' theorem. For the two remaining terms of \eqref{eq1} we first note $d(\rho_T \eta) = \rho_T'(t) dt \wedge \eta + \rho_T d\eta$ and  $\rho_T'(t)=0$ outside $[T,T+1]$. We obtain for these two terms
\begin{multline}\label{eq2}
-  \int_{X}  (d \eta) \wedge \left( \tilde{\psi}_1(n) - f \pi^{\ast} \phi_{0,1}(n) \right) \\ - \lim_{T\to \infty} \int_T^{T+1} \int_{e'(P)} \rho_T'(t)dt \wedge \eta \wedge \left( \tilde{\psi}_1(n) - f  \pi^{\ast}\phi_{0,1}(n)\right). 
\end{multline}
It remains to compute the second term in the previous equation. For $T$ sufficiently large we have $f \equiv 1$. Furthermore by Theorem~\ref{psitilderes} and its proof we have $\tilde{\psi}_1(n) = \pi^{\ast} \tilde{\psi}_{0,1}(n) + O(e^{-Ct})$. As
$\phi_{0,1}(n) = \tilde{\psi}_{0,1}(n)+\tilde{\psi}'_{0,1}(n)$, we can replace  
$\tilde{\psi}_1(n) - f  \pi^{\ast}\phi_{0,1}(n)$ by $-\pi^{\ast} \tilde{\psi'}_{0,1}(n)$. Since $\eta$ is special it does not depend on the $t$-variable near the boundary. For the last term in \eqref{eq2}
\[
 \lim_{T\to \infty} \int_T^{T+1}  \rho_T'(t)dt \int_{e'(P)} \eta \wedge \pi^{\ast} \tilde{\psi'}_{0,1}(n) = -  \int_{e'(P)} \eta \wedge \tilde{\psi'}_{0,1}(n) = -
\left(\int_{A_n} \eta \right) e^{-2\pi n}.
\]
Indeed, for $\eta = \Omega$ this is Remark~\ref{youwillneedthis}. Otherwise, $\eta$ is exact with special primitive $\omega$, and it is not hard to see that the proof of Proposition~\ref{finalintegral} carries over to this situation. Since $C_n^c = C_n \coprod (-A_n)$ collecting all terms completes the proof of Theorem~\ref{newcurrenteq}. 








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