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\title{Hirzebruch's Proportionality}
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\begin{document}
\maketitle

\vspace{1cm}
\Large
(References)
\\ \null \\
F. Hirzebruch, {\it Characteristic numbers of homogeneous domains}, (1957). 
\\ \null \\
F. Hirzebruch, {\it Automorphic formen und der Satz von Riemann-Roch}, (1958).
\newpage
\noindent
1. Let $X$ be a complex manufold of dimension $n$.
The total Chern class of $X$ is given by 
\begin{equation*}
c(X) = 1 + c_1 + c_2 + \cdots + c_n \cong det( I + \frac{\sqrt{-1}}{2 \pi} \Theta),
\end{equation*}
where $\Theta$ is the curvature matrix of $X$, i.e.,\\
$\Theta$ is a local expression of the curvature $R$ of $X$.\\
\vspace{1cm}

\noindent
2. Let $X$ be compact. \\
For any partition $\ld = ( i_1 , i_2 , \cdots, i_r )$ of $n$, a Chern number $c_{\ld} (X) \in H^{2n} (X, \Z)$ is defined by the relation
\begin{equation*}
c_{\ld} \cdot h_X = c_{i_1} c_{i_2} \cdots c_{i_r}, 
\end{equation*}
where $h_X$ is the generator of $H^{2n} (X, \Z)$. \\
\vspace{1cm}

\noindent
3. Two compact complex manifolds $X$ and $\tilde{X}$ of dimension $n$ are called {\it proportional} if for any two partitions $\ld$ and $\mu$ of $n$, the following proportionality holds:
\begin{equation*}
\frac{c_{\ld} (X)}{c_{\mu} (X)} = \frac{c_{\ld} (\tilde{X})}{c_{\mu} (\tilde{X})}.
\end{equation*}
\newpage

For example, any two compact manifolds of dimension $n$ are proportional if there is a local diffeomorphism.\\
\vspace{0.5cm}

\noindent
$X$: noncompact irreducible Hermitian symmetric space,  \\
$\tilde{X}$: the compact dual Hermitian symmetric space, \\
$\Gamma$: discrete subgroup of $Aut(X)$ acting on $X$ without fixed points. \\
( $\Rightarrow$ $\Gamma \backslash X$ is a compact manifold of dimension $n$.)
\\ \null \\
{\rm \bf \underline{Theorem}} (Hirzebruch) \hspace{0.5cm} $\tilde{X}$ and $\Gamma \backslash X$ are proportional. \\
\vspace{1cm}

\underline{Compact irreducible Hermitian symmetric spaces}  
\\ \null \\
\noindent
${\bf SU} (p+q)/{\bf S}( {\bf U} (p) \times {\bf U} (q))$, Grassmannians  \\ 
$ {\bf SO} (2n)/ {\bf U} (n)$, maximal orthogonal Grassmannians \\
$ {\bf SP} (n)/ {\bf U} (n)$,  Lagrangian Grassmannians \\
$ {\bf SO} (n+2)/( {\bf SO} (2) \times {\bf SO} (n))$, quadric hypersurfaces \\
$ E_6 / Spin(10) \times T^1$ \\
$ E_7 / E_6 \times T^1$
\newpage

Hirzebruch's proportionality gives information on the Chern numbers of the compact quotients of bounded symmetric domains:
\begin{equation*}
\frac{c_{\ld} (\Gamma \backslash X)}{c_n (\Gamma \backslash X)} = \frac{c_{\ld} (\tilde{X})}{c_n (\tilde{X})},
\end{equation*}
hence 
\begin{equation*}
c_{\ld} (\Gamma \backslash X) = \frac{\chi (\Gamma \backslash X)}{\chi({\tilde{X}})} c_{\ld} (\tilde{X}).
\end{equation*}
\\ \null \\
\noindent
Let $\bP = \bP (\C)$, which is dual to the ball ${\rm \bf B}^2$. Then,
\begin{equation*}
c(\bP) = (1 + H)^3 = 1 + 3H + 3(H^2), 
\end{equation*}
where $H$ is the hyperplane of $\bP$ which generates $H^2 (\bP , \Z)$.\\
Hence $c_1^2 = 3c_2$ and this relation holds for any surface 
given by $M = \Gamma \backslash {\bf B}^2$ by Hirzebruch's proportionality. 
\\ \null \\
{\rm \bf \underline{Theorem}} (Yau, 1977) \hspace{0.5cm} For a K\"{a}hler surface $M$ whose canonical line bundle is ample,\\
(i) $3c_2 \ge c_1^2$. \\
(ii) The equality holds if and only if $M$ is covered by ${\bf B}^2$.\\
Higher dimensional analogue.
\newpage

(Sketch of proof)
\\ \null \\
\noindent
$X$:  bounded symmetric domain $\subset \C$ \\
$\tilde{X}$: compact dual manifold \\
$\Gamma \backslash X$: compact quotient 
\vspace{1cm}

\noindent
1. \underline{$X$ and $\Gamma \backslash X$}.
\\ \null \\
We must define the Chern numbers of $X$ (noncompact).\\
The Bergmann metric of $X$ is given by 
\begin{equation*}
ds^2 = \sum \frac{\partial^2 logK}{\partial z_{\alpha} \partial \bar{z_{\beta}}} dz_{\alpha} \otimes \bar{dz_{\beta}} 
= \sum g_{\alpha \bar{\beta}} dz_{\alpha} \otimes \bar{dz_{\beta}},
\end{equation*}
where $K$ is the Bergmann's kernel function.\\
We have the associates closed (1,1)-form 
\begin{equation*}
\omega = \frac{\sqrt{-1}}{2 \pi} \sum  g_{\alpha \bar{\beta}} dz_{\alpha} \wedge \bar{dz_{\beta}}
\end{equation*}
and $\omega^n / n!$ is the real volume element.
\newpage

For any partition $\ld = ( i_1 , i_2 , \cdots, i_r )$ of $n$, the $2n$-form $c_{i_1} c_{i_2} \cdots c_{i_r}$ is a multiple of the volume element  $\omega^n / n!$. The Chern number $c_{\ld}$ of $X$ is defined by the relation 
\begin{equation*}
c_{\ld} \cdot \frac{\omega^n}{n!} = (-1)^n c_{i_1} c_{i_2} \cdots c_{i_r}. 
\end{equation*}

Since the covering map $p : X \to \Gamma \backslash X$ is a local isometry, we get 
\begin{equation*}
p^* (c_i (\Gamma \backslash X)) = c_i (X), \ \forall \ i.
 \end{equation*}
Hence it is clear that $X$ and $\Gamma \backslash X$ are proportional.
\vspace{1.5cm}

2. \underline{$\tilde{X}$ and $X$}.
\\ \null \\
$\tilde{X} = G/H$, \\
$\frak{g}$ and $\frak{h}$: Lie algebras of $G$ and $H$. \\
Cartan decomposition $\frak{g} = \frak{h}+\frak{m}$, where $\frak{m} = T_e X$. \\
$C(G)$ : the complex extension of $G$ \\
\indent with Lie algebra $\frak{c} = \frak{h} + i \frak{h} + \frak{m} + i \frak{m}$. \\
$\tilde{G} \le C(G)$ : the Lie subgroup \\
\indent corresponding to $\tilde{\frak{g}} = \frak{h} + i \frak{m} \subset \frak{c}$. \\

The space $\tilde{X} = \tilde{G} /H$ is just the compact dual of $X$. \\
We get the isomorphism of the tangent spaces:
\begin{eqnarray*}
T_e X = \frak{m} & \cong & i \frak{m} = T_{\tilde{e}} \tilde{X}, \\
v & \mapsto & iv. 
\end{eqnarray*}

For symmetric spaces $X$ and $\tilde{X}$, the curvature tensor is given by 
\begin{equation*}
R(x,y)z = - [[x,y],z], \ \null \  x,y,z \in \frak{m} \ ({\rm or} \ i \frak{m}),
\end{equation*}
hence 
\begin{equation*}
\tilde{R} (ix, iy) (iz) = i [[x,y],z] = -i R(x,y)z.
\end{equation*} 
Recall that the total Chern class is given by 
\begin{equation*}
c(X) = det( I + \frac{\sqrt{-1}}{2 \pi} R).
\end{equation*}
These two formulas shows that $X$ and $\tilde{X}$ are proportional.
\\ \null \\
\noindent
From 1. and 2., we conclude that $\tilde{X}$ and $\Gamma \backslash X$ are proportional. $\Box$


\end{document}
