Teichmüller's lemma and the complex structure of Teichmüller spaces

Consider the Fricke homeomorphism $ F:T(X) \to {\mathbb{R}}^{6g-6+2n}$ (given by Fricke coordinates on Teichmüller space) and let $ \widetilde{F}:
B(L^\infty_{(-1,1)}(X)) \to {\mathbb{R}}^{6g-6+2n}$ be a lift of $ F$.

Theorem 2.24 (Teichmüller's Lemma)   The set

$\displaystyle A_2(X)^\perp = \left\{\nu\in L^\infty_{(-1,1)}(X)_1; \int_X\nu \varphi
= 0, \mathrm{for all }\varphi\in A_2(X) \right\}
$

is the kernel of $ d_0\widetilde{F}$, the differential of $ \widetilde{F}$ at the origin $ 0 \in L^\infty_{(-1,1)}$.

This theorem can be used to give a complex structure to the Teichmüller space $ T(X)$. For that we need some general facts on mappings between complex and real manifolds, and when such mappings induce complex structures (on the real manifold). Let $ f:X\to Y$ be a $ C^1$ submersion from a complex Banach manifold $ X$ onto a real $ C^1$ manifold $ Y$. We say that $ f$ induces a well-defined almost complex structure on $ Y$ if for all $ y\in Y$ the space $ T_yY$ inherits via the differential $ d_xf$ a unique complex structure, independent of the choice of $ x \in f^{-1}(y)$.

Theorem 2.25   Let $ f:X\to Y$ be a surjective $ C^1$ submersion from a complex manifold $ X$ onto a $ 2d$-dimensional $ C^1$ manifold $ Y$. Then there is a complex structure on $ Y$, compatible with the $ C^1$ structure, making $ f$ holomorphic if and only if $ f$ induces a well-defined almost complex structure on $ Y$. The complex structure, if it exists,is unique.

We can apply the above result to the mapping $ \widetilde{F} = F\circ\Phi$, define from the unit ball of $ L^\infty({\mathbb{H}},G)$ to $ Im(F) \subset
{\mathbb{R}}^{6g-6+2n}$, where $ F$ is the Fricke mapping. To check that $ \widetilde{F}$ induces a well-defined almost complex structure one needs to use the right-translation mappings, $ R_\theta$, from the unit ball of $ L^\infty({\mathbb{H}},G)$ to itself, defined by

\begin{displaymath}
\begin{split}
R_\theta: B(L^\infty({\mathbb{H}},G)) & \to B(...
...m{complex dilatation of }(w_\lambda\circ
w_\theta).
\end{split}\end{displaymath}

Theorem 2.26   For a Riemann surface $ X$ of type $ (g,n)$, with $ 2g-2+n > 0$, there is a unique complex structure of a $ (3g-3+n$)-dimensional complex manifold on $ Im(F)$ such that it is compatible with the real-analytic structure of the open domain $ Im(F) \subset
{\mathbb{R}}^{6g-6+2n}$ and that makes $ \widetilde{F}$ a holomorphic submersion.

The details of the proof of this result can be found in [39, Chapter 3].

The Fricke homeomorphism $ F:T(X) \to Im(F)$ gives then a complex structure to Teichmüller space $ T(X)$. The mapping $ F$ becomes biholomorphic and the projection $ \Phi$ is then a holomorphic submersion.

The discontinuous action of the modular group on Teichmüller space gives us some information about the moduli space of surfaces.

Theorem 2.27   The moduli space $ M(g,n)$ of compact surfaces of type $ (g,n)$ ( $ 2g-2+n > 0$) is a normal complex space of dimension $ 3g-3+n$. The space $ M(g,0)$ is simply connected.

Pablo Ares Gastesi 2005-08-31