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

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

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

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

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.