# Explicit local expression for Bers embedding in genus 2

Let $\mathcal T_{g,n}$ be the Teichmüller space of genus g compact Riemann surfaces with $n$ marked points. According to Riemann, this is a complex manifold of complex dimension 3g-3+n.

Bers constructed a map $B_{g,n}: \mathcal T_{g,n}\rightarrow \mathbb C^{3g-3+n}$ that is an holomorphic embedding.

Let's focus on the case when $g=2$.

It is well known that there is an isomorphism $\mathcal T_2\simeq \mathcal T_{0,6}$ and that $\mathcal T_{0,6}$ is nothing but the universal covering of the moduli space $\mathcal M_{0,6}$ of projective configurations of 6 points on $\mathbb P^1$. The later is isomorphic to the set of $(z_i)_{i=1}^{n-3}\in (\mathbb C\setminus \{0,1\})^{n-3}$ such that $z_i\neq z_j$ for $i\neq j$. From this it follows that one can consider the $z_i$'s as local holomorphic coordinates on $\mathcal T_2\simeq \mathcal T_{0,6}$.

Question: what is the local expression of $B_{0,6}$ in the coordinates $z_i$'s?

On the other hand, the period map $\tau: \mathcal T_2\rightarrow \mathcal H_2$ is holomorphic and injective and allows to see $\mathcal T_2$ as a dense open subset of Siegel upper half-plane $\mathcal H_2$ (I hope this is correct! A precise reference would be welcome!).

(I wouldn't be surprised if such a local expression can be found in classical papers or books on Teichmüller theory...)

Question: how are related Bers embedding $B_2:\mathcal T_2\rightarrow \mathbb C^3$ and the period map $\tau: \mathcal T_2\rightarrow \mathcal H_2\subset \mathbb C^3$?

(Again, it is certainly very classical but I didn't find it in the books I looked at)

Thanks for any help on these (certainly elementary) questions on Teichmüller theory.

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I do not understand what you are asking for. Do you want to see "explicit solution" of Schwartzian differential equation in the case if 6 poles? There is none. The second question is a bit better, but it is unclear what kind if relation you are asking beyond the definition since the two maps are completely different. –  Misha Dec 18 '13 at 16:38