Let $\mathcal{L}_g$ be the space of abelian differentials on Riemann surfaces of genus $g\ge 2$ and $\mathcal{TH}_g:=\mathcal{L}_g/Diff_0^+(S_g)$ be the Teichmuller space of abelian differentials on Riemann surfaces as it is defined in this article http://arxiv.org/pdf/1311.2758v1.pdf by Giovanni Forni and Carlos Matheus. Let $\mathcal{TH}(k)$ be a stratum of $\mathcal{TH}_g$.

At page 20 they state: "It is possible to prove that every $\omega_0\in \mathcal{TH}(k)$ has an open neighborhood $U_0\subset\mathcal{TH}(k)$ such that, after identifying, for all $\omega\in U_0$, the cohomology $H^1(M,\Sigma(\omega),\mathbb{C})$ with the fixed complex vector space $H^1(M,\Sigma(\omega_0),\mathbb{C})$ via the Gauss-Manin connection (i.e., through identification of the integer lattices $H^1(M,\Sigma(\omega),\mathbb{Z}+i\mathbb{Z})$ and $H^1(M,\Sigma(\omega_0),\mathbb{Z}+i\mathbb{Z})$), the period map $\Theta : U_0 \rightarrow H^1(M,\Sigma(\omega_0),\mathbb{C})$ that is, the map defined by the formula

$\Theta(\omega) :=( \gamma \mapsto\int_\gamma\omega )\in Hom(H^1(M, \Sigma(\omega), \mathbb{Z}); \mathbb{C}) \simeq H^1(M, \Sigma(\omega), \mathbb{C}) \simeq H^1(M, \Sigma(\omega_0), \mathbb{C})$ ,

is a local homeomorphism."

In fact in the next page they also state that the period map $\Theta$ gives $\mathcal{TH}(k)$ the structure of a complex manifold.

My question is really silly, but **I'm not sure which are the complex coordinates given by $\Theta$.**

My guess is this: fix $\omega_0$ and $U_0$. Then choose a basis $\{\tau_1,\dots,\tau_{2g+\sigma-1}\}$ of $H_1(M,\Sigma(\omega_0),\mathbb{Z})$. Then for every $\omega\in U_0$, $\Theta(\omega)=(\int_{\tau_1}\omega,\dots,\int_{\tau_{2g+\sigma-1}}\omega)\in\mathbb{C}^{2g+\sigma-1}$. So the complex coordinates are $(\int_{\tau_1}\omega,\dots,\int_{\tau_{2g+\sigma-1}}\omega)$ and a change of atlas consists in a change of basis of $H_1(M,\Sigma(\omega_0),\mathbb{Z})$.

**Am I right?**