Question on period map, Gauss-Manin connection and complex coordinates of $\mathcal{H}^1(k)$

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?

• What about asking the authors ? – Thomas Jun 16 '16 at 16:21
• it's a standard construction, they're not the only ones doing it – user0029 Jun 16 '16 at 16:22

Yes, $(\int_{\tau_1}\omega,...,\int_{\tau_{2g+\sigma-1}}\omega)$ are the complex coordinates 'given by $\Theta$'. In fact, the authors prove (using the Moser trick) that these integrals are locally injective. To prove that $\Theta$ maps to an open subset in $\mathbb C^{2g+\sigma-1}$ one observes that there exist closed complex-valued $\alpha_1,..,\alpha_{2g+\sigma-1}$ 1-forms such that they vanish identically on a neighborhood of the given set of zeros, and such that $(\int_{\tau_j}\alpha_i)$ is invertible.
Coordinate changes are then given by either changing the basis of $H^1(M,\Sigma(\omega_0),\mathbb Z)$ or by changing $\Sigma(\omega_0)$, i.e. the Riemann surface and the abelian differential.
• could you explain why you say "To prove that $\Theta$ maps to an open set in $\mathbb{C}^{2g+\sigma-1}$ one observes that there exist closed complex forms which vanish identically on a neighborhood of the zeroes? Why is the condition of vanishing in a neighborhood of the zeroes important? This also seems very relevant to this question mathoverflow.net/questions/309498/… – User28341 Sep 4 '18 at 21:13
• It can be used to define a Riemann surface structure: Take $\omega+t\alpha_k$ for $t\in\mathbb C$ small enough such that the zeros of $\omega+t\alpha_k$ are the same as the zeros of $\omega$. Then, away from the zeros of $\omega+t\alpha_k$, this closed form is the differential of a local diffeomorphism onto $\mathbb C$, which can be used to define a new Riemann surface structure on $M\setminus\{\text{ zeros of } \omega\}$.Near the zeros of $\omega$, we can use the old complex structure (for which $\omega$ is holomorphic), and the vanishing of $\alpha_k$ near the zeros shows compatibility. – Sebastian Sep 5 '18 at 10:01
• Thank you! So, if I understand correctly, the purpose of vanishing in a neighborhood of the zeroes is that then each complex differential $\omega+t\alpha_k$ admits a complex structure with respect to which $\omega+t\alpha_k$ is an Abelian differential. Am I right to say that this proves that the Period map $\Theta:U_0\rightarrow H^1(M,\Sigma(\omega_0),\mathbb{C})$ is surjective? – User28341 Sep 5 '18 at 14:39