# Question on local coordinates in the moduli space $\mathcal{H}_g(1)$

Fix $g\ge 2$ and denote by $\mathcal{H}_g(1)$ the moduli space of holomorphic 1-forms on a closed Riemann surface of genus $g$ with one zero of order $2g-2$. Given any point $\omega\in \mathcal{H}_g(1)$, the corresponding tangent space $T_{\omega}\mathcal{H}_g(1)$ can be identified with the cohomology group $H^1(X,Z(\omega),\mathbb{C})$, where $X$ is the Riemann surface where $\omega$ is holomorphic and $Z(\omega)$ is the zero of $\omega$.

Let $B=\{\phi_1,\dots,\phi_{2g}\}$ be a basis of closed complex forms on $X$ which vanish in a neighborhood of $Z(\varphi)$, from what we just said one can deduce that each $\phi_i$ represents a tangent direction to $\mathcal{H}_g(1)$ in $\omega$.

It is possible to endow $\mathcal{H}_g(p)$ of an hermitian metric $h$ which on each tangent space $T_{\omega}\mathcal{H}_g(1)$ is defined as $h_{\omega}(\phi,\psi):=\frac i 2 \int_X\phi\wedge \overline \psi$ for every $\phi,\psi \in T_\omega\mathcal{H}_g(1)$.

Since the local period maps give local coordinates on $\mathcal{H}_g(1)$, we can write all elements of $\mathcal{H}_g(1)$ sufficiently close to $\omega$ as $\omega_z:=\omega+ \sum_{i=1}^{2g}z_i\phi_i$ and so $z=(z_1,\dots,z_{2g})\in \mathbb{C}^{2g}$ are complex coordinates near $\omega$.

I would like to write down explicitly the hermitian metric $h$ in these coordinates $z$, i.e. I would like to compute the coefficients $h_{ij}(\omega_z)$ of $h=\sum_{i,j}h_{ij}(\omega_z)dz_idz_j$. I've tried to do so, and I obtain constant coefficients $h_{ij}(\omega_z)=\int_X\phi_i\wedge \overline{\phi_j}$, since the wedge product is the same in each point $\omega_z$.

Am I right to say that the coefficients $h_{ij}(\omega_z)$ are constant? If not, can you point me out where I'm wrong?

Thank you!