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!


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.