Hermitian form on $H^1(X,\mathbb{C})$ and on the moduli space of holomorphic differentials Given any Riemann surface $X$, on $H^1(X,\mathbb{C})$ it is possible to define the hermitian form $h$
$$h(\omega,\tau):=\int_X\omega\wedge* \overline{\tau}$$
for every $\omega,\tau\in H^1(X,\mathbb{C})$, where $*$ is the Hodge product of $X$. 
This form can then be extended to an hermitian form on the moduli space of abelian differentials with one zero, since its tangent space in every point is $H^1(X,\mathbb{C})$.
I have two questions:
1) Does this hermitian form have a "name"? What is it commonly referred to as?
2) Can you give me some references where I could read more about this hermitian form?
 A: It's actually more useful to ignore the Hodge product here. (To have a Hodge product we need to fix a Kahler metric, and I assume you're taking "the" canonical one on a given Riemann surface; the one of constant curvature.) Without the Hodge product, this just becomes
$$
\langle \sigma, \tau \rangle = \int_X \sigma \cup \overline \tau.
$$
This is a perfectly fine inner product on $H^1(X,\mathbb C)$, and it respects the Hodge decomposition of the cohomology group.
On a general compact complex manifold $X$ of dimension $n$, we get "the same" inner product
$$
\langle \sigma, \tau \rangle = \int_X \sigma \cup \overline \tau,
$$
where this time $\sigma, \tau$ are elements of $H^n(X,\mathbb C)$. We can use this inner product when considering complex deformations of $X$ over a smooth base $S$ to prove that the Gauss-Manin connection on the vector bundle
$$
E^n \to S, \quad E^n_s = H^n(X_s, \mathbb C)
$$
is not only flat, but Hermitian flat.
Now, since this inner product again respects the Hodge decomposition of the cohomology group (if $X$ is Kahler), it also defines an inner product also on $H^{n,0}(X, \mathbb C)$. This in turn gives a smooth Hermitian metric on the holomorphic subbundle
$$
E^{n,0} \to S, \quad E^{n,0}_s = H^{n,0}(X_s, \mathbb C)
$$
of $E^n \to S$. General complex differential geometry then tells us that the Hermitian metric on this bundle is non-positively curved.
This last bit can be inspected further, and in some cases (like when $c_1(X) = 0$) yields a smooth Kahler metric on the base of deformations $S$, called the Weil-Peterson metric. That metric has been investigated quite a lot, and is often used to prove things like that the moduli space of some manifolds is hyperbolic.
For more about these kinds of things, you can have a look at Tian's construction of the Weil-Peterson metric in Smoothness of the universal deformation space of compact Calabi-Yau manifolds
and its Petersson-Weil metric, or Griffith's series of papers on Periods of Integrals on Algebraic Manifolds, and the citation trail leading out from either one.
