On the symplectic structure over the moduli space of projective structures on a surface

The moduli space of projective structures on a compact oriented surface $\Sigma$ has a holomorphic symplectic structure, which is constructed by pulling back, using the monodromy map, the Atiyah--Bott--Goldman symplectic form on the character variety ${\rm Hom}(\pi_1(\Sigma), \text{PSL}(2, {\mathbb C}))/\!\!/\text{PSL}(2, {\mathbb C})$. We produce another construction of this symplectic form. This construction shows that the symplectic form on the moduli space is actually algebraic. Note that the monodromy map is only holomorphic and not algebraic, so the first construction does not give algebraicity of the pulled back form.