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

TL;DR

This paper presents an algebraic construction of the holomorphic symplectic structure on the moduli space of projective structures on a surface, clarifying its algebraic nature beyond the monodromy map.

Contribution

It introduces a new algebraic construction of the symplectic form on the moduli space, demonstrating its algebraic property.

Findings

The symplectic form on the moduli space is algebraic.

The new construction bypasses the non-algebraic monodromy map.

The approach clarifies the algebraic structure of the symplectic form.

Abstract

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.