The complex symplectic geometry of the deformation space of complex projective structures

TL;DR

This paper explores the complex symplectic geometry of deformation spaces of complex projective structures on surfaces, comparing different symplectic forms and extending known results in the field.

Contribution

It clarifies and generalizes the relationship between Schwarzian and Goldman symplectic structures, and extends results on quasi-Fuchsian reciprocity and Fenchel-Nielsen coordinates.

Findings

Comparison of Schwarzian and Goldman symplectic structures

Generalization of quasi-Fuchsian reciprocity

Description of symplectic geometry in Hamiltonian form

Abstract

This article investigates the complex symplectic geometry of the deformation space of complex projective structures on a closed oriented surface of genus at least 2. The cotangent symplectic structure given by the Schwarzian parametrization is studied carefully and compared to the Goldman symplectic structure on the character variety, clarifying and generalizing a theorem of S. Kawai. Generalizations of results of C. McMullen are derived, notably quasi-Fuchsian reciprocity. The symplectic geometry is also described in a Hamiltonian setting with the complex Fenchel-Nielsen coordinates on quasi-Fuchsian space, recovering results of I. Platis.