The complex-symplectic geometry of SL(2,C)-characters over surfaces

TL;DR

This paper explores the complex-symplectic geometry of SL(2,C)-character varieties over surfaces, revealing invariant functions, Hamiltonian flows, and integrable systems linked to surface decompositions and complex projective structures.

Contribution

It introduces a holomorphic integrable system on the character variety and connects complex Hamiltonian flows to known structures like Fenchel-Nielsen coordinates.

Findings

Invariant meromorphic functions are constant under the mapping class group.

Complex Hamiltonian flows generalize Fenchel-Nielsen and quakebend flows.

A pants decomposition yields a holomorphic integrable system.

Abstract

The SL(2)-character variety X of a closed surface M enjoys a natural complex-symplectic structure invariant under the mapping class group G of M. Using the ergodicity of G on the SU(2)-character variety, we deduce that every G-invariant meromorphic function on X is constant. The trace functions of closed curves on M determine regular functions which generate complex Hamiltonian flows. For simple closed curves, these complex Hamiltonian flows arise from holomorphic flows on the representation variety generalizing the Fenchel-Nielsen twist flows on Teichmueller space and the complex quakebend flows on quasi-Fuchsian space. Closed curves in the complex trajectories of these flows lift to paths in the deformation space of complex-projective structures between different projective structures with the same holonomy (grafting). A pants decomposition determines a holomorphic completely integrable system on X. This integrable system is related to the complex Fenchel-Nielsen coordinates on quasi-Fuchsian space developed by Tan and Kourouniotis, and relate to recent formulas of Platis and Series on complex-length functions and complex twist flows.