The Complex-symplectic Geometry of Sl(2,c)-characters over Surfaces

The SL(2,C)-character variety X of a closed surface M enjoys a natural complex-symplectic structure invariant under the mapping class group Γ of M . Using the ergodicity of Γ on the SU(2)-character variety, we deduce that every Γ-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 Teichmüller space and the complex quakebend flows on quasiFuchsian space. Closed curves in the complex trajectories of these flows lift to paths in the deformation space CP(M) of complexprojective structures between different CP-structures with the same holonomy (grafting). If P is a pants decomposition, then the trace map τP : X −→ C defines a holomorphic completely integrable system. Furthermore, if ΓP is the subgroup of Γ preserving P , then every ΓP-invariant holomorphic function X −→ C factors through τP . This holomorphic integrable system is related to the complex Fenchel-Nielsen coordinates on quasi-Fuchsian space QF(M) developed by Tan and Kourouniotis, and relate to recent formulas of Platis and Series on complex-length functions and complex twist flows on QF(M).