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The complex-symplectic geometry of \(\text{SL}(2,\mathbb{C})\)-characters over surfaces. (English) Zbl 1089.53060
Dani, S. G. (ed.) et al., Algebraic groups and arithmetic. Proceedings of the international conference, Mumbai, India, December 17–22, 2001. New Delhi: Narosa Publishing House/Published for the Tata Institute of Fundamental Research (ISBN 81-7319-618-4/hbk). 375-407 (2004).
This is a survey paper on the \(\text{SL}(2, {\mathbb C})\)-character variety of a closed surface \(M\) from the view point of complex-symplectic structure. There is a natural complex-symplectic structure on it which commutes with the action of the mapping class group \(\Gamma\). Let \(\pi\) be the fundamental group of \(M\), and \(\operatorname{Hom}(\pi, \text{SL}(2, {\mathbb C}))^s\) be the set of irreducible homomorphisms which are stable points of \(\operatorname{Hom}(\pi, \text{SL}(2, {\mathbb C}))\). The group \(\text{SL}(2, {\mathbb C})\) acts freely and properly on \(\operatorname{Hom}(\pi, \text{SL}(2, {\mathbb C}))^s\) and the quotient \(X= \operatorname{Hom}(\pi, \text{SL}(2, {\mathbb C}))^s / SL(2, {\mathbb C})\) is a \((6g-6)\)-dimensional complex manifold, where \(g\) is the genus of \(M\).
This paper treats many topics. For example, using ergodicity of the \(\Gamma\) action, it is shown that any \(\Gamma\) invariant meromorphic function \(X \to {\mathbb C}P^1\) is constant. From the view point of Hamiltonian dynamics, for each free homotopy class \(\alpha\) of closed curves on \(M\), a complex regular function \(f_{\alpha}: X \to {\mathbb C}\) is given. The natural complex-symplectic structure associates to these functions complex Hamiltonian vector fields Ham\((f_{\alpha})\). From the study of such vector fields passing through Poincare duality, it is shown that \(f_{\alpha}\) and \(f_{\beta}\) are mutually Poisson-commuting, whenever \(\alpha\) and \(\beta\) are disjoint. Periods and the relations with Dehn twists of these Hamiltonian vector fields are also studied. There is also a detailed account of deformation spaces of \({\mathbb C}P^1\) structures.
For the entire collection see [Zbl 1067.00014].

53D35 Global theory of symplectic and contact manifolds