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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].

##### MSC:
 53D35 Global theory of symplectic and contact manifolds