Complex Symplectic Geometry of Quasi-Fuchsian Space IOANNIS D. PLATIS Department of Mathematics, University of Crete, GR 71409 Heraklion, Crete, Greece. e-mail: jplatis@math.uch.gr (Received: November 1999; in ¢nal form: 8 June 2000) Abstract. We study the complex symplectic geometry of the space QFS of quasi-Fuchsian structures of a compact orientable surface S of genus g > 1: We prove that QFS is a complex symplectic manifold. The complex symplectic structure is the complexi¢cation of the Weil^Petersson symplectic structure of Teichmu« ller space and is described in terms which look natural from the point of view of hyperbolic geometry. Mathematics Subject Classi¢cation (2000). 30F40. Key words. quasi-Fuchsian space, complex symplectic. 1. Introduction and Statement of Results Let S be a compact orientable smooth surface of negative Euler characteristic. In this article we study the complex symplectic geometry of the Quasi-Fuchsian space QFS of S: This space is the set of equivalence classes of quasi-Fuchsian representations of p 1 S into PSL2; C: Images of such representations are quasi-Fuchsian groups: Such a group G is quasi-conformally congruent to a Fuchsian group, its limit set is a Jordan curve, and if D G is the region of discontinuity of G in the complex plane C and U 3 is the upper semispace, then the quasi-Fuchsian 3-manifold M G U 3 [ D G =G is diffeomorphic to S 0; 1: QFS can be viewed as the space of marked quasi-Fuchsian manifolds, where a marking of M G is a choice of isomorphism between p 1 S and p 1 M G : Quasi-Fuchsian space is a complex manifold of complex dimension 6g  6; [B] and the Teichmu«ller space TeichS of S; the space of marked Riemann surfaces, can be regarded as the 6g  6 real submanifold FS of QFS consisting of equivalence classes of Fuchsian representations. Complex holomorphic coordinates for QFS are provided by the complex length functions which correspond to certain simple closed geodesics on S [K3, T]. These are the extension to Quasi-Fuchsian space of the geodesic length functions, which provide real analytic (Fenchel^Nielsen) coordinates for TeichS [W3]. Teichmu«ller space also admits a natural complex structure and endowed with the Weil^Petersson Geometriae Dedicata 87: 17^34, 2001. 17 # 2001 Kluwer Academic Publishers. Printed in the Netherlands.