ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

4. Algebraic Geometry

1991 MS Classification: 30F60 Huisman, Johannes, Institut Mathematique de Rennes, Universite de Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France Uniformization of real algebraic curves and complex analytic coordinates on Teichmüller spaces Let X be a complete nonsingular real algebraic curve of genus g ≥ 2. Let Σ be the Galois group of ℂ/ℝ and let 𝔻 be the double half-plane ℂ\ℝ. There is a Kleinian subgroup G of PGL_{2}(ℝ) acting freely on 𝔻 such that X(ℂ) and 𝔻/G are Σ-equivariantly biholomorphic. This is uniformization of the real algebraic curve X. Its interest lies in the fact that the group G is the equivariant fundamental group of X(ℂ). If the real algebraic curve X does not have any real points, the group G has a particularly simple presentation. This leads to a nice and new construction of a global system of complex analytic coordinates on the Teichmüller space of complex algebraic curves of genus g.