ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

4. Algebraic Geometry

1991 MS Classification: 14, 30, 57 Natanzon Sergei, Independent University of Moscow, Russia Topological structure of spaces of holomorphic morphisms A holomorphic morphism is a triple (P̃, f, P), where P̃, P are compact Riemann surfaces and f : P̃ → P is a holomorphic map. Here (P̃_{1}, f_{1}, P_{1}) = (P̃_{2}, f_{2}, P_{2}) if there exist biholomorphic maps φ̃: P̃_{1} → P̃_{2} and φ : P_{1} → P_{2} such that φf_{1} = f_{2}φ̃. Morphisms (P̃_{1}, f_{1}, P_{1}) and (P̃_{2}, f_{2}, P_{2}) are topologically equivalent if there exist homeomorphisms φ̃: P̃_{1} → P̃_{2} and φ : P_{1} → P_{2} such that φf_{1} = f_{2}φ̃. Theorem. Let (P̃_{0}, f_{0}, P_{0}) be a holomorphic morphism such that 2g + n > 2, where g is the genus of P and n is the number of critical values of f_{0}. Consider the set H of all holomorphic morphisms (P̃, f, P) that are topologically equivalent to (P̃_{0}, f_{0}, P_{0}). Then H is homeomorphic to ℝ^{6g+2n−6}/Mod, where Mod is a discrete group.