2012/11/05 by Alexander Hanysz, Hanysz, Alexander · 1 citation
Mathematics · #32Q28 (Primary) 32H02 #32Q55 #54C35 #58D15 (Secondary) #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #math.CV #msc:32H02 #msc:32Q28 #msc:32Q55 #msc:54C35 #msc:58D15
paper · pdf · doi:10.48550/arxiv.1211.0765
37 pages. Version 2 with minor corrections
openalex publication_date 2012/11/05 · arxiv created 2012/11/12 · arxiv updated 2012/11/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For each natural number d, the space Rd of rational maps of degree d on the Riemann sphere has the structure of a complex manifold. The topology of these manifolds has been extensively studied. The recent development of Oka theory raises some new and interesting questions about their complex structure. We apply geometric invariant theory to the cases of degree 2 and 3, studying a double action of the Möbius group on Rd. The action on R2 is transitive, implying that R2 is an Oka manifold. The action on R3 has C as a categorical quotient; we give an explicit formula for the quotient map and describe its structure in some detail. We also show that R3 enjoys the holomorphic flexibility properties of strong dominability and C-connectedness.