2003/01/13 by Gregery T. Buzzard, Buzzard, Gregery T., Sergei Merenkov +1
Mathematics · #32H02 #32H50 #Advanced Topics in Algebra #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Holomorphic and Operator Theory #Homotopy and Cohomology in Algebraic Topology #math.CV #math.DS #msc:32H02 #msc:32H50
paper · pdf · doi:10.48550/arxiv.math/0301120
10 pages
openalex publication_date 2003/01/13 · arxiv created 2003/11/10 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
If f is a bijection from Cn onto a complex manifold M, which conjugates every holomorphic map in Cn to an endomorphism in M, then we prove that f is necessarily biholomorphic or antibiholomorphic. This extends a result of A. Hinkkanen to higher dimensions. As a corollary, we prove that if there is an epimorphism from the semigroup of all holomorphic endomorphisms of Cn to the semigroup of holomorphic endomorphisms in M, or an epimorphism in the opposite direction for a doubly-transitive M, then it is given by conjugation by some biholomorphic or antibiholomorphic map. We show also that there are two unbounded domains in Cn with isomorphic endomorphism semigroups but which are neither biholomorphically nor antibiholomorphically equivalent.