Transcendental holomorphic maps between real algebraic manifolds in a complex space

Abstract

We give an example of a real algebraic manifold embedded in a complex space that does not satisfy the Nash-Artin approximation Property. This Nash-Artin approximation Property is closely related to the problem of determining when the biholomorphic equivalence for germs of real algebraic manifolds coincides with the algebraic equivalence. This example is an elliptic Bishop surface, and its construction is based on the functional equation satisfied by the generating series of some walks restricted to the quarter plane.2010 Mathematics Subject Classification. Primary: 32H02, Secondary: 05A15, 14P05, 32C05, 32V40, 39B32.Key words and phrases. holomorphic map, algebraic map, Bishop surface, lattice walk. The author is deeply grateful to the UMI LASOL of the CNRS where this project has been carried out.