Real algebraic surfaces biholomorphically equivalent but not algebraically equivalent

Abstract

We answer in the negative the long-standing open question of whether biholomorphic equivalence implies algebraic equivalence for germs of real algebraic manifolds in ${\mathbb{C}}^n$. More precisely we give an example of two germs of real algebraic surfaces in ${\mathbb{C}}^2$ that are biholomorphic, but not by means of an algebraic biholomorphism.