Real algebraic surfaces biholomorphically equivalent but not algebraically equivalent

2025/03/11 by Rond, Guillaume
#05A15 #14P05 #32C05 #32H02 #32V40 #39B32 #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.2503.08957

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 Cⁿ. More precisely we give an example of two germs of real algebraic surfaces in \mathbb C² that are biholomorphic, but not via an algebraic biholomorphism. In fact we even prove that the components of any biholomorphism between these two surfaces are never solutions of polynomial differential equations. The proof is based on enumerative combinatorics and differential Galois Theory results concerning the nature of the generating series of walks restricted to the quarter plane.

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