Real algebraic surfaces biholomorphically equivalent but not algebraically equivalent

Guillaume Rond

Real algebraic surfaces biholomorphically equivalent but not algebraically equivalent

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arxiv 2503.08957 v4 pith:ZEAQDDYY submitted 2025-03-11 math.CV math.AGmath.CO

Guillaume Rond This is my paper · ORCID

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keywords algebraicrealbiholomorphicequivalenceequivalentgermsmathbbsurfaces

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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.

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