# Fake real planes: exotic affine algebraic models of $\mathbb{R}^{2}$

Abstract : We study real rational models of the euclidean plane $\mathbb{R}^{2}$ up to isomorphisms and up to birational diffeomorphisms. The analogous study in the compact case, that is the classification of real rational models of the real projective plane $\mathbb{R}\mathbb{P}^{2}$ is well known: up to birational diffeomorphisms, there is only one model. A fake real plane is a nonsingular affine surface defined over the reals with homologically trivial complex locus and real locus diffeomorphic to $\mathbb{R}^2$ but which is not isomorphic to the real affine plane. We prove that fake planes exist by giving many examples and we tackle the question: does there exist fake planes whose real locus is not birationally diffeomorphic to the real affine plane?
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https://hal.archives-ouvertes.fr/hal-01175154
Submitted on : Friday, July 10, 2015 - 1:34:07 PM
Last modification on : Monday, March 9, 2020 - 6:15:58 PM

### Citation

Adrien Dubouloz, Frédéric Mangolte. Fake real planes: exotic affine algebraic models of $\mathbb{R}^{2}$. Selecta Mathematica (New Series), Springer Verlag, 2017, 23 (3), pp.1619 - 1668. ⟨10.1007/s00029-017-0326-6⟩. ⟨hal-01175154⟩

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