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

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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: do there exist fake planes whose real locus is not birationally diffeomorphic to the real affine plane?

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Correspondence to Frédéric Mangolte.

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This project was partially funded by ANR Grant “BirPol” ANR-11-JS01-004-01.

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Dubouloz, A., Mangolte, F. Fake real planes: exotic affine algebraic models of \(\mathbb {R}^{2}\) . Sel. Math. New Ser. 23, 1619–1668 (2017). https://doi.org/10.1007/s00029-017-0326-6

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