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Selecta Mathematica

, Volume 23, Issue 3, pp 1619–1668 | Cite as

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

  • Adrien Dubouloz
  • Frédéric Mangolte
Article

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?

Keywords

Real algebraic model Affine surface Rational fibration Birational diffeomorphism Affine complexification 

Mathematics Subject Classification

14R05 14R25 14E05 14P25 14J26 

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Copyright information

© Springer International Publishing 2017

Authors and Affiliations

  1. 1.IMB UMR5584, CNRSUniv. Bourgogne Franche-ComtéDijonFrance
  2. 2.LAREMA UMR6093, CNRSUniv. Angers, Univ. Bretagne-LoireAngersFrance

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