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

Abstract : We study real rational models of the euclidean affine 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, $\mathbb{P}^{2}(\mathbb{R})$ is the only model. A fake real plane is a smooth geometrically integral surface $S$ defined over $\mathbb{R}$ not isomorphic to $\mathbb{A}^2_\mathbb{R}$, whose real locus $S(\mathbb{R})$ is diffeomorphic to $\mathbb{R}^2$ and such that the complex surface $S_\mathbb{C}(\mathbb{C})$ has the rational homology type of $\mathbb{A}^2_\mathbb{C}$. We prove that fake planes exist by giving many examples and we tackle the question: does there exist fake planes $S$ such that $S(\mathbb{R})$ is not birationally diffeomorphic to $\mathbb{A}^2_\mathbb{R}(\mathbb{R})$?
Keywords :
Type de document :
Pré-publication, Document de travail
2015

https://hal.archives-ouvertes.fr/hal-01175154
Soumis le : vendredi 10 juillet 2015 - 13:34:07
Dernière modification le : lundi 27 février 2017 - 16:50:50

### Identifiants

• HAL Id : hal-01175154, version 1
• ARXIV : 1507.01574

### Citation

Adrien Dubouloz, Frédéric Mangolte. Fake Real Planes: exotic affine algebraic models of $\mathbb{R}^{2}$. 2015. <hal-01175154>

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