# Real frontiers of fake planes

* Corresponding author
Abstract : In Dubouloz and Mangolte (Fake real planes: exotic affine algebraic models of $\mathbb{R}^{2}$, arXiv:1507.01574, 2015) we define and partially classify fake real planes, that is, minimal complex surfaces with conjugation whose real locus is diffeomorphic to the euclidean real plane $\mathbb{R}^{2}$. Classification results are given up to biregular isomorphisms and up to birational diffeomorphisms. In this note, we describe in an elementary way numerous examples of fake real planes and we exhibit examples of such planes of every Kodaira dimension $\kappa\in \{-\infty,0,1,2\}$ which are birationally diffeomorphic to $\mathbb{R}^{2}$.
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https://hal.archives-ouvertes.fr/hal-01188470
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Submitted on : Friday, June 10, 2022 - 2:17:51 PM
Last modification on : Wednesday, June 22, 2022 - 3:41:54 AM

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### Citation

Adrien Dubouloz, Frédéric Mangolte. Real frontiers of fake planes. European Journal of Mathematics, Springer, 2016, Special Issue: Spitsbergen Volume, 2 (1), pp.140-168. ⟨10.1007/s40879-015-0087-8⟩. ⟨hal-01188470v2⟩

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