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

TL;DR

This paper investigates exotic affine algebraic surfaces called fake real planes, which resemble the real affine plane topologically but are not isomorphic to it, providing examples and exploring their properties.

Contribution

The paper constructs multiple examples of fake real planes and examines whether their real loci can differ birationally from the standard affine plane.

Findings

Existence of fake real planes demonstrated through explicit examples.

Some fake planes have real loci not birationally diffeomorphic to the affine plane.

The study advances understanding of real algebraic surfaces with unusual topological and algebraic properties.

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?