Resumen de Fake real planes: exotic affine algebraic models of R2
Adrien Dubouloz, Frédéric Mangolte
We study real rational models of the euclidean plane R2 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 RP2 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 R2 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?
