Fake real planes: exotic affine algebraic models of \(\mathbb {R}^{2}\). (English) Zbl 1487.14131
Summary: 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 \(\mathbb {RP}^{2}\) 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?
MSC:
| 14R05 | Classification of affine varieties |
| 14E05 | Rational and birational maps |
| 14J26 | Rational and ruled surfaces |
| 14P05 | Real algebraic sets |
| 14R10 | Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem) |
| 14R25 | Affine fibrations |
Keywords:
real algebraic model; affine surface; rational fibration; birational diffeomorphism; affine complexificationReferences:
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