Frederic Mangolte

Abstract: We study topologically minimal complexifications of the Euclidean affine plane R² up to isomorphism and up to birational diffeomorphism. A fake real plane is a smooth geometrically integral surface S defined over R such that: The real locus S(R) is diffeomorphic to R^2; The complex surface S_C(C) has the rational homology type of A^2_C.; S is not isomorphic to A^2_R as surfaces defined over R.  The analogous study in the compact case, that is th classification of complexifications of the real projective plane P^2(R) with the rational homology of the complex projective plane is well known: P^2_C is the only one. We prove that fake real planes exist by giving many examples and we tackle the question: does there exist fake planes S such that S(R) is not birationally diffeomorphic to A^2_R(R)? (Joint work with Adrien Dubouloz.)

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