Géométrie algébrique réelle de certaines variétés de dimension 2 et 3

Abstract : The results presented here deal with the geometry and the topology of real algebraic varieties. A large part is devoted to surfaces (2-dimensional varieties). We present also a new program about 3-dimensional varieties. The selected results are organized in three directions:

1. Algebraic cycles on surfaces.

2. Topology of real algebraic varieties.

3. Approximation of smooth maps by regular maps.

1. We study the group of homology classes representable by real algebraic curves. This group is a geometric invariant which is an important tool, for example for some approximation problems. In a serie of papers, one of which with J. van Hamel, we achieved the classification of totally algebraic surfaces among all special surfaces.

2. The systematic study of the topology of real algebraic varieties was initiated in 1900 by D. Hilbert in the XVIth problem of his famous list. The main result here is the proof, with J. Huisman, of a conjecture of J. Kollar:
Every orientable Seifert 3-manifold is a real component of a uniruled algebraic variety.
This is an important step towards a classification of real algebraic uniruled threefolds.

3. Let X, Y be two nonsingular real algebraic varieties, X compact, the question is: "when is the space of regular maps R(X,Y) dense in the space of smooth maps C(X,Y) ?" (Cf. The Stone-Weierstrass Theorem when Y=R).
Here Y is the usual sphere. In a serie of two papers, one of which with with N. Joglar, we achieved the case when X is a surface of strictly negative Kodaira dimension: when X is homeomorphic to a torus, the only approximable maps are homotopically trivial, in all other cases with X connected, we have density.
An amazing fact is the existence of a unique intermediate rational case between triviality and density which is a real Del Pezzo surface of degree 2 with 4 connected components.
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Frédéric Mangolte. Géométrie algébrique réelle de certaines variétés de dimension 2 et 3. Mathématiques [math]. Université de Savoie, 2004. ⟨tel-00006900⟩

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