Topology of real algebraic three-dimensional manifolds

We know since Nash (1952) and Tognoli (1973) that any compact smooth manifold M admits a real algebraic model. Namely, given a manifold M, there exists polynomials with real coefficients whose locus of common real zeroes is diffeomorphic to M. Bochnak and Kucharz proved later that there exists in fact an infinite number of distinct models for a given M. We try therefore to find “simpler” algebraic models than the others in a meaning to be specified. In this talk, I will describe the state-of-the-art concerning this research program about “simple” real algebraic models for low-dimensional varieties.
The situation for curves and surfaces is quite well understood now, and the surface case is already interesting. For real algebraic threefolds, János Kollár opened in 1999 a direction of research thanks to his solution of Minimal Model Program over the reals. We will discuss several Kollár’s conjectures solved since then.