Abstract : Smooth, complex Fano 4-folds are not classified, and we still lack a good understanding of their general properties. We focus on Fano 4-folds with large second Betti number $b_{2}$, studied via birational geometry and the detailed analysis of their contractions and rational contractions (we recall that a contraction is a morphism with connected fibers onto a normal projective variety, and a rational contraction is given by a sequence of flips followed by a contraction). The main result that we want to present is the following: let $X$ be a Fano 4-fold having a nonconstant rational contraction $X --> Y$ of fiber type. Then either $b_{2}(X)$ is at most 18, with equality only for a product of surfaces, or $Y$ is $\mathbb{P}^{1}$ or $\mathbb{P}^{2}$. The proof is achieved by reducing to the case of "special" rational contractions of fiber type. We will explain this notion and give an idea of the techniques that are used.

Keywords : Fano varieties; 4-folds; birational geometry