Sur une conjecture de Mukai

Generalizing a question of Mukai, we conjecture that a Fano manifold $X$ with Picard number ${\rho }_X$ and pseudo-index ${\iota }_X$ satisfies $\rho_X (\iota_X-1) \le \dim(X)$\mathrm{.}Weprovethisinequalityinseveralsituations:$X$ is a Fano manifold of dimension $\le 4$, $X$ is a toric Fano manifold of dimension $\le 7$or$X$isatoricFanomanifoldofarbitrarydimensionwith$\iota_X \ge \dim(X)/3+1$. Finally, we offer a new approach to the general case.