Sur une conjecture de Mukai

Abstract

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

21 pages, in french