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) <= dim(X). We prove this inequality in several situations: X is a Fano manifold of dimension <= 4, X is a toric Fano manifold of dimension <= 7 or X is a toric Fano manifold of arbitrary dimension with ${\iota }_X$ >= dim(X) / 3 + 1. Finally, we offer a new approach to the general case.