The generalized Mukai conjecture on Picard number and pseudoindex of Fano manifolds

Theorem 1.1: every smooth connected complex projective Fano manifold of positive dimension $n$ has $\rho_X(\iota_X-1)\le n$, with equality if and only if $X\cong(\mathbb P^{\iota_X-1})^{\rho_X}$. Consequently Mukai's original index version also holds. The paper treats smooth complex Fano manifolds only: singular ($\mathbb Q$-factorial Gorenstein) Fano varieties beyond the known toric, horospherical and spherical cases, and positive characteristic, are not covered.

Theorem 1.1: every smooth connected complex projective Fano manifold of positive dimension $n$ has $\rho_X(\iota_X-1)\le n$, with equality if and only if $X\cong(\mathbb P^{\iota_X-1})^{\rho_X}$. Consequently Mukai's original index version also holds. The paper treats smooth complex Fano manifolds only: singular ($\mathbb Q$-factorial Gorenstein) Fano varieties beyond the known toric, horospherical and spherical cases, and positive characteristic, are not covered.