Abstract. We classify Fano fivefolds of index two which are projectivization of rank two vector bundles over four dimensional manifolds.

We prove that Generalized Mukai Conjecture holds for Fano manifolds $X$ of pseudoindex $i_X \geq (dim X + 3)/3$. We also give different proofs of the conjecture for Fano fourfolds and fivefolds

No abstract

Abstract. We prove a relative version of the theorem of Cho, Miyaoka and Shepherd-Barron : a Mori fibre space of maximal length is birational to a projective bundle.