Uniform Vector Bundles on Fano Manifolds and an Algebraic Proof of Hwang-Mok Characterization of Grassmannians

Abstract

A projective manifold X is called Fano if its anticanonical divisor − K x is ample. Fano manifolds form a very distinguished class: in each dimension there is only a finite number of deformation classes of them and they are classified in dimension ≤ 3, the case dim X = 3 due to Fano, Roth, Iskovskih and Shokurov. In dimension ≥ 4 not much is known about Fano manifolds in general. However, due to results of Mori, Kawamata and Shokurov, Fano manifolds with Picard number ρ(X) bigger than 1 admit special morphisms, called Fano-Mori contractions, which can be used to study the structure of such Fano’s. The case ρ(X) = 1 seems to be harder to approach, see [IP] for an overview on Fano varieties.