Rational curves, Dynkin diagrams and Fano manifolds with nef tangent bundle

A Fano manifold X with nef tangent bundle is of Flag-Type if it has the same kind of elementary contractions as a complete flag manifold. In this paper we present a method to associate a Dynkin diagram D(X) with any such X, based on the numerical properties of its contractions. We then show that D(X) is the Dynkin diagram of a semisimple Lie group. As an application we prove that Campana–Peternell conjecture holds when X is a Flag-Type manifold whose Dynkin diagram is An, i.e. we show that X is the variety of complete flags of linear subspaces in Pn.