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

Abstract

A Fano manifold (Formula presented.) 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 (Formula presented.) with any such (Formula presented.), based on the numerical properties of its contractions. We then show that (Formula presented.) is the Dynkin diagram of a semisimple Lie group. As an application we prove that Campana–Peternell conjecture holds when (Formula presented.) is a Flag-Type manifold whose Dynkin diagram is (Formula presented.), i.e. we show that (Formula presented.) is the variety of complete flags of linear subspaces in (Formula presented.).