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

Abstract

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 \(\mathcal {D}(X)\) with any such \(X\), based on the numerical properties of its contractions. We then show that \(\mathcal {D}(X)\) 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 \(A_n\), i.e. we show that \(X\) is the variety of complete flags of linear subspaces in \(\mathbb {P}^n\).