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

TL;DR

This paper links Fano manifolds with nef tangent bundles to Dynkin diagrams, showing these diagrams correspond to semisimple Lie groups and applying this to prove the Campana-Peternell conjecture for certain flag-type manifolds.

Contribution

It introduces a method to associate Dynkin diagrams to Fano manifolds with nef tangent bundles and proves their correspondence to semisimple Lie groups.

Findings

Dynkin diagrams can be associated with Fano manifolds with nef tangent bundle.

These diagrams correspond to semisimple Lie groups.

The Campana-Peternell conjecture is proved for flag-type manifolds with Dynkin diagram A_n.

Abstract

A Fano manifold $X$ with nef tangent bundle is of flag-type if it has the same type 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 $A_n$ ($X$ is shown to be the complete flag in $\mathbb{P}^n$).