Flag bundles on Fano manifolds

TL;DR

This paper proves that certain Fano manifolds with specific rational curve families are homogeneous, using flag bundle analysis and Grothendieck's theorem, extending understanding of their geometric structure.

Contribution

It establishes the homogeneity of Fano manifolds with particular rational curve configurations by analyzing flag bundles and minimal sections.

Findings

Fano manifolds with specified rational curves are homogeneous.

Flag bundles can be characterized via special sections and Grothendieck's theorem.

The approach links rational curve families to the global symmetry of the manifold.

Abstract

As an application of a recent characterization of complete flag manifolds as Fano manifolds having only ${\mathbb P}^1$-bundles as elementary contractions, we consider here the case of a Fano manifold $X$ of Picard number one supporting an unsplit family of rational curves whose subfamilies parametrizing curves through a fixed point are rational homogeneous, and we prove that $X$ is homogeneous. In order to do this, we first study minimal sections on flag bundles over the projective line, and discuss how Grothendieck's theorem on principal bundles allows us to describe a flag bundle upon some special sections.