Uniform families of minimal rational curves on Fano manifolds

TL;DR

This paper explores how strong uniformity conditions on minimal rational curves on Fano manifolds of Picard number one can imply the homogeneity of the manifold, extending known results about rational homogeneous manifolds.

Contribution

It establishes that certain strong uniformity conditions on minimal rational curves imply the homogeneity of Fano manifolds of Picard number one.

Findings

Strong uniformity conditions lead to manifold homogeneity

Extension of uniformity results from rational homogeneous manifolds

Provides criteria for homogeneity based on rational curves

Abstract

It is a well-known fact that families of minimal rational curves on rational homogeneous manifolds of Picard number one are uniform, in the sense that the tangent bundle to the manifold has the same splitting type on each curve of the family. In this note we prove that certain --stronger-- uniformity conditions on a family of minimal rational curves on a Fano manifold of Picard number one allow to prove that the manifold is homogeneous.