Uniform families of minimal rational curves on Fano manifolds
Gianluca Occhetta, Luis E. Sol\'a Conde, Kiwamu Watanabe

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.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Geometry and complex manifolds · Geometric Analysis and Curvature Flows
