Simplicity of tangent bundles of smooth horospherical varieties of Picard number one

TL;DR

This paper proves that the tangent bundle of smooth horospherical Fano varieties with Picard number one is simple, using the existence of special rational curves, thus supporting a weaker stability conjecture.

Contribution

It establishes the simplicity of tangent bundles for a class of horospherical varieties, advancing understanding of their geometric properties.

Findings

Tangent bundle of these varieties is simple.

Existence of a family of unbendable rational curves.

Supports the weaker conjecture on tangent bundle simplicity.

Abstract

Recently, Kanemitsu has discovered a counterexample to the long-standing conjecture that the tangent bundle of a Fano manifold of Picard number one is (semi)stable. His counterexample is a smooth horospherical variety. There is a weaker conjecture that the tangent bundle of a Fano manifold of Picard number one is simple. We prove that this weaker conjecture is valid for smooth horospherical varieties of Picard number one. Our proof follows from the existence of an irreducible family of unbendable rational curves whose tangent vectors span the tangent spaces of the horospherical variety at general points.