On simplicity and stability of tangent bundles of rational homogeneous varieties

TL;DR

This paper proves that tangent bundles of rational homogeneous varieties of type ADE are simple and stable, using quiver representations and the Hitchin-Kobayashi correspondence, advancing understanding of their geometric and algebraic properties.

Contribution

It establishes the simplicity and stability of tangent bundles on G/P for type ADE, linking homogeneous vector bundles to quiver representations.

Findings

All tangent bundles T_{G/P} are simple.

Tangent bundles are stable with respect to the anticanonical polarization.

Utilizes quiver category equivalence to prove main results.

Abstract

Given a rational homogeneous variety G/P where G is complex simple and of type ADE, we prove that all tangent bundles T_{G/P} are simple, meaning that their only endomorphisms are scalar multiples of the identity. This result combined with Hitchin-Kobayashi correspondence implies stability of these tangent bundles with respect to the anticanonical polarization. Our main tool is the equivalence of categories between homogeneous vector bundles on G/P and finite dimensional representations of a given quiver with relations.