Vector Bundles on Rational Homogeneous Spaces

TL;DR

This paper studies vector bundles on rational homogeneous spaces, showing under certain conditions they split into line bundles or are unstable, with implications for the structure of bundles on generalized Grassmannians.

Contribution

It provides new splitting criteria for vector bundles on rational homogeneous spaces, extending previous results and offering explicit bounds using Chow rings and tangent bundle calculations.

Findings

Bundles split as direct sums of line bundles under certain conditions.

Improved bounds for the Grauert-Mülich-Barth theorem on these spaces.

Explicit calculations of relative tangent bundles.

Abstract

We consider a uniform $r$-bundle $E$ on a complex rational homogeneous space $X$ %over complex number field $\mathbb{C}$ and show that if $E$ is poly-uniform with respect to all the special families of lines and the rank $r$ is less than or equal to some number that depends only on $X$, then $E$ is either a direct sum of line bundles or $\delta_i$-unstable for some $\delta_i$. So we partially answer a problem posted by Mu\~{n}oz-Occhetta-Sol\'{a} Conde. In particular, if $X$ is a generalized Grassmannian $\mathcal{G}$ and the rank $r$ is less than or equal to some number that depends only on $X$, then $E$ splits as a direct sum of line bundles. We improve the main theorem of Mu\~{n}oz-Occhetta-Sol\'{a} Conde when $X$ is a generalized Grassmannian by considering the Chow rings. Moreover, by calculating the relative tangent bundles between two rational homogeneous spaces, we give explicit bounds for the generalized Grauert-M\"{u}lich-Barth theorem on rational homogeneous spaces.