We consider a uniform r-bundle E on a complex rational homogeneous space X 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 unstable with respect to some numerical class of a line. So we partially answer a problem posted by Muñoz et al. (Eur J Math 6:430-452, 2020). In particular, if X is a generalized Grassmannian 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. So we improve the main theorem of Muñoz et al. (J Reine Angew Math (Crelles J) 664:141-162, 2012, Theorem 3.1) when X is a generalized Grassmannian. Moreover, by calculating the relative tangent bundles between two rational homogeneous spaces, we give explicit bounds for the generalized Grauert-Mülich-Barth theorem on rational homogeneous spaces.

中文翻译：

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有理齐次空间上的向量丛。

我们考虑复有理齐次空间 X 上的均匀 r-束 E，并证明如果 E 对于所有特殊的线族是多均匀的，并且秩 r 小于或等于某个仅取决于 X 的数，那么 E 要么是线束的直接和，要么是关于线的某个数值类的不稳定。因此，我们部分回答了 Muñoz 等人发布的问题。(Eur J Math 6:430-452, 2020)。特别是，如果 X 是广义格拉斯曼 G 并且秩 r 小于或等于某个仅取决于 X 的数，则 E 分裂为线束的直接和。因此，我们改进了 Muñoz 等人的主要定理。(J Reine Angew Math (Crelles J) 664:141-162, 2012, Theorem 3.1) 当 X 是广义的 Grassmannian 时。此外，通过计算两个有理齐次空间之间的相对切丛，