Uniform Bundles on Generalized Grassmannians

Abstract Let $E$ be a uniform bundle on an arbitrary generalized Grassmannian $X$ defined over $\mathbb{C}$. We show that if the rank of $E$ is at most the so-called effective good divisibility of the variety of minimal rational tangents ($e.d.(\textrm{VMRT})$ for short), then $E$ necessarily splits. For some generalized Grassmannians, we prove that the upper bounds $e.d.(\textrm{VMRT})$ are optimal and classify all unsplit uniform bundles of minimal ranks. Under some special assumptions, we show that morphisms to some generalized flag varieties must be constant, which partially answers a conjecture of Kumar.