Uniform bundles on generalised Grassmannians

Abstract

Let $E$ be a uniform bundle on an arbitrary generalised Grassmannian $X$ defined over $\mathbb{C}$. We show that if the rank of $E$ is at most $e.d.(\mathrm{VMRT})$, then $E$ necessarily splits. For some generalised Grassmannians, we prove that the upper bounds $e.d.(\mathrm{VMRT})$ are optimal and classify all unsplit uniform bundles of minimal ranks. Under some special assumptions, we show that morphisms to some generalised flag varieties must be constant, which partially answered a conjecture of Kumar.