Threshold rank for homogeneity of uniform bundles on generalized Grassmannians

Determine the largest integer k(X), depending only on a generalized Grassmannian X, such that every uniform vector bundle on X of rank at most k(X) is homogeneous.

Background

The paper studies the relationship between uniform and homogeneous vector bundles on generalized Grassmannians. Although every homogeneous bundle is uniform, the converse fails in general, and the rank at which non-homogeneous uniform bundles first occur depends on the underlying variety.

The stated problem asks for the precise threshold k(X) for each generalized Grassmannian X. The paper establishes the upper bound k(X)≤dim X−1 when X is not a projective space and determines the value for odd-dimensional quadrics, but the problem is posed in general for arbitrary generalized Grassmannians.