Ampleness equivalence and dominance for vector bundles

Abstract

Hartshorne in “Ample vector bundles” proved that E is ample if and only if ${\mathcal O}_{P(E)}(1)$ is ample. Here we generalize this result to flag manifolds associated to a vector bundle E on a complex projective manifold X: For a partition a we show that the line bundle $Q_a^s$ on the corresponding flag manifold $\mathcal {F}l_s(E)$ is ample if and only if ${\mathcal S}_aE$ is ample. In particular $\det Q$ on ${G}_r(E)$ is ample if and only if $\wedge ^rE$ is ample. We give also a proof of the Ampleness Dominance theorem that does not depend on the saturation property of the Littlewood–Richardson semigroup.