Ampleness equivalence and dominance for vector bundles

Abstract

Hartshorne in "Ample vector bundles" proved that $E$ is ample if and only if $\OOO_{P(E)}(1)$ is ample. Here we generalize this result to flag manifolds associated to a vector bundle $E$ on a complex 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 $ \SSS_aE $ is ample. In particular $\det Q$ on $G_r(E)$ is ample if and only if ${\wedge }^{r}E$ 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.