# Ampleness equivalence and dominance for vector bundles

## TL;DR

This paper extends Hartshorne's ampleness criterion from vector bundles to flag manifolds, establishing equivalences for ampleness of associated line bundles and vector bundle dominance, with a new proof of the Ampleness Dominance theorem.

## Contribution

It generalizes ampleness equivalence to flag manifolds and provides a novel proof of the Ampleness Dominance theorem independent of Littlewood-Richardson saturation.

## Key findings

- Ampleness of line bundles on flag manifolds corresponds to ampleness of associated vector bundles.
- The determinant line bundle on Grassmannians is ample if and only if the wedge power of the bundle is ample.
- A new proof of the Ampleness Dominance theorem that does not rely on saturation properties.

## 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 $\it Q_a^s$ 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 $\it{G}_r(E)$ 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.