A note on Chow stability of the Projectivisation of Gieseker Stable Bundles

TL;DR

This paper studies the Chow stability of projective bundles over manifolds with constant scalar curvature, showing conditions for stability and providing examples where stability fails.

Contribution

It demonstrates Chow stability of projective bundles over certain manifolds and identifies cases where asymptotic Chow stability does not hold.

Findings

(P(E),L) is Chow stable for suitable polarisations L

Examples are provided where (P(E),L) is not asymptotically Chow stable

Conditions for stability depend on the properties of the bundle E and the base manifold

Abstract

We investigate Chow stability of projective bundles P(E) where E is a strictly Gieseker stable bundle over a base manifold that has constant scalar curvature. We show that, for suitable polarisations L, the pair (P(E),L) is Chow stable and give examples for which it is not asymptotically Chow stable.