Balanced metrics and chow stability of projective bundles over Riemann surfaces

TL;DR

This paper extends Morrison's 1980 result by proving that slope stability of higher rank vector bundles over compact Riemann surfaces implies Chow stability of their projectivizations for certain polarizations, under specific geometric conditions.

Contribution

It generalizes Morrison's theorem to higher rank bundles over manifolds with constant scalar curvature and discrete automorphisms, providing a simplified proof for specific polarizations.

Findings

Slope stability implies Chow stability for higher rank bundles.

The proof applies to projectivizations over Riemann surfaces of genus ≥ 2.

Results hold for polarizations with large tensor powers.

Abstract

In 1980, I. Morrison proved that slope stability of a vector bundle of rank 2 over a compact Riemann surface implies Chow stability of the projectivization of the bundle with respect to certain polarizations. We generalized Morrison's result to higher rank vector bundles over compact algebraic manifolds of arbitrary dimension that admit constant scalar curvature metric and have discrete automorphism group. In this article, we give a simple proof for polarizations $\mathcal{O}_{\mathbb{P}E^*}(d)\otimes \pi^* L^k$, where $d$ is a positive integer, $k \gg 0$ and the base manifold is a compact Riemann surface of genus $g \geq 2$.