Balanced metrics and Chow stability of projective bundles over Riemann surfaces

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. In a previous work, we generalized Morrison’s result to higher rank vector bundles over compact algebraic manifolds of arbitrary dimension that admit a constant scalar curvature metric and have a 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 $g \geq 2$.