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Balanced metrics and chow stability of projective bundles over Riemann surfaces

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 $\mathcal{O}_{\mathbb{P}E^*}(d)\otimes π^* L^k$, where $d$ is a positive integer, $k \gg 0$ and the base manifold is a compact Riemann surface of genus $g \geq 2$.