Balanced Metrics and Chow Stability of Projective Bundles over Kähler Manifolds II

Abstract

In the previous article (Seyyedali, Duke Math. J. 153(3):573–605, 2010), we proved that slope stability of a holomorphic vector bundle E over a polarized manifold (X,L) implies Chow stability of $(\mathbb{P}E^{*},\mathcal{O}_{\mathbb{P}E^{*}}(1)\otimes\pi^{*} L^{k})$ for k≫0 if the base manifold has no nontrivial holomorphic vector field and admits a constant scalar curvature metric in the class of 2πc 1(L). In this article, using asymptotic expansions of the Bergman kernel on Sym d E, we generalize the main theorem of Seyyedali (Duke Math. J. 153(3):573–605, 2010) to polarizations $(\mathbb{P}E^{*},\mathcal {O}_{\mathbb{P}E^{*}}(d)\otimes\pi^{*} L^{k})$ for k≫0, where d is a positive integer.