Balanced Metrics and Chow Stability of Projective Bundles over K\"ahler Manifolds II

In the previous article (\cite{S}), 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 $k \gg 0$ if the base manifold has no nontrivial holomorphic vector field and admits a constant scalar curvature metric in the class of $2\pi c_{1}(L)$. In this article using asymptotic expansions of Bergman kernel on $\textrm{Sym}^d E$, we generalize the main theorem of \cite{S} to polarizations $(\mathbb{P}E^*,\mathcal{O}_{\mathbb{P}E^*}(d)\otimes \pi^* L^k)$ for $k \gg 0$, where $d$ is a positive integer.