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

Abstract

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 $(\mathbb{P}{E}^{\ast },{\mathcal{O}}_{\mathbb{P}{E}^{\ast }}(1)\otimes {\pi }^{\ast }{L}^k)$ 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 ${\text{Sym}}^{d}E$, we generalize the main theorem of \cite{S} to polarizations $(\mathbb{P}{E}^{\ast },{\mathcal{O}}_{\mathbb{P}{E}^{\ast }}(d)\otimes {\pi }^{\ast }{L}^k)$ for $k\gg 0$, where $d$ is a positive integer.