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

TL;DR

This paper extends previous results linking slope stability and Chow stability of projective bundles over Kähler manifolds by using Bergman kernel asymptotics to handle more general polarizations involving higher tensor powers.

Contribution

It generalizes the main theorem to include polarizations with higher tensor powers, broadening the scope of stability implications for projective bundles.

Findings

Established asymptotic expansion formulas for Bergman kernels on symmetric powers of vector bundles.

Proved that slope stability implies Chow stability for a wider class of polarizations.

Extended stability results to cases with higher tensor powers of the tautological line bundle.

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 $(\mathbb{P}E^*,\mathcal{O}_{\mathbb{P}E^*}(1)\otimes \pi^* L^k)$ 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.