A Note on Chow Stability of the Projectivization of Gieseker Stable Bundles

We investigate Chow stability of projective bundles ℙ(E), where E is a strictly Gieseker stable bundle over a base manifold that has constant scalar curvature. We show that, for suitable polarizations \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{L}$\end{document}, the pair \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(\mathbb{P}(E),\mathcal{L})$\end{document} is Chow stable and give examples for which it is not asymptotically Chow stable.