About canonical Kahler metrics on Mumford semistable projective bundles over a curve

We study the existence of canonical Kahler metrics on the projectivization of strictly Mumford semistable vector bundles over a curve. We show the existence of almost balanced and almost constant scalar curvature Kahler metrics for such ruled manifolds. The most canonical metric on such manifolds is actually a constant scalar curvature Kahler metric with conical singularities that appears from a natural construction of parabolic stable structure. Eventually, we address some extra results on the log-stability of the obtained ruled manifold.