About projectivisation of Mumford semistable bundles over a curve

TL;DR

This paper investigates the existence and characterization of canonical Kähler metrics on the projectivization of Mumford semistable bundles over complex curves, linking differential geometry with algebraic geometry.

Contribution

It provides new insights into when such canonical metrics exist and offers an algebro-geometric characterization of these metrics for Mumford semistable bundles.

Findings

Existence criteria for canonical Kähler metrics on projectivized bundles.

Algebro-geometric characterization of these metrics.

Connections between stability conditions and metric properties.

Abstract

We study the existence of canonical K\"ahler metrics on the projectivisation of strictly Mumford semistable holomorphic vector bundles over a complex curve. We also provide an algebro-geometric characterization of these metrics.