# Relative K-polystability of projective bundles over a curve

## TL;DR

This paper characterizes when a projective bundle over a curve admits an extremal Kähler metric, linking geometric stability to the bundle's decomposition into stable components.

## Contribution

It provides a criterion connecting the existence of extremal Kähler metrics on projective bundles with their relative K-polystability and bundle decomposition.

## Key findings

- Existence of extremal Kähler metrics is equivalent to relative K-polystability.
- Bundle decomposes into a direct sum of stable bundles when extremal metrics exist.
- Characterization applies to projectivizations over complex curves.

## Abstract

Let $P(E)$ be the projectivization of a holomorphic vector bundle $E$ over a compact complex curve $C$. We characterize the existence of an extremal K\"ahler metric on the ruled manifold $P(E)$ in terms of relative K-polystability and the fact that $E$ decomposes as a direct sum of stable bundles.