Extremal K\"ahler metrics on projectivised vector bundles

TL;DR

This paper demonstrates the existence of extremal K"ahler metrics on certain unstable projectivised vector bundles over cscK-manifolds, expanding the understanding of geometric structures on complex fiber bundles.

Contribution

It constructs extremal, non-constant scalar curvature K"ahler metrics on specific unstable bundles with split stable subbundles and different slopes, in certain K"ahler classes.

Findings

Existence of extremal K"ahler metrics on unstable bundles.

Metrics are non-constant scalar curvature and extremal.

Applicable to bundles with split stable subbundles of different slopes.

Abstract

We prove the existence of extremal, non-csc, K\"ahler metrics on certain unstable projectivised vector bundles $\P (E) \to M$ over a cscK-manifold $M$ with discrete holomorphic automorphism group, in certain adiabatic K\"ahler classes. In particular, the vector bundles $E \to M$ under consideration are assumed to split as a direct sum of stable subbundles $E=E_1 \oplus \dots \oplus E_s$ all having different Mumford-Takemoto-slope, e.g. $\mu(E_1) > \dots > \mu(E_s)$.