Extremal Kähler metrics on projectivised vector bundles

We prove the existence of extremal, non-csc, Kähler metrics on certain unstable projectivised vector bundles $¶(E) \to M$ over a cscK-manifold $M$ with discrete holomorphic automorphism group, in certain adiabatic Kähler 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. $μ(E_1) > \dots > μ(E_s)$.

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