Extremal Kähler metrics on projectivised vector bundles

Abstract

We prove the existence of extremal, non-csc, Kähler metrics on certain unstable projectivised vector bundles $\P (E)\to M$ 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 $E=E_1\oplus \cdots \oplus E_s$ all having different Mumford-Takemoto-slope, e.g. $\mu (E_1)>\cdots >\mu (E_s)$.

39 pages

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