Extremal Kähler metrics on projectivized vector bundles

Abstract

We prove the existence of extremal, nonconstant–scalar curvature, Kähler metrics on certain unstable projectivized vector bundles $\mathbb{P}\left(E\right)\to M$ over a compact constant–scalar curvature Kähler manifold $M$ with discrete holomorphic automorphism group, in certain adiabatic Kähler classes. In particular, the vector bundles $E\to M$ 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, for example, $\mu \left(E_1\right)>\cdots >\mu \left(E_s\right)$.