We prove the existence of extremal, nonconstant–scalar curvature, Kähler metrics on certain unstable projectivized vector bundles over a compact constant–scalar curvature Kähler manifold with discrete holomorphic automorphism group, in certain adiabatic Kähler classes. In particular, the vector bundles are assumed to split as a direct sum of stable subbundles all having different Mumford–Takemoto slope, for example, .
"Extremal Kähler metrics on projectivized vector bundles." Duke Math. J. 164 (2) 195 - 233, 1 February 2015. https://doi.org/10.1215/00127094-2860166