In this talk, we construct Kaehler metrics with constant scalar curvature on the projectivisation of certain holomorphic vector bundles. When the vector bundle is slope-stable and the base admits a constant scalar curvature metric, it is a classical result of Hong that the total space of the projectivisation admits a constant scalar curvature metric in adiabatic classes. We extend their result to slope-semistable vector bundles: we show that if E -> B is slope-semistable and the total space of the projectivisation is K-polystable then it admits a constant scalar curvature metric in adiabatic classes.
This is joint work with L.M. Sektnan.
