Constant scalar curvature K\"ahler metrics and semistable vector bundles

We give a necessary and sufficient condition for the projectivisation of a slope semistable vector bundle to admit constant scalar curvature K\"ahler (cscK) metrics in adiabatic classes, when the base admits a constant scalar curvature metric. More precisely, we introduce a stability condition on vector bundles, which we call adiabatic slope stability, which is a weaker version of K-stability and involves only test configurations arising from subsheaves of the bundle. We prove that, for a simple vector bundle with locally free graded object, adiabatic slope stability is equivalent to the existence of cscK metrics on the projectivisation, which solves a problem that has been open since work of Ross--Thomas. In particular, this shows that the existence of cscK metrics is equivalent to K-stability in this setting. We provide a numerical criterion for the Donaldson-Futaki invariant associated to said test configurations in terms of Chern classes of the vector bundle. This criterion is computable in practice and we present an explicit example satisfying our assumptions which is coming from a vector bundle that does not admit a Hermite-Einstein metric.