A central question in Kähler geometry is if a Kähler manifold admits a canonical metric, such as a Kähler-Einstein metric or more generally a constant scalar curvature Kähler (cscK) metric, in a given Kähler class. The Yau-Tian-Donaldson conjecture predicts that this is equivalent to an algebraic notion of stability.
In this talk, I will discuss a necessary and sufficient condition for the projectivisation of a slope semistable vector bundle to admit cscK metrics in adiabatic classes, when the base admits a cscK metric. In particular, this shows that the existence of cscK metrics is equivalent to K-stability in this setting. Moreover, our construction reduces K-stability to a finite dimensional criterion in terms of intersection numbers associated to the vector bundle. This is joint work with Annamaria Ortu.