Abstract

In this talk, I will discuss a correspondence between the problem of finding a constant scalar curvature K\"ahler metric (cscK for short) on a polarized complex manifold and the CR-Yamabe problem on the associated circle bundle. I will then use a CR version of the Einstein-Hilbert functional to show K-semistability of polarised complex manifolds admitting cscK metrics. This is a joint work with Eveline Legendre and Carlo Scarpa.