Extremal Kähler metrics on modifications

I will present the invariance of extremal Kahler manifolds under a suitable class of bimeromorphic morphisms.

Focusing mostly on the cscK (constant scalar curvature Kähler) case, I will show how the main result is obtained as a consequence of a general uniform coercivity estimate for the Mabuchi energies of perturbed Kähler classes. I will then explain how such variational approach applies more generally to the class of weighted extremal metrics, modulo a log-concavity assumption on the first weight, and to any equivariant resolution of singularities of Fano type of a compact Kähler klt space whose weighted Mabuchi energy is assumed to be coercive.

This is a joint work with S. Boucksom and M. Jonsson.