Abstract：

The weighted extremal Kähler metrics introduced by Lahdili provide a vast generalisation of Calabi's extremal Kähler metrics, encompassing many examples of canonical metrics in geometry. In this talk, I will give a quick introduction to these metrics, and discuss the proof that weighted extremal manifolds are relatively weighted K-polystable, in a suitable sense. The proof is along the lines of Stoppa--Szekelyhidi's argument that an extremal polarised manifold is relatively K-polystable, which in particular exploits the existence of an extremal metric on the blowup of an extremal manifold at a suitable point.