| Résume | The Yau–Tian–Donaldson conjecture predicts that the existence of an extremal metric (in the sense of Calabi) in a given Kähler class of Kähler manifold is equivalent to a certain algebro-geometric notion of stability of this class. In this talk, we will discuss a resolution of this conjecture for a certain type of toric fibrations, called semisimple principal toric fibrations. One of the main assets of these fibrations is that they come equipped with a connection which allows defining, from any Kähler metrics on the toric fiber X, a Kähler metric on the total space Y. After an introduction to the Calabi Problem for general compact Kähler manifolds, we will focus on the weighted toric setting. Then, I will explain how to translate the Calabi problem on Y, to a weighted cscK problem on the corresponding toric fiber X (arxiv paper: arXiv:2108.12297 |
