| Résume | The Yau-Tian-Donaldson (YTD) 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 the resolution of this conjecture for a certain class of toric fibrations, called semisimple principal toric fibrations. After an introduction to the Calabi Problem for general Kähler manifolds, we will focus on the toric setting. Then we will see how to reduce the Calabi problem on the total space of a semisimple principal toric fibration to a weighted constant scalar curvature Kähler problem on the toric fibers. If the time allows, I will giveelements of proof. (preprint: arXiv:2108.12297) |
