Séance Séminaire

Séminaire Algèbre Géométrie Algébrique Topologie Algébrique

The Calabi problem and K-moduli spaces of Fano 3-folds

The Calabi problem asks which compact complex manifolds can be endowed with a special metric that satisfies both an algebraic condition (being Kähler) and the Einstein (partial differential) equation. These are called Kähler-Einstein metric, and they are unique when they exist. A necessary condition for the existence of such a metric is that the canonical class of the manifold has a definite sign. Yau and Aubin/Yau confirmed Calabi's prediction and showed that manifolds with zero or positive canonical class always admit a Kähler-Einstein metric in the 1970s. By contrast, the Calabi problem is much more subtle for manifolds with negative canonical class: Fano manifolds may or may not admit a Kähler-Einstein metric. The Calabi problem for Fano manifolds was the focus of much research in the last decades, leading to the formulation and proof of the Yau-Tian-Donaldson conjecture. This conjecture, now a theorem, states that a Fano manifold admits a Kähler-Einstein metric precisely when it satisfies a sophisticated algebro-geometric condition called K-polystability. Surprisingly, the notion of K-polystability also sheds some light on another poorly understood aspect of the geometry of Fano varieties: how they behave in families. We have known for a long time that the set of all Fano varieties does not form a reasonable moduli space, but recent works have shown that the set of K-polystable Fano varieties does, and these are called K-moduli spaces. Our explicit understanding of K-polystability is still partial, and few examples of K-moduli spaces are known. In dimension 3, the known classification of Fano manifolds were classified into 105 deformation families by Mori-Mukai and Iskovskikh. is an invitation to investigate K-moduli spaces for these families. In the first part of the talk, I will present an overview of the Calabi problem, and discuss its solution for surfaces. In the second part of the talk, I will present results in dimension 3. Knowing - as we do - which families in the classification of Fano 3-folds have K-polystable members is a starting point to investigate the corresponding K-moduli spaces. I will describe explicitly some of their components of small dimension.