[Topology.beta] Talk on K-stability by Anne-Sophie Kaloghiros at UvA on May 14

[Benedetti, G. (Gabriele)]

Dear all,

I forward the invitation to a talk at the UvA on K-stability for anyone who might be interested.

Best wishes,

Gabriele

Anne-Sophie Kaloghiros<https://sites.google.com/view/anne-sophie-kaloghiros/home> (Brunel University London)
Wednesday 14th of May at 11am
KdVI, seminar room F3.20 (on the third floor)
Title: Explicit K-stability and moduli construction of Fano 3-folds

Abstract: The Calabi problem asks which compact complex manifolds are Kähler-Einstein (KE)- i.e. can be endowed with a canonical metric that satisfies both an algebraic condition (being Kähler) and the Einstein (partial differential) equation. Such manifolds always have a canonical class of definite sign. The existence of such a metric on manifolds with positive and trivial canonical class (general type and Calabi Yau) was proved in the 70s. In the case of Fano manifolds the situation is more subtle - Fano manifolds may or may not have a KE metric.

We now know that a Fano manifold admits a KE metric precisely when it satisfies a sophisticated algebro-geometric condition called K-polystability. Surprisingly, K-polystability also sheds some light on another important problem: while Fano varieties do not behave well in families, K-polystable Fano varieties do and form K-moduli spaces.

Our explicit understanding of K-polystability is still partial, and few examples of K-moduli spaces are known. In dimension 3, Fano manifolds were classified into 105 deformation families by Mori-Mukai and Iskovskikh. In this talk, I will present an overview of the Calabi problem, and discuss its solution 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 K-moduli spaces of Fano 3-folds.
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