K-stability of Fano 3-folds

Abstract: Fano varieties are geometric shapes which are positively curved. They arise in a wide array of fields from theoretical physics to phylogenetic trees. In fact, every geometric shape which can be parametrised (or covered ) is - up to surgery - a family of Fano varieties. There are rich interactions between differential geometric and algebro-geometric properties of Fano manifolds (and more generally of Kahler manifolds). An instance of this phenomenon was conjectured by Yau Tian and Donaldson ( and proved by Donaldson, Chen and Sun): they proved that on Fano manifolds the existence of special canonical metrics is equivalent to a stability property. This is an equivalence between properties that are subtle, and still little understood. I will discuss algebro-geometric approaches to this problem and will present recent developments and their applications to our understanding of Fano surfaces and 3-folds.

Mathematics

Audience: researchers in the discipline

MESS (Mathematics Essex Seminar Series)

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