Exceptional Singularities and Fano Varieties

Research Group Cheltsov

August 1 - September 4, 2012

Organizers: Ivan Cheltsov, Jihun Park, Constantin Shramov

The research area lies in algebraic geometry, which is a constantly developing branch of mathematics with numerous applications in other areas of mathematics, including mathematical physics, differential geometry, topology and many others.

A basic object of study in algebraic geometry is an algebraic variety. Classically these were usually supposed smooth, but during the last decades the progress in the area - including the development of the Minimal Model Program in the works of Mori, Shokurov, Kawamata, McKernan, Hacon, Birkar and others - both gave tools and posed problems to deal with singular varieties. One of the basic tasks is to classify the relevant singularities appearing on the algebraic varieties in connection with Minimal Model Program.

Fano varieties play a central role in Minimal Model Program, and generalized cones over Fano varieties give many examples of Kawamata log terminal singularities. The notion of exceptional Fano varieties and exceptional Kawamata log terminal singularities was introduced by Shokurov in his famous paper of 3-fold log flips. They are closely related to the deep problem of existence of Kähler-Einstein metrics on Fano orbifolds. The research group studied Kähler-Einstein metrics on Fano orbifolds, birational automorphisms of Fano orbifolds, and non-rationality of Fano hypersurfaces.