Borisov-Alexeev-Borisov conjecture and its applications

In August 2018, Caucher Birkar got Fields Medal for his proof of the famous Borisov-Alexeev-Borisov conjecture. This result has many applications. One of them is the proof, due to Prokhorov and Shramov, of the Jordan property for the Cremona group Cr(n) of birational automorphisms of the n-dimensionl projective space. This is a generalization of the classical result of Serre on the structure of Cr(2), the Cremona group of the plane.

 

Another application is due to Blanc, Lamy and Zimmermann (see https://arxiv.org/abs/1901.04145), who generalized the classical result of Cantat and Lamy to higher dimensions.

 

In my talk I will introduce the classical framework and motivations in a geometric and accessible way. Then I will explain what the Borisov-Alexeev-Borisov conjecture is and will present its new application, which is a work in progress with Blanc, Duncan and Prokhorov.