Boundedness results for singular Fano varieties and applications to Cremona groups

Abstract

This survey paper reports on work of Birkar, who confirmed a long-standing conjecture of Alexeev and Borisov-Borisov: Fano varieties with mild singularities form a bounded family once their dimension is fixed. Following Prokhorov-Shramov, we explain how this boundedness result implies that birational automorphism groups of projective spaces satisfy the Jordan property, answering a question of Serre in the positive.

Final version. To appear in the proceedings of the January 2019 edition of the Séminaire Nicolas Bourbaki

References in corpus (9)

• Locally nilpotent derivations on affine surfaces with a $\C^*$-action
• Quotients of higher dimensional Cremona groups
• Automorphisms of pointless surfaces
• ACC for log canonical thresholds
• Jordan property for algebraic groups and automorphism groups of projective varieties in arbitrary characteristic
• Boundedness of varieties of Fano type with alpha-invariants and volumes bounded below
• Three plots about the Cremona groups
• Shokurov's Rational Connectedness Conjecture
• Summary on the proof of BAB Conjecture

Cited by in corpus (4)

• A categorical approach to injective envelopes
• Finite subgroups of the birational automorphism group are 'almost' nilpotent of class at most two
• On Character Variety of Anosov Representations
• Finite subgroups of the birational automorphism group are `almost' nilpotent