Jordan property for groups of birational selfmaps

TL;DR

This paper explores the Jordan property in groups of birational selfmaps, proving boundedness of finite automorphism subgroups under certain conjectural assumptions and examining which varieties satisfy this property.

Contribution

It establishes boundedness of finite automorphism groups assuming a specific case of the Borisov--Alexeev--Borisov conjecture and analyzes the Jordan property for birational selfmaps of algebraic varieties.

Findings

Finite subgroups of automorphism groups have bounded orders under certain conjectural assumptions.

Investigation into algebraic varieties with groups of birational selfmaps satisfying the Jordan property.

Conditions under which the Jordan property holds for these groups.

Abstract

Assuming a particular case of Borisov--Alexeev--Borisov conjecture, we prove that finite subgroups of the automorphism group of a finitely generated field over Q have bounded orders. Further, we investigate which algebraic varieties have groups of birational selfmaps satisfying the Jordan property.