Abstract:
Groups of birational automorphisms of algebraic varieties is one of the
oldest and quite difficult to study object of algebraic geometry. Much
attention has been given both by the classics and especially recently to the
study of the Cremona groups, i.e., the groups of birational automorphisms of
projective spaces of dimension n. They are infinite-dimensional (for n> 1)
which distinguishes them sharply from the algebraic groups most studied in
algebraic geometry. Nevertheless, a number of constructions and properties
of algebraic groups can be carried over to these groups. The main result
consists in proving the existence of Borel subgroups in any Cremona group.
This is one of the themes of the series of papers to which the talk is
devoted. Also in these papers the concepts of the Jordan property of a group
and its Jordan constant are introduced and studied. This caused a stream of
papers both here and abroad. This technique has found applications outside
of algebraic geometry. In particular, it was possible to prove the Jordan
property of any connected real Lie group and, as a consequence, that of
automorphism groups of many topological varieties.