Simple groups of birational transformations in dimension two

TL;DR

This paper classifies simple groups acting birationally on complex surfaces and shows that finitely generated simple groups acting non-trivially on projective surfaces are finite.

Contribution

It provides a classification of simple groups acting birationally on complex surfaces and establishes finiteness for finitely generated simple groups over arbitrary fields.

Findings

Classification of simple groups acting on complex Kähler surfaces

Finitely generated simple groups acting non-trivially are finite

Results apply over arbitrary fields

Abstract

We classify simple groups that act by birational transformations on compact complex K\"ahler surfaces. Moreover, we show that every finitely generated simple group that acts non-trivially by birational transformations on a projective surface over an arbitrary field is finite.