Cohomology of finite subgroups of the plane Cremona group

TL;DR

This paper extends the computation of the first cohomology group of the Picard module, an invariant of finite group actions on varieties, from cyclic groups to more general groups using advanced algebraic tools.

Contribution

It introduces a new method leveraging the Brauer group of quotient stacks to generalize previous results on cohomology computations for finite group actions.

Findings

Extended cohomology computations to broader group actions

Connected cohomology invariants with the equivariant Burnside group

Provided new insights into the geometry of fixed point loci

Abstract

An equivariant stable birational invariant of an action of a finite group on a smooth projective variety is the first cohomology group of the Picard module. Bogomolov-Prokhorov and Shinder computed this for actions of cyclic groups on rational surfaces, with maximal stabilizers, in terms of the geometry of the fixed point locus. Using the Brauer group of the quotient stack, we extend the computation to more general actions and relate it to the equivariant Burnside group formalism.