The Bogomolov-Prokhorov invariant of surfaces as equivariant cohomology

TL;DR

This paper establishes a uniform proof connecting the Bogomolov-Prokhorov invariant of complex surfaces with equivariant cohomology, extending previous results to a broader class of surfaces with finite cyclic group actions.

Contribution

It provides a new, unified proof of the isomorphism between the invariant cohomology group and fixed divisors' cohomology using equivariant methods, generalizing prior work.

Findings

Proves the isomorphism using equivariant cohomology techniques.

Extends the Bogomolov-Prokhorov invariant results to all smooth projective surfaces with cyclic group actions.

Provides a unified framework for understanding invariants related to surface automorphisms.

Abstract

For a complex smooth projective surface $M$ with an action of a finite cyclic group $G$ we give a uniform proof of the isomorphism between the invariant $H^1(G, H^2(M, {\mathbb Z}))$ and the first cohomology of the divisors fixed by the action, using $G$-equivariant cohomology. This generalizes the main result of Bogomolov and Prokhorov "On stable conjugacy of finite subgroups of the plane Cremona group, I".