On the Artin vanishing theorem for Stein spaces

TL;DR

This paper investigates Artin vanishing theorems for Stein spaces, providing new results on cohomology group vanishing in various contexts, including relative and equivariant settings.

Contribution

It introduces novel positive and negative results on Artin vanishing for Stein spaces, especially relative to Runge open subsets and Gal(C/R)-equivariant cohomology.

Findings

Vanishing results for cohomology groups in Stein spaces relative to Runge open subsets.

An Artin vanishing theorem for Gal(C/R)-equivariant cohomology.

New insights into the behavior of cohomology in complex and real algebraic contexts.

Abstract

Artin vanishing theorems for Stein spaces refer to the vanishing of some of their (co)homology groups in degrees higher than the dimension. We obtain new positive and negative results concerning Artin vanishing for the cohomology of a Stein space relative to a Runge open subset. We also prove an Artin vanishing theorem for the Gal(C/R)-equivariant cohomology of a Gal(C/R)-equivariant Stein space relative to the fixed locus.