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On the Artin vanishing theorem for Stein spaces

Olivier Benoist

TL;DR

The paper extends Artin vanishing to Stein spaces by combining stratified Morse theory with weakly constructible coefficients and $ ext{O}(S)$-convexity, establishing vanishing for relative cohomology $H^k(S,K, ext{F})$ when $k>n$ and for $H^k(S,S^Grac{K}{ ext{F}})$ in the $G$-equivariant setting for $k>n$. It analyzes Runge pairs to address the subtle case $k=n+1$ and constructs 1D counterexamples showing nonvanishing in certain Runge relative cohomology, while proving general $G$-equivariant vanishing theorems and consequences for alterations. The results generalize Artin–SGA43 to analytic Stein spaces, connect topological and analytic methods, and have potential implications for arithmetic questions about meromorphic function fields on (equivariant) Stein spaces. The methods highlight the role of $ ext{O}(S)$-convex subsets, Runge theory, and fixed-point loci in controlling equivariant cohomology.

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.

On the Artin vanishing theorem for Stein spaces

TL;DR

The paper extends Artin vanishing to Stein spaces by combining stratified Morse theory with weakly constructible coefficients and -convexity, establishing vanishing for relative cohomology when and for in the -equivariant setting for . It analyzes Runge pairs to address the subtle case and constructs 1D counterexamples showing nonvanishing in certain Runge relative cohomology, while proving general -equivariant vanishing theorems and consequences for alterations. The results generalize Artin–SGA43 to analytic Stein spaces, connect topological and analytic methods, and have potential implications for arithmetic questions about meromorphic function fields on (equivariant) Stein spaces. The methods highlight the role of -convex subsets, Runge theory, and fixed-point loci in controlling equivariant cohomology.

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.
Paper Structure (14 sections, 16 theorems, 16 equations)

This paper contains 14 sections, 16 theorems, 16 equations.

Table of Contents

  1. Artin vanishing theorems for Stein spaces
  2. The homotopy type of a Stein space
  3. Weakly constructible coefficients
  4. O(S)-convex subsets
  5. Relative cohomology of Runge pairs
  6. Gal(C/R)-equivariant Artin vanishing for Stein spaces
  7. Relative cohomology of Runge pairs
  8. Surfaces of finite genus
  9. A surface of infinite genus
  10. Artin vanishing for G-equivariant Stein spaces
  11. Artin vanishing relative to an O(S)-convex compact subset
  12. Artin vanishing relative to the set of G-fixed points
  13. G-equivariant Artin vanishing
  14. $G$-equivariant vanishing on alterations

Key Result

Theorem 2

Let $S$ be a connected Stein manifold of dimension $1$ that has finite genus. Let $U\subset S$ be a Runge open subset. Then $H^k(S,U,A)=0$ for any $k>1$ and any abelian group $A$.

Theorems & Definitions (29)

  • Theorem 2: Corollary \ref{['corcoho']}
  • Theorem 3: Corollary \ref{['ex1']}
  • Corollary 4
  • Theorem 5: Theorem \ref{['thArtin']}
  • Lemma 1.1
  • proof
  • Remark 1.2
  • Theorem 1.3
  • proof
  • Corollary 1.4
  • ...and 19 more