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Étale cohomology of algebraic varieties over Stein compacta

Olivier Benoist

TL;DR

This work develops a robust bridge between étale cohomology of algebraic varieties over Stein algebras and singular cohomology of their analytifications, enabling a transfer of topological information to algebraic context. The authors establish absolute and relative Artin comparison theorems in the Stein setting, refine the comparison via the qfh topology, and prove cohomological-dimension bounds for fields of meromorphic functions near Stein compacta. They further exploit a Galois (G)–equivariant framework to derive quantitative real-analytic Hilbert 17th problem results, including sums of squares for nonnegative real-analytic functions on Stein spaces and Pfister-type bounds. These results illuminate the interplay between complex-analytic, real-analytic, and algebraic geometry, with concrete implications for cohomology, function fields, and real-analytic positivity problems.

Abstract

We prove a comparison theorem between the étale cohomology of algebraic varieties over Stein compacta and the singular cohomology of their analytifications. We deduce that the field of meromorphic functions in a neighborhood of a connected Stein compact subset of a normal complex space of dimension $n$ has cohomological dimension $n$. As an application of $\textrm{Gal}(\mathbb{C}/\mathbb{R})$-equivariant variants of these results, we obtain a quantitative version of Hilbert's 17th problem on compact subsets of real-analytic spaces.

Étale cohomology of algebraic varieties over Stein compacta

TL;DR

This work develops a robust bridge between étale cohomology of algebraic varieties over Stein algebras and singular cohomology of their analytifications, enabling a transfer of topological information to algebraic context. The authors establish absolute and relative Artin comparison theorems in the Stein setting, refine the comparison via the qfh topology, and prove cohomological-dimension bounds for fields of meromorphic functions near Stein compacta. They further exploit a Galois (G)–equivariant framework to derive quantitative real-analytic Hilbert 17th problem results, including sums of squares for nonnegative real-analytic functions on Stein spaces and Pfister-type bounds. These results illuminate the interplay between complex-analytic, real-analytic, and algebraic geometry, with concrete implications for cohomology, function fields, and real-analytic positivity problems.

Abstract

We prove a comparison theorem between the étale cohomology of algebraic varieties over Stein compacta and the singular cohomology of their analytifications. We deduce that the field of meromorphic functions in a neighborhood of a connected Stein compact subset of a normal complex space of dimension has cohomological dimension . As an application of -equivariant variants of these results, we obtain a quantitative version of Hilbert's 17th problem on compact subsets of real-analytic spaces.
Paper Structure (35 sections, 51 theorems, 44 equations)

This paper contains 35 sections, 51 theorems, 44 equations.

Table of Contents

  1. Hilbert's 17th problem
  2. The real-analytic variant
  3. From real-analytic to complex-analytic geometry
  4. Cohomological dimension of fields of meromorphic functions
  5. Étale and singular cohomology
  6. Proof of the comparison theorem
  7. Structure of the article
  8. Acknowledgements
  9. Geometry of Stein spaces
  10. Complex spaces and coherent sheaves
  11. Runge domains
  12. O(S)-convex compact subsets
  13. Stein compact subsets
  14. Meromorphic functions and ramified coverings
  15. Killing cohomology classes on finite coverings
  16. ...and 20 more sections

Key Result

Theorem 1

Let $M$ be a normal real-analytic variety of pure dimension $n$. Let $K\subset M$ be compact. Let $f:M\to \mathbb R$ be a nonnegative real-analytic function. Then there exists an open neighborhood $U$ of $K$ in $M$ such that $f|_U$ is a sum of $2^n$ squares of real-analytic meromorphic functions on

Theorems & Definitions (96)

  • Theorem 1: Theorem \ref{['sosreal']}
  • Corollary 2
  • Theorem 3: Theorem \ref{['soscx']}
  • Theorem 4: Corollary \ref{['cdfieldcor']} and Remark \ref{['remcd']} (ii)
  • Theorem 5: Theorem \ref{['Artincompcx']}
  • Theorem 6: Theorem \ref{['relArtin']}
  • Lemma 1.1
  • proof
  • Proposition 1.2
  • proof
  • ...and 86 more