Cohomological Dimension of Meromorphic Function Fields

Definitions and mathematical statement

Let $S$ be a connected normal Stein complex space of finite complex dimension $n$, and let $K=\mathcal M(S)$ be its field of global meromorphic functions. Here normality means that every local analytic ring is an integrally closed domain. Interpret cohomological dimension in the standard Galois-cohomological sense: choose a separable closure $K^{\mathrm{sep}}$, put $G_K=\operatorname{Gal}(K^{\mathrm{sep}}/K)$, and define \[ \operatorname{cd}(K)=\inf\left\{d\in{\mathbb{Z}}_{\geq 0}: H^q_{\mathrm{cont}}(G_K,A)=0\text{ for every integer }q>d\text{ and every discrete torsion }G_K\text{-module }A\right\}, \] with value $\infty$ if no such $d$ exists. The action on each module is required to be continuous.

The question is whether, for every such $S$, \[ \operatorname{cd}\left(\mathcal M(S)\right)=\dim_{{\mathbb{C}}}S=n. \] The supplied dated assessment identifies $n\geq 3$ as the remaining general range, and states that the question remains open already for $S={\mathbb{C}}^n$ in that range. It also states that the equality has been proved for connected normal Stein spaces of dimension two. These are supplied literature-status claims, not independently verified conclusions of this draft. The brief original excerpt does not define $\operatorname{cd}$; the Galois-cohomological convention made explicit above requires confirmation against the report.

Short English statement

Does every connected normal Stein complex space have a meromorphic function field whose Galois cohomological dimension equals its complex dimension? The supplied assessment reports the dimension-two case as proved and the general case in dimensions at least three as open, including complex affine space. This is a statement-only draft requiring human mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005956$, title Cohomological Dimension of Meromorphic Function Fields, exact source problem code catalogue entry; collection: Oberwolfach Reports Open Problems. The record gives proposed year $2024$ and names no proposer.
• Supplied source citation: Algebraic Geometry: Wall Crossing and Moduli Spaces, Varieties and Derived Categories (2025), Oberwolfach Reports, pp. 1787--1854, DOI 10.4171/owr/2024/32, https://doi.org/10.4171/owr/2024/32. The original excerpt points to p. 1807 and asks whether $\operatorname{cd}(\mathcal M(S))=n$. This URL is also the supplied literature source labelled "original OWR report."
• Supplied literature source labelled "Benoist primary paper": https://arxiv.org/abs/2410.16809. No paper title or more detailed bibliographic attribution is supplied.
• Literature assessment checked at 2026-08-22: the supplied curation labels the problem partially solved, reports the equality proved in dimension two, and reports the general dimension-at-least-three case open, already for complex affine space. This is supplied dated curation, not independently verified literature assessment.
• The registry labels the statement corrected_verified and says it was reconstructed and checked against the official Oberwolfach report. This is a supplied verification claim; no source was independently visited or checked for this draft.