A Serre-Style Proof of the Jordan Property

Definitions and mathematical statement

A group $\Gamma$ has the Jordan property if there exists an integer $J\geq 1$ such that every finite subgroup $G\leq\Gamma$ contains an abelian normal subgroup $A\trianglelefteq G$ with $[G:A]\leq J$.

For an integral variety $X$ over a field $k$, write $\operatorname{Bir}_k(X)$ for the group of birational self-maps defined over $k$. For a smooth projective geometrically integral $k$-variety, rational connectedness means that, after extension to an algebraic closure $\overline{k}$, two general points can be joined by a rational curve.

Qualified mathematical target. The reference to Cremona groups suggests interpreting the requested Jordan property as that of $\operatorname{Bir}_k(X)$. Under this interpretation, and using smooth projective geometrically integral varieties to make rational connectedness precise, the property to be proved is \[ \begin{gathered} \text{for every field }k\text{ with }\operatorname{char}(k)=0 \text{ and every rationally connected }k\text{-variety }X,\\ \exists J_{k,X}\in{\mathbb{Z}}_{\geq1}\quad \forall\text{ finite }G\leq\operatorname{Bir}_k(X)\quad \exists A\trianglelefteq G:\quad A\text{ is abelian and }[G:A]\leq J_{k,X}. \end{gathered} \] Here $X$ ranges over the smooth projective geometrically integral varieties just described. The bound is independent of $G$; the supplied statement does not specify whether it should also be uniform in $k$ or $X$ at fixed dimension.

Requested proof problem. Give an alternative proof of this Jordan property by extending Serre's Minkowski-style approach to finite subgroups of rank-two Cremona groups, rather than using the boundedness argument attributed in the supplied record to Prokhorov and Shramov. The supplied record describes the Jordan property itself as known; the requested target is a proof by the indicated approach.

Unresolved qualifications. The source excerpt does not explicitly name the group whose Jordan property is intended, specify the geometric conventions for varieties over nonclosed fields, or describe precisely which ingredients constitute Serre's approach and which boundedness arguments are to be avoided. Thus the displayed formulation is an explicitly qualified birational interpretation, not a resolution of these source omissions. Its smooth projective convention does not establish the intended scope for singular or nonproper varieties.

Short English statement

The problem asks for a Serre-style alternative proof that finite birational symmetry groups of rationally connected varieties in characteristic zero contain abelian normal subgroups of uniformly bounded index for each fixed variety and field. The supplied record treats the Jordan property as known and labels the alternative-proof question open. The intended group, scope of uniformity, and permitted proof ingredients require clarification.

Sources

• Local UnsolvedMath registry, permanent ID $30003865$, title A Serre-Style Proof of the Jordan Property, source problem code catalogue entry. The supplied record belongs to the Oberwolfach Reports Open Problems collection and records proposed year 2018; no individual proposer is supplied.
• Supplied source citation: Subgroups of Cremona Groups (2019), Oberwolfach Reports, pp. 1685--1743, DOI 10.4171/owr/2018/28. https://doi.org/10.4171/owr/2018/28. The supplied background describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source: Oberwolfach Report 28/2018. https://ems.press/content/serial-article-files/46750.
• Literature assessment dated 2026-08-21: the supplied curation labels the question open, states that the Jordan property for rationally connected varieties is known, and reports that no general alternative proof following the cited approach was identified. This is supplied dated curation, not an independently verified literature assessment; the record explicitly requests independent verification of time-sensitive status and cited claims.
• The supplied formulation attributes the proposed Minkowski-style method to Serre and the existing boundedness approach to Prokhorov and Shramov. It supplies no separate bibliographic references for those works. The record labels the statement corrected_verified and reports comparison with the official report; that verification claim is supplied metadata and has not been independently checked here.