Smooth Projective Realization of Connected Algebraic Groups

Definitions and mathematical statement

Let $k$ be a field and let $G$ be a smooth connected algebraic group over $k$, meaning a smooth connected group scheme of finite type over $k$. No restriction on the characteristic of $k$, and no affineness assumption on $G$, is supplied.

For a projective $k$-scheme $X$, write $\operatorname{Aut}_{X/k}$ for its automorphism group scheme, representing the functor \[ T\longmapsto \operatorname{Aut}_T(X\times_k T) \] on $k$-schemes $T$. Write $\operatorname{Aut}_{X/k}^{0}$ for its identity component.

Under the interpretation that "automorphism group" means the full automorphism group scheme, the supplied question is: for every field $k$ and every smooth connected algebraic group $G$ over $k$, does there exist a smooth projective $k$-variety $X$ such that \[ \operatorname{Aut}_{X/k}\cong G \qquad\text{as group schemes over }k? \] The source record does not explicitly distinguish this interpretation from an isomorphism of groups of $k$-rational automorphisms. It also does not specify the integrality or geometric integrality convention intended by "$k$-variety." These qualifications remain unresolved; the displayed formulation is conditional on the group-scheme interpretation.

The supplied dated assessment describes an identity-component realization result for projective geometrically integral schemes and labels the stronger smooth-projective full-automorphism formulation incomplete. Replacing $\operatorname{Aut}_{X/k}$ by $\operatorname{Aut}_{X/k}^{0}$, or dropping smoothness of $X$, changes the stated target. The assessment is supplied curation, not an independently verified theorem or status determination.

Short English statement

Given a smooth connected algebraic group over any field, can it be realized as the full automorphism group scheme of a smooth projective variety over that same field? The intended meaning of automorphism group and the convention for variety require confirmation. The supplied record labels the problem partially solved, distinguishing identity-component realization from the stronger question stated here.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30006249$, source code catalogue entry, title "Smooth Projective Realization of Connected Algebraic Groups." The supplied record belongs to the Oberwolfach Reports Open Problems collection and gives proposed year $2025$; no individual proposer is identified.
• Supplied report citation: Algebraic Groups (2025), Oberwolfach Reports, pp. 941--1010, DOI 10.4171/owr/2025/20. https://doi.org/10.4171/owr/2025/20. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature reference: Brion--Schröer, "inverse Galois problem." https://doi.org/10.1007/s00031-024-09865-0.
• Supplied literature reference: Blanc--Brion, "smooth projective abelian cases." https://doi.org/10.5802/afst.1746.
• Supplied literature reference labeled "source formulation": https://doi.org/10.4171/OWR/2025/20.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem "partially solved," identifies it with the inverse Galois problem for connected algebraic groups, describes identity-component realization as proved for projective geometrically integral schemes, and describes the stronger smooth-projective full-automorphism formulation as incomplete. This is supplied dated curation derived from the OWR material, not independently verified here; its status and cited claims require independent verification.
• The supplied statement status is corrected_verified. Its verification note reports that the official report was checked and that the question was isolated from surrounding exposition with notation repaired. This is a source-record assertion, not a verification performed for this draft. The field labeled original statement contains only an incomplete introductory sentence and a contextual retention note, rather than the mathematical question itself.