Restrictions of Toric Vector Bundles to Embedded Varieties

Definitions and mathematical statement

Let $Y$ be a variety and let $E$ be a finite-rank algebraic vector bundle on $Y$. The base field and the precise convention for "variety" are not specified in the supplied record. All varieties, bundles, and morphisms below are to be taken over the same base field.

For a toric variety $X_\Sigma$ with acting algebraic torus $T$, a toric vector bundle $F$ is an algebraic vector bundle equipped with an algebraic $T$-action lifting the action on $X_\Sigma$, such that the induced maps between fibres are linear. Write $i^*F$ for the pullback vector bundle along a morphism $i:Y\to X_\Sigma$.

The problem is to characterize precisely those pairs $(Y,E)$ for which \[ \exists\,X_\Sigma\;\exists\,i:Y\hookrightarrow X_\Sigma\;\exists\,F \quad\text{such that}\quad \begin{cases} X_\Sigma\text{ is a toric variety},\\ i\text{ is an embedding},\\ F\text{ is a toric vector bundle on }X_\Sigma,\\ E\cong i^*F\text{ as algebraic vector bundles on }Y. \end{cases} \] The ambient toric variety, embedding, and toric vector bundle may depend on both $Y$ and $E$. No torus action on $Y$ or equivariant structure on $E$ is specified or required by the supplied formulation.

The source calls for a "good embedding" but does not define an additional goodness condition. It also does not specify whether the embedding must be closed or may be locally closed. These choices remain unresolved; the displayed existence condition retains "embedding" in the intended but unspecified source sense. No smoothness, projectivity, or completeness hypotheses are supplied.

Short English statement

Determine which algebraic vector bundles on a variety can be obtained by restricting an equivariant vector bundle from some ambient toric variety. The ambient variety and embedding may vary with the bundle. The base-field assumptions and the precise requirements on the embedding remain unspecified.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30006307$, source problem code catalogue entry, title "Restrictions of Toric Vector Bundles to Embedded Varieties."
• Supplied source citation: Toric Geometry (2025), Oberwolfach Reports, DOI $10.4171/owr/2025/19$, pp. 883--940. The supplied excerpt labels the target as Question 8 in the discussion "Restrictions of toric vector bundles." Source URL and supplied literature-source link, labelled "Oberwolfach report": https://doi.org/10.4171/owr/2025/19. No individual proposer is identified in the record; the supplied proposed year is 2025.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and states that the cited toric-bundle classification does not resolve this general extension problem. This is supplied dated curation, not an independently verified literature assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry marks the statement as corrected_verified and reports repairs only to formula delimiters, notation, grammar, and typography, with mathematical content and scope preserved. This is a supplied editorial qualification, not independent mathematical verification of this draft.