Sheaf Models for Modular Category O

Definitions and mathematical statement

A full subcategory $\mathcal S$ of an abelian category $\mathcal A$ is a Serre subcategory if it is closed under subobjects, quotients, and extensions. Its Serre quotient $\mathcal A/\mathcal S$ is an abelian category equipped with an exact functor $Q:\mathcal A\to\mathcal A/\mathcal S$ that annihilates $\mathcal S$ and is universal among exact functors from $\mathcal A$ that annihilate $\mathcal S$.

The supplied question asks whether the modular representation category $\mathcal O$, described as a Serre quotient, admits a description as an appropriate category of sheaves. Schematically, the requested description would be an equivalence \[ \mathcal O=\mathcal A/\mathcal S\simeq\mathcal C_{\mathrm{sh}}, \] where $\mathcal C_{\mathrm{sh}}$ denotes the intended sheaf category. This notation only displays the requested form: the record does not specify $\mathcal A$, $\mathcal S$, the modular representation data, or $\mathcal C_{\mathrm{sh}}$. In particular, it supplies neither the underlying space and coefficients for the sheaves nor the precise categorical sense of the proposed equivalence.

Unresolved source qualification. The record labels the problem open, but also marks its statement as unrecoverable. Its supplied statement-verification note says that the report immediately answers the question by presenting a motivic-sheaf model and a categorical equivalence, and that the question is rhetorical. Consequently, the supplied evidence does not determine a precise unresolved mathematical assertion. The displayed equivalence is a schematic transcription of the extracted question, not a newly formulated conjecture or a verified theorem.

Short English statement

The extracted question asks for a sheaf description of a modular representation category $\mathcal O$ defined by a Serre quotient. The quotient and sheaf category are unspecified, and a supplied verification note says the report already answers the question. A precise open problem cannot be recovered from this record alone.

Sources

• Local UnsolvedMath registry, permanent ID $30004170$, title "Sheaf Models for Modular Category O," source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives proposed year 2019 and identifies no proposer.
• Supplied report citation: Algebraic K-theory (2020), Oberwolfach Reports, DOI 10.4171/owr/2019/29, pp. 1737--1790. Supplied source URL and literature-source label "OWR primary report": https://doi.org/10.4171/owr/2019/29. This is supplied primary-source attribution; its contents have not been independently checked here.
• Supplied literature-source label "geometric modular category O": https://arxiv.org/abs/1609.05956. No further bibliographic attribution is supplied.
• Literature assessment dated 2026-08-22: the supplied curation reports that geometric and graded realizations of modular category $\mathcal O$ have been constructed, while calling the broad question with an unspecified sheaf category imprecise and open as stated. This is supplied dated curation, not an independently verified assessment; the background explicitly calls for independent verification of time-sensitive status and cited claims.
• The supplied statement-verification note says that the official report immediately answers the rhetorical question with a motivic-sheaf model and categorical equivalence. The record therefore retains the original question for provenance, marks the statement unrecoverable, and promotes no clean statement. This supplied verification claim has not been independently checked here.