Characteristic Varieties of Rigid-Analytic D-Modules

Definitions and mathematical statement

Let $X$ be a smooth rigid-analytic space. The supplied record does not specify its ground field, the precise sheaf of differential operators, or the completion used to define co-admissible modules.

For terminology, if a local operator algebra has a Fréchet--Stein presentation $A=\varprojlim_n A_n$, an $A$-module $M$ is co-admissible relative to that presentation when it arises as $M\cong\varprojlim_n M_n$, where each $M_n$ is a finitely generated $A_n$-module and the transition maps induce isomorphisms \[ A_n\otimes_{A_{n+1}}M_{n+1}\cong M_n. \] This explains the module-theoretic terminology; the source does not identify the local operator algebras or presentations intended on $X$.

The comparison requested by the source is with algebraic characteristic varieties. For a smooth algebraic variety $Y$ over a field of characteristic zero and a coherent algebraic $\mathcal D_Y$-module $N$, a good filtration $F$ gives an associated graded module over $\operatorname{gr}\mathcal D_Y\cong\operatorname{Sym}_{\mathcal O_Y}\mathcal T_Y$. Its support in the cotangent bundle defines \[ \operatorname{Char}(N)=\operatorname{Supp}_{T^*Y}(\operatorname{gr}_F N), \] independently of the good filtration. This is the comparison model, not an additional hypothesis on the rigid-analytic spaces in the source.

Problem. Develop a satisfactory theory of characteristic varieties for co-admissible $\mathcal D$-modules on smooth rigid-analytic spaces, analogous to the characteristic-variety theory for algebraic $\mathcal D$-modules.

The source states a research objective rather than a fully specified existence conjecture: it supplies neither the precise rigid-analytic module category nor a target space and axioms for the proposed characteristic varieties. In particular, a cotangent-bundle support construction and particular analogues of algebraic properties cannot be imposed as source-stated requirements without further evidence.

Short English statement

Construct a theory assigning characteristic varieties to co-admissible differential-operator modules on smooth rigid-analytic spaces, comparable to the algebraic theory. The source leaves the ground-field hypotheses, operator completion, target space, and required properties unspecified, so this remains a research-program statement requiring mathematical review.

Sources

• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $30002886$, source problem code catalogue entry, title "Characteristic Varieties of Rigid-Analytic D-Modules." The supplied record attributes the problem to Oberwolfach workshop discussions, records proposed year 2015, and names no individual proposer.
• Supplied report citation: Enveloping Algebras and Geometric Representation Theory (2016), Oberwolfach Reports, pp. 1385--1447, DOI 10.4171/owr/2015/25. https://doi.org/10.4171/owr/2015/25. The citation's 2016 year and the record's 2015 provenance are retained without reconciliation.
• Ardakov--Wadsley, D-modules on rigid analytic spaces I. https://arxiv.org/abs/1501.02215.
• Ardakov--Bode--Wadsley, D-modules on rigid analytic spaces III. https://arxiv.org/abs/1904.13280.
• Oberwolfach Report 2015/25. https://publications.mfo.de/bitstream/handle/mfo/3472/OWR_2015_25.pdf?isAllowed=y&sequence=1.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that dimension theory, weak holonomicity, and Kashiwara-type results are available, while a satisfactory general characteristic-variety theory for coadmissible $\mathcal D$-modules remains unresolved. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment; its status and cited claims require independent checking.