Geometry and Mutation of Quiver Grassmannians

Definitions and mathematical statement

The supplied registry record is marked unrecoverable. Its verification note states that the extraction concatenates seven independent numbered conjectures and problems, and no clean statement is supplied. Consequently, the following records the recoverable themes and their missing qualifications; it does not reconstruct seven precise source statements or assert a single combined conjecture.

For notation only, let $k$ be a field, let $Q=(Q_0,Q_1,s,t)$ be a finite quiver, and let $M$ be a finite-dimensional $k$-representation of $Q$. Thus each vertex $i$ has a vector space $M_i$, and each arrow $a$ has a linear map $M_a:M_{s(a)}\to M_{t(a)}$. For integers $0\leq e_i\leq\dim_k M_i$, the quiver Grassmannian $\operatorname{Gr}_{\mathbf e}(M)$ is the closed subscheme of \[ \prod_{i\in Q_0}\operatorname{Gr}_{e_i}(M_i) \] parametrizing subspaces $U_i\subseteq M_i$ satisfying $M_a(U_{s(a)})\subseteq U_{t(a)}$ for every arrow $a$. The representation $M$ is rigid if \[ \operatorname{Ext}^{1}_{\operatorname{Rep}_k(Q)}(M,M)=0. \] This notation does not settle the source's intended hypotheses: the extraction does not specify the field, characteristic, admissible quivers, or precise representation category.

Write $C(M)$ and $S(M)$ only as placeholders for the undefined source properties $(C)$ and $(S)$. No meaning for these properties is supplied. The extraction identifies the following six themes, without recovering their division into the seven numbered source items:

1. Whether every rigid $Q$-representation has property $(C)$. Schematically, in the intended but unspecified setting, the question is whether \[ \forall M,\qquad \operatorname{Ext}^{1}(M,M)=0\ \Longrightarrow\ C(M). \] The scope of $Q$ and $M$, and the definition of $C$, remain unresolved.
2. Whether every representation of an extended Dynkin quiver has property $(C)$. Schematically, the proposed universal assertion ranges over the intended extended Dynkin quivers $Q$ and their intended representations $M$, and asks for $C(M)$. The allowed orientations, finiteness hypotheses, and field are not specified.
3. Whether property $(C)$, or property $(S)$, is preserved by Fomin--Zelevinsky mutation. The extraction does not specify the objects carrying these properties, a mutation operation on those objects, admissible mutation vertices, or how representations and dimension vectors correspond across mutation. Hence a precise preservation implication cannot be written from this record.
4. Whether quiver Grassmannians of rigid representations have additional geometric properties, with being Fano given as an example. No precise universal Fano assertion is recoverable: the intended additional hypotheses, dimension vectors, and treatment of empty, singular, or disconnected Grassmannians are absent.
5. How to determine coefficients in a decomposition into simple perverse sheaves. The relevant geometric map, complex, coefficient system, shifts, and decomposition are not supplied, so no multiplicity formula or determination problem can be specified.
6. How cellular decomposition behaves under degeneration. The relevant family, fibers, degeneration, meaning of cellular decomposition, and proposed relationship between decompositions are unspecified.

The supplied literature assessment additionally mentions flag multiplicities, without defining the flags or the multiplicities involved. These omissions prevent a precise, independently usable mathematical formulation of the complete problem set.

Short English statement

This record combines several questions about quiver Grassmannians, rigidity, mutation, perverse-sheaf multiplicities, and degeneration. The registry reports seven independent source items, but the extraction gives only six broad themes and omits essential definitions and hypotheses. The supplied label is open; the formulation is marked unrecoverable. This is a statement-only draft requiring human mathematical review.

Sources

• Local UnsolvedMath registry: permanent ID $30004369$, source code catalogue entry, title Geometry and Mutation of Quiver Grassmannians, in the Oberwolfach Reports Open Problems collection. The record supplies proposed year $2020$, status open, and statement status unrecoverable; no individual proposer is named. Its verification note reports that seven independent numbered conjectures and problems were concatenated and that no separate clean statement was promoted. This verification note is supplied registry evidence, not an independent verification here.
• Supplied report citation: Representation Theory of Quivers and Finite Dimensional Algebras (2021), Oberwolfach Reports, pp. 143--230, DOI 10.4171/owr/2020/3. https://doi.org/10.4171/owr/2020/3. The background attributes the material to mathematical workshop open-problem discussions documented in Oberwolfach Reports.
• Supplied literature source, labelled "Oberwolfach report, quiver Grassmannians": https://ems.press/content/serial-article-files/46842.
• Literature assessment checked at 2026-08-22: the supplied curation labels the problem set open and reports unresolved directions concerning property $(C)$, mutation, quiver Grassmannians, and flag multiplicities, with no complete resolution known to that curation. This is supplied dated curation, not independently verified. The background explicitly requests independent verification of time-sensitive status and cited claims.