Allowed Differential Forms and Monodromy Eigenvalues

Definitions and mathematical statement

The supplied record asks whether allowed differential forms can capture precisely the monodromy eigenvalues of a function $f$ through poles of associated topological zeta functions. It does not specify the class of functions, the ambient space, the local or global setting, or the meaning of "allowed." Consequently, it does not determine a complete mathematical conjecture.

The following is a conditional formulation of one possible intended target. Once the missing geometric setting and definitions have been fixed, let $M_f\subseteq{\mathbb{C}}^\times$ denote the relevant set of monodromy eigenvalues, let $\mathcal A(f)$ be a nonempty family of allowed differential forms, and let \[ P_{f,\omega}=\{s\in{\mathbb{C}}:s\text{ is an actual pole of }Z_{\mathrm{top}}(f,\omega;s)\} \] for each $\omega\in\mathcal A(f)$. Here $Z_{\mathrm{top}}(f,\omega;s)$ is a placeholder for the associated topological zeta function; its construction and localization are not supplied.

If the intended correspondence sends a pole $s$ to $\exp(2\pi i s)$, and if completeness is intended across the family of forms, the question becomes whether one can specify an allowed family satisfying \[ \bigcup_{\omega\in\mathcal A(f)} \{\exp(2\pi i s):s\in P_{f,\omega}\}=M_f. \] Equivalently, the two requirements are \[ \forall\omega\in\mathcal A(f)\ \forall s\in P_{f,\omega},\qquad \exp(2\pi i s)\in M_f, \] and \[ \forall\lambda\in M_f\ \exists\omega\in\mathcal A(f)\ \exists s\in P_{f,\omega},\qquad \lambda=\exp(2\pi i s). \] This expresses absence of extraneous poles under the chosen correspondence and recovery of every eigenvalue across the family. It asserts neither a bijection nor equality of multiplicities.

The exponential correspondence, its sign convention, and the family interpretation are explicit unresolved interpretations, not details established by the supplied excerpt. The record also does not determine whether the assertion should hold for every function in some class, whether a single form must recover all eigenvalues, or whether every form meeting an independently defined allowedness criterion must satisfy the first requirement. Its background mentions a monomial example but gives no example or hypotheses from which to specialize this template.

Short English statement

The question asks whether suitable differential forms make the poles of topological zeta functions detect every relevant monodromy eigenvalue without producing extraneous values. The supplied excerpt leaves the geometric setting, allowedness criterion, pole correspondence, and scope of the eigenvalues unspecified, so the displayed formulation is conditional and requires human mathematical review.

Sources

• Local UnsolvedMath problem registry, permanent ID $30001350$, exact source problem code catalogue entry, title "Allowed Differential Forms and Monodromy Eigenvalues." The supplied record gives proposed year 2009 and names no proposer.
• Supplied report citation: Singularities (2010), Oberwolfach Reports, DOI 10.4171/owr/2009/43, pp. 2405--2470. Original Oberwolfach report: https://doi.org/10.4171/owr/2009/43. The supplied title year 2010 and the 2009 year metadata are retained without reconciliation; the report has not been independently consulted here.
• Supplied literature source, labeled "introduction to the monodromy conjecture": https://arxiv.org/abs/2403.03343. No author or further bibliographic identification is supplied.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that the monomial example belongs to a broader allowed-form and monodromy program, that the general analogue remains open, and that the extracted fragment is not a resolved theorem. This is supplied dated curation derived from the OWR material, not an independently verified assessment; time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified, explaining this as meaning-preserving editorial and LaTeX cleanup with no source-dependent content added. This supplied editorial label does not establish mathematical completeness or independent source verification.