Harmonic Metrizations of Line Bundles

Definitions and mathematical statement

Let $M$ denote a line bundle on the space $X$, and let $\overline{M}$ denote $M$ equipped with a metrization. Write $c_1(\overline{M})$ for the curvature object associated with that metrization in the intended non-Archimedean theory. The supplied record does not specify that theory, the hypotheses on $X$, the background object $\omega$, or the dimension parameter $n$.

The source calls the curvature $\omega$-harmonic when it is $\omega$-bounded and satisfies \[ c_1(\overline{M})\,\omega^{n-1}=\lambda(M)\,\omega^n. \] Here $\omega$-boundedness, the products, their ambient space of equality, and the scalar $\lambda(M)$ are not defined in the supplied record.

Question, conditional on the missing setup. For every line bundle $M$ on $X$, does there exist a metrization $\overline{M}$ whose curvature is $\omega$-harmonic, and is this metrization unique up to scale? Under the usual norm interpretation, uniqueness up to scale means that any two admissible metrizations differ by multiplication by a single positive real constant on all of $X$; the intended metric convention remains to be specified.

Unresolved qualifications. A precise conjecture requires the underlying valued field, the class and hypotheses of $X$, the meaning and positivity assumptions of $\omega$, the relation of $n$ to $X$, the admissible class of metrics, the curvature and product constructions, the definition of $\omega$-boundedness, and the prescription of $\lambda(M)$. These cannot be recovered from the supplied statement alone.

Short English statement

The question asks whether every line bundle admits a metric, unique up to a constant scale, whose curvature is bounded relative to $\omega$ and satisfies the displayed proportionality equation. The geometric setting and several essential definitions are missing, so this is a conditional statement-only draft requiring mathematical review.

Sources

• Local UnsolvedMath registry, supplied record: permanent ID $30003015$, title Harmonic Metrizations of Line Bundles, exact source problem code catalogue entry; collection: Oberwolfach Reports Open Problems. No individual proposer is identified.
• Supplied report citation: Non-Archimedean Geometry and Applications (2016), Oberwolfach Reports, pp. 3271--3331, DOI 10.4171/owr/2015/57. Supplied source URL and literature-source link: https://doi.org/10.4171/owr/2015/57. The registry gives the proposed year as 2015; no specific passage within the report is supplied.
• Supplied literature assessment, checked 2026-08-21: the registry labels the problem partially solved and reports that the canonical harmonic metrization conjecture is proved for curves, model metrics, certain residue-characteristic-zero dual-complex cases, and polarized dynamical systems, but not in the stated generality. This is supplied dated curation derived from the OWR material, not an independently verified assessment. The supplied background expressly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and says that its mathematical content was preserved while notation, typography, or omitted definitions were repaired for self-contained publication. This is a supplied editorial assertion; the present record still omits essential mathematical setup, and the report has not been independently consulted here.