Adelic Harmonic Metrics on Line Bundles

Definitions and mathematical statement

Let $M$ denote a line bundle on the source's space $X$. An adelic metrization $\overline M$ denotes $M$ equipped with metrics at the places of the source's field $K$. The supplied background describes the intended assertion as one over global fields, but the statement does not specify the hypotheses on $K$ or $X$.

Source assertion, with unresolved parameters. Does every line bundle $M$ on $X$ admit an admissible adelic metrization $\overline M$ whose metric at every place of $K$ has harmonic curvature and for which \[ \overline M\cdot\omega^n=\lambda(M)\,\omega^{n+1}, \] with this metrization unique up to multiplication by a degree-zero element of $\operatorname{Div}(K)$?

Unresolved qualifications. This is a transcription of the intended assertion, not a fully specified conjecture. The record does not define $X$, $n$, $\omega$, or $\lambda(M)$, nor does it specify the intersection product or the space in which the displayed equality holds. In particular, it does not state whether $n=\dim X$, what geometric or metric structure $\omega$ carries, or how $\lambda(M)$ is determined. The notions of admissibility and harmonic curvature, including their meaning at each type of place, are also unspecified. Finally, the definition of $\operatorname{Div}(K)$, its degree map, and its stated multiplicative action on metrizations must be supplied to make the uniqueness clause precise. These missing choices cannot be inferred from the displayed formula alone.

Short English statement

The source asks whether every line bundle admits an adelic metric with harmonic curvature at every place, satisfying the displayed intersection identity and unique modulo the stated degree-zero divisor action. Essential geometric hypotheses, definitions, and normalizations are missing, so a precise standalone formulation requires review of the cited report. The supplied status is partially solved; it is not independently verified here.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30003016$, title Adelic Harmonic Metrics on Line Bundles, collection Oberwolfach Reports Open Problems, source problem code catalogue entry.
• 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. No individual proposer is identified; the registry gives proposed year 2015.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially solved and reports uniqueness and existence in important special cases, while describing the global-field assertion for arbitrary varieties and line bundles as open. The special cases and supporting results are not identified. This is supplied dated curation, not an independently verified literature assessment; the record itself requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and says mathematical content was preserved while notation, typography, or omitted definitions were repaired for self-contained publication. This supplied editorial label does not resolve the missing hypotheses and definitions listed here. The cited report has not been independently consulted for this draft.