Adelic Harmonic Metrizations over Global Fields

Definitions and mathematical statement

Let $K$ be a global field, let $X$ be a projective variety over $K$ of dimension $n$, and let $\overline L$ be a positive adelic metrized line bundle on $X$, with underlying line bundle $L$. Retain the source's normalization \[ \omega=\frac{c_1(\overline L)}{(\deg L)^{1/n}}. \] Here $c_1(\overline L)$ denotes the first Chern object associated with the metrized line bundle. The displayed normalization requires $n>0$ and a nonzero denominator; the supplied statement does not explicitly state the dimension restriction or define its positivity and degree conventions.

Source conjecture, with unresolved terminology retained. For every line bundle $M$ on $X$, does there exist an adelic metrization $\overline M$ of $M$ which is admissible, has harmonic curvature at every place of $K$, and satisfies \[ \overline M\cdot\omega^n=\lambda(M)\,\omega^{n+1}? \] The proposed uniqueness is modulo tensoring with the pullback of a degree-zero adelic divisor from $K$: writing $\pi:X\to\operatorname{Spec}K$ for the structure morphism, any two such metrizations should differ by a factor $\pi^*\overline D$, where $\overline D$ is an adelic divisor on $K$ with $\deg\overline D=0$. This expresses the source's stated equivalence; its precise realization as an equivalence of metrizations of the fixed line bundle $M$ requires clarification.

Unresolved qualifications. The supplied record does not define admissibility, the meaning of harmonic curvature at each archimedean and nonarchimedean place, or the coefficient $\lambda(M)$. In particular, it does not specify whether $\lambda(M)$ is prescribed by an omitted formula or is an additional unknown. It also does not identify the intersection theory or the space in which the displayed identity holds. The conventions for positivity, $\deg L$, and the normalized adelic form $\omega$ require confirmation. No smoothness, regularity, or geometric integrality hypothesis beyond the word "variety" is supplied. Accordingly, this is a transcription of the proposed existence and uniqueness assertion with its missing specifications exposed, rather than a fully determined conjecture.

Short English statement

The supplied conjecture asks whether every line bundle on a projective variety over a global field has an admissible adelic metric with harmonic curvature at every place and the stated normalized intersection identity, uniquely modulo degree-zero adelic divisors pulled back from the base field. The definitions of admissibility, harmonicity, and $\lambda(M)$, together with the intersection conventions and dimension restriction, remain unresolved.

Sources

• Supplied local project registry record, permanent UnsolvedMath ID $30003193$, title "Adelic Harmonic Metrizations over Global Fields," source problem code catalogue entry, in the collection "Oberwolfach Reports Open Problems." No individual proposer is supplied; the proposed year is recorded as 2016.
• Supplied report citation: Arithmetic Geometry (2017), Oberwolfach Reports, pp. 2171--2224, DOI 10.4171/owr/2016/38. Source URL and supplied literature-source link, labeled "OWR Arithmetic Geometry report": https://doi.org/10.4171/owr/2016/38. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. The citation's 2017 designation is retained alongside its 2016 DOI and proposed year without resolving their relationship.
• Supplied literature assessment, checked on 2026-08-21: the record labels the problem open, states that the report formulates a general admissible metrization conjecture and records only special cases, and reports that no general proof or disproof was found. This is supplied dated curation, not an independently verified literature assessment. The supplied background explicitly requests independent verification of time-sensitive status and cited claims; no particular special cases are identified in the record.
• The registry labels the statement corrected_verified and reports that the official report was checked to restore the global-field setup, normalized adelic form, intersection identity, and uniqueness qualification. These are supplied verification claims, not verification performed for this draft.