Geometric Chern Classes in Arakelov Cobordism

Definitions and mathematical statement

Supplied problem. Give a geometric interpretation of the Chern classes in Arakelov algebraic cobordism.

The supplied record associates this question with Rodriguez's weak Arakelov algebraic cobordism theory, but does not define that theory or identify it explicitly as the theory intended in the original question. It also does not specify the admissible base schemes or geometric objects, the bundles and any additional data carrying Chern classes, the groups or operators in which those classes occur, or the range of Chern indices.

Accordingly, the requested construction is a geometric description of the Chern classes in the intended Arakelov algebraic cobordism theory. The record does not specify what constitutes a geometric interpretation or which compatibility properties it must satisfy. These missing specifications prevent a fully quantified mathematical formulation; choosing them would add hypotheses absent from the supplied source. This is a statement-only draft requiring human mathematical review.

Short English statement

The problem asks for a geometric meaning for Chern classes in Arakelov algebraic cobordism. The intended theory, its domain, and the required form of the interpretation need clarification. The supplied record labels the problem open; this status has not been independently verified.

Sources

• Local registry: data/unsolvedmath/problems.json and supplementary catalog/problems.json; supplied record with permanent UnsolvedMath ID $30003773$, source problem code catalogue entry, titled Geometric Chern Classes in Arakelov Cobordism. The record supplies proposed year 2018 and names no proposer. Registry file contents were not independently inspected.
• Supplied report citation: Mini-Workshop: Arithmetic Geometry and Symmetries around Galois and Fundamental Groups (2019), Oberwolfach Reports, DOI 10.4171/owr/2018/17, pp. 1021--1080. https://doi.org/10.4171/owr/2018/17. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. The citation's 2019 date differs from the record's proposed year 2018 and its label "OWR 2018 report"; the discrepancy is unresolved.
• Supplied literature source: Rodriguez, weak Arakelov cobordism. https://arxiv.org/abs/1608.04191. No fuller bibliographic attribution is supplied.
• Supplied literature source labeled "OWR 2018 report": https://doi.org/10.4171/owr/2018/17.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that Rodriguez constructs a weak Arakelov algebraic cobordism theory, while a geometric interpretation of its Chern classes is still not supplied by the available primary literature. This is supplied dated curation, not independently verified. The background explicitly calls it a dated literature triage derived from the supplied OWR curation and requests independent verification of time-sensitive status and cited claims.