Motivic Albanese and Walker Abel–Jacobi Targets

Definitions and mathematical statement

Let $X$ be a complex algebraic variety, and let $p$ denote the index occurring in the supplied source notation $CH_g^p(X)$. Write $T_{\mathrm W}(X,p)$ solely as a placeholder for the target of Walker's morphic Abel--Jacobi map referred to in that source. The proposed identification is \[ \operatorname{Alb}\left(CH_g^p(X)\right)({\mathbb{C}}) \overset{?}{\cong} T_{\mathrm W}(X,p), \] with the isomorphism required to be canonical.

The supplied record presents this as a conjectural identification for a complex algebraic variety. It does not define $CH_g^p(X)$, the Albanese construction applied to it, or Walker's morphic Abel--Jacobi map and its target. It also does not specify the range of $p$, whether additional hypotheses on $X$ are required, or the category and compatibility conditions intended by a canonical isomorphism. Consequently, the displayed question preserves the source notation but remains schematic pending those definitions and qualifications; no smoothness, properness, projectivity, or index restrictions have been added.

Short English statement

The source asks whether the complex points of the Albanese object associated with $CH_g^p(X)$ are canonically isomorphic to the target of Walker's morphic Abel--Jacobi map. The constructions and their domains require clarification before this can be a fully specified conjecture.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001288$, exact source code catalogue entry, title: Motivic Albanese and Walker Abel--Jacobi Targets. The record assigns the proposed year 2009 and supplies no named proposer; Walker is named in connection with the map.
• Algebraic $K$-Theory and Motivic Cohomology (2009), Oberwolfach Reports, pp. 1731--1774, DOI: https://doi.org/10.4171/owr/2009/31. The record describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. This report is the supplied authoritative source citation; no precise internal location for this problem is given.
• Ayoub--Barbieri-Viale on 1-motivic sheaves: https://arxiv.org/abs/math/0607738. This is the additional supplied literature reference.
• Literature assessment checked at 2026-08-21: the supplied curation labels the problem open, describes the Walker morphic Abel--Jacobi identification as a conjecture, and states that the cited motivic Albanese work does not prove the general identification. This is supplied dated curation derived from the Oberwolfach report, not an independently verified assessment; time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and reports that its mathematical content was checked against the official Oberwolfach report, with notation, context, and question form normalized without changing the source-level claim. This is supplied verification metadata, not verification performed for this draft.