Coarse moduli of motivic augmentations
Abstract
A variety X over a suitable base Z gives rise to a highly structured algebra C*(X) in motives over Z. In turn, a Z-point gives rise to an augmentation C*(X) → 1. This assignment X(Z) → Aug(C*(X)) factors the so-called ``unipotent Kummer map'' to torsors under the unipotent fundamental group in realizations. In one direction, this suggests the possibility of ``performing'' Chabauty-Kim theory motivically without waiting for a motivic t-structure. In a different (largely independent) direction, we may hope to extract arithmetic information for use in bounding sets of integral points from the full rational homotopy type going beyond π1. In both directions, the coarse space for motivic augmentations Aug(C*(X)) would benefit from a structure of finite type ℚ-variety, and similarly, the coarse space Aug(C*Fϕ(X)) of augmentations in filtered ϕ modules would benefit from a structure of finite type ℚp-variety. We establish two criteria for representability, compute these spaces in several examples, and construct a comparison with Selmer varieties. Finally, we demonstrate how these constructions lead to K-theoretic finiteness criteria in an example.
Related
- Motivic Euler characteristics of moduli spaces of curves
- Heavy-Light Wall Crossings and Motivic Asymptotics of Moduli Spaces of Curves. Part I: Stabilization Graphs and Euler Characteristics
- Existence of coarse moduli superspaces for Deligne--Mumford superstacks
- Turán densities of hypergraph augmentations
- Schematic Functorialities of Birational Motivic Homotopy Categories
BibTeX
@misc{dancohen2026coarse,
title = {{Coarse moduli of motivic augmentations}},
author = {Ishai Dan-Cohen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05670},
eprint = {2609.05670},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05670},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Coarse moduli of motivic augmentations
AU - Ishai Dan-Cohen
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2609.05670
UR - https://arxiv.org/abs/2609.05670
AB - A variety X over a suitable base Z gives rise to a highly structured algebra C^*(X) in motives over Z. In turn, a Z-point gives rise to an augmentation C^*(X) → 1. This assignment X(Z) → Aug(C^*(X)) factors the so-called ``unipotent Kummer map'' to torsors under the unipotent fundamental group in realizations. In one direction, this suggests the possibility of ``performing'' Chabauty-Kim theory motivically without waiting for a motivic t-structure. In a different (largely independent) direction, we may hope to extract arithmetic information for use in bounding sets of integral points from the full rational homotopy type going beyond π_1. In both directions, the coarse space for motivic augmentations Aug(C^*(X)) would benefit from a structure of finite type ℚ-variety, and similarly, the coarse space Aug(C^*_Fϕ(X)) of augmentations in filtered ϕ modules would benefit from a structure of finite type ℚ_p-variety. We establish two criteria for representability, compute these spaces in several examples, and construct a comparison with Selmer varieties. Finally, we demonstrate how these constructions lead to K-theoretic finiteness criteria in an example.
ER -