Harmonic norms on quiver representations without spherical completeness (Haiden-Katzarkov-Kontsevich-Pandit Question 6.24)
Haiden, Katzarkov, Kontsevich and Pandit (Towards Categorical Kähler Geometry, arXiv:2609.00978, Theorem 6.23) prove that over a spherically complete non-archimedean field , a polystable finite-dimensional representation of a finite quiver admits a harmonic norm, and a representation admitting one is semistable. Their Question 6.24 asks: can the spherical completeness assumption on be omitted in Theorem 6.23?
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-assisted
- Method
- Argument
- Field
- Representation theory; non-archimedean geometry
- Posed by
- Fabian Haiden, Ludmil Katzarkov, Maxim Kontsevich and Pranav Pandit, Towards Categorical Kähler Geometry (arXiv:2609.00978, September 2026), Question 6.24
- Year posed
- 2026
- Years open
- 0y
- Solved
- 2026-10-01
- Model
- ChatGPT-6.0 Sol
- Vendor
- OpenAI
- Collaborators
- Oren Ben-Bassat
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 8 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Yes, for finite quivers: over any complete non-archimedean field, every polystable finite-dimensional representation admits a split harmonic norm, and a representation admitting a harmonic norm is semistable, so both implications of Theorem 6.23 hold without spherical completeness. The norm produced is split (diagonalisable), which is the natural form here, since without spherical completeness norms need not diagonalise.
What the AI did
This article was written through extensive conversations with OpenAI’s
ChatGPT-6.0 Sol, beginning with the author’s original arguments, which contained substantial gaps. The ideas and development of the article represent roughly equal contributions of the author and the AI. OpenAI’s ChatGPT, Codex, and Prism were used to discuss and revise the arguments and to proofread and typeset the manuscript. The examples were chosen by the author.
Verification
Read by this site on 6 October 2026: Question 6.24 was checked in the source of arXiv:2609.00978, where it follows Theorem 6.23 directly, and the paper's main theorem was compared with it and covers both implications for every complete non-archimedean field. The proof was not refereed here. Single-author preprint, posted 1 October 2026; no expert reaction or later discussion found.
Source
- PaperarXiv
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