Bondal-Polishchuk Conjecture for a Smooth Projective Variety
Bondal and Polishchuk conjectured in 1993 that the braid group acts transitively on the set of full exceptional collections in a triangulated category. Chang, Haiden and Schroll disproved it for partially wrapped Fukaya categories, but no counterexample of the form for a smooth projective variety was known. A particular weak Fano threefold provides one.
- Result
- Disproved(see note)
- Status
- Resolved
- AI contribution
- AI-assisted
- Method
- Construction
- Field
- Algebraic geometry
- Posed by
- Alexei Bondal, Alexander Polishchuk
- Year posed
- 1993
- Years open
- 33y
- Solved
- 2026-07-28
- Model
- ChatGPT 5.6
- Vendor
- OpenAI
- Collaborators
- Anya Nordskova
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 25 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
first counterexample of the form D^b(X) for X smooth projective
What the AI did
The AI disclosure is carefully bounded and worth quoting in full: the model most significantly found numerous mistakes in earlier attempted constructions, leading to the successful modifications, and it located the reference containing the threefold used. The author states that it did not produce the actual counterexample and was not used to generate any of the text.
Verification
Single-author arXiv preprint; not yet peer-reviewed.
Source
arXiv:2607.25391 - A smooth projective counterexample to Bondal-Polishchuk's conjecture