Breakthrough Tracker record
A claimed smooth-projective counterexample to the Bondal–Polishchuk conjecture
A preprint gives a weak Fano threefold whose bounded derived category has two full exceptional collections in different braid-group orbits, claimed as the first smooth-projective counterexample to the Bondal–Polishchuk conjecture. Its expanded second version also reports infinitely many connected components in the threefold’s Bridgeland stability space.
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- Stable ID
math-bondal-polishchuk-smooth-projective-counterexample-2026- Revision
math-bondal-polishchuk-smooth-projective-counterexample-2026.v2- Field
- Mathematics · Algebraic geometry and derived categories
- Evidence
- Tier 1 · Peer reviewed: No
- Record state
- Provisional · Provisional claimed counterexample
- Last checked
AI role
The author says ChatGPT 5.6 found mistakes in earlier constructions and a mistaken reference, but did not produce the counterexample or generate the manuscript text.
Record details
- Problem or result
- Bondal and Polishchuk’s 1993 transitivity conjecture for full exceptional collections
- Authors
- Anya Nordskova
- Institutions
- Kavli Institute for the Physics and Mathematics of the Universe, University of Tokyo
- Result date
- Preprint submitted July 28; substantially expanded second version posted August 3, 2026
Why it matters
Earlier published counterexamples lived in partially wrapped Fukaya categories. The construction targets the smooth-projective setting that motivated the original conjecture, and the added stability-space result would give the first known disconnected Bridgeland stability space for a smooth projective variety.
Limits
This is an 18-page second-version preprint without located independent correctness assessment or peer review. The expanded manuscript retains the same core counterexample and adds refinements and stability-condition results; the broader conjecture had already failed in other triangulated categories.
Sources
- Primary: Anya Nordskova, A smooth projective counterexample to Bondal–Polishchuk’s conjecture
- Published context: Chang, Haiden and Schroll, published counterexamples in other triangulated categories
Correction and revision history
- 2026-07-29 — Added as a provisional claimed counterexample with the smooth-projective scope and disclosed error-checking role of AI stated precisely.
- 2026-08-08 — Reviewed the substantially expanded v2, including new Bridgeland-stability results; the core counterexample and provisional status are unchanged.
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