Browse › Algebraic and complex geometry › 055 Gepner symmetry and large-volume stability on threefolds
Prescribed large-volume charges on threefolds with trivial canonical bundle
Open PDFCite Report a problem Add a reference or comment Interested in working on this Start a discussion
Edit NOTES.md · Existing issues · Source on GitHub
Abstract
Let X be a smooth projective complex threefold with trivial canonical bundle. We construct numerical Bridgeland stability conditions with the exact ordinary and square-root-Todd central charges at every sufficiently large volume. One volume threshold works on an open set of real twists and ample directions. The resulting stability conditions have the support property on the full numerical Grothendieck group and stable point sheaves. Separately, on every smooth projective complex threefold we prove a strong tilt inequality above a volume threshold uniform in the object and twist along a fixed polarization.
If the PDF does not display here, open it directly: Open PDF.
Notes
No notes yet. Add a reference, a comment or your name with the buttons above, or edit NOTES.md on GitHub.
Automatically collected notes (unreviewed)
work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Formalization
- OpenAI, NumericalWalls.lean (OpenAI math release, accessed 2026-10-21). This Lean source develops real-valued character, discriminant, and fixed-branch wall-transport identities and inequalities supporting the numerical argument, rather than the construction of Bridgeland stability conditions.
Related prior literature mentioned by others
- Chunyi Li, A Remark on Stability Conditions on Smooth Projective Varieties (arXiv:2601.22994, 2026-01-30). MathDB’s discussion of the existence problem cites Li’s general construction as background to its entry on this manuscript, whose stated target additionally prescribes the large-volume charges.
Possibly relevant literature (found by the agent, not checked)
- Naoki Koseki, Stability conditions on Calabi-Yau double/triple solids (arXiv:2007.00044, Forum Math. Sigma). Proves a stronger Bogomolov-Gieseker inequality on weighted Calabi-Yau hypersurfaces in P(1,1,1,1,2) and P(1,1,1,1,4) and builds stability conditions from it; these are earlier special cases of the setting of Theorem 1.1 ($K_X\simeq\mathcal O_X$).
- Howard Nuer, Stability conditions on some families of Calabi-Yau threefolds via orbifolding (arXiv:2501.06207, 2024). Shows that Bridgeland stability conditions descend under orbifolding to further families of Calabi-Yau threefolds, including the quintic mirror; these are earlier existence results in the setting of Theorem 1.1.
Digests and expositions
- AI Proved What?, A fifth of a turn? Gepner symmetry and large-volume stability on threefolds (episode 055, October 2026; updated through 2026-10-19). This video and transcript outline the proposed Gepner construction and the companion prescribed-charge claims, with a correction distinguishing numerical Lean lemmas from formalization of the main theorems. [AI-generated per source] [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 055
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Prescribed-large-volume-charges-on-threefolds-with-trivial-canonical-bundle-September-24-2026,
author = {{OpenAI}},
title = {{Prescribed large-volume charges on threefolds with trivial canonical bundle}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Prescribed-large-volume-charges-on-threefolds-with-trivial-canonical-bundle-September-24-2026/paper.pdf}{OAI:Prescribed-large-volume-charges-on-threefolds-with-trivial-canonical-bundle-September-24-2026}},
year = {2026}
}Family 055: Gepner symmetry and large-volume stability on threefolds
Proves Toda's Gepner conjecture for every smooth complex quintic threefold, constructing a numerical Bridgeland stability condition with the prescribed phase shift 2/5. Also constructs numerical Bridgeland stability conditions at every sufficiently large volume on all smooth projective complex threefolds with trivial canonical bundle, with the exact ordinary and square-root-Todd central charges.
Other papers in this family:
- A Gepner stability condition on every smooth quintic threefold · September 24, 2026