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A Gepner stability condition on every smooth quintic threefold
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Abstract
We prove Toda's normalized quintic Gepner conjecture. On every smooth complex quintic threefold, we construct a numerical Bridgeland stability condition for which tensoring by the hyperplane bundle, followed by the spherical twist at the structure sheaf, increases phase by 2/5.
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Related prior literature mentioned by others
- Yukinobu Toda, Gepner point and strong Bogomolov-Gieseker inequality for quintic 3-folds (arXiv:1305.0345, 2013-05-02). The AI Proved What? exposition identifies this as the source of the Gepner stability conjecture and candidate tilt construction addressed by the manuscript.
- Chunyi Li, On stability conditions for the quintic threefold (arXiv:1810.03434, 2018-10-08; Inventiones Mathematicae, 2019). AIProofWatch identifies Li’s existence theorem as prior work, distinguishing it from the manuscript’s additional prescribed Gepner symmetry.
- Chunkai Xu, Stronger Bogomolov–Gieseker type inequality on quintic threefold (arXiv:2511.21288, 2025-11-26; revised 2026-01-04). The AI Proved What? exposition identifies Xu’s stronger slope-sheaf inequality as the geometric input used in the manuscript’s construction.
Possibly relevant literature (found by the agent, not checked)
- Yukinobu Toda, Gepner type stability condition via Orlov/Kuznetsov equivalence (arXiv:1308.3791, 2013). Constructs Gepner-type stability conditions on graded matrix factorization categories for general cubic fourfolds containing a plane: an existence result of the Gepner type that Theorem 1.1 establishes for the quintic, in a different geometry.
- So Okada, Quintic periods and stability conditions via homological mirror symmetry (arXiv:1304.0937, 2013). Introduces Bridgeland-type stability conditions on the Fermat quintic, via homological mirror symmetry and the monodromy around the Gepner point; this bears on the Gepner symmetry $Z(\Phi E)=e^{2\pi i/5}Z(E)$ in Theorem 1.1.
- Howard Nuer, Stability conditions on some families of Calabi-Yau threefolds via orbifolding (arXiv:2501.06207, 2024). Proves that the quintic mirror admits Bridgeland stability conditions, extending the existence results for quintic-type Calabi-Yau threefolds that the paper builds on (Li 2019).
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:A-Gepner-stability-condition-on-every-smooth-quintic-threefold-September-24-2026,
author = {{OpenAI}},
title = {{A Gepner stability condition on every smooth quintic threefold}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/A-Gepner-stability-condition-on-every-smooth-quintic-threefold-September-24-2026/paper.pdf}{OAI:A-Gepner-stability-condition-on-every-smooth-quintic-threefold-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:
- Prescribed large-volume charges on threefolds with trivial canonical bundle · September 24, 2026