Result 055, Algebraic and complex geometry

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.

Proof

The bigger picture

Why it matters

Stability rules help geometers organize bundles and related objects on complex three-dimensional spaces. These manuscripts claim rules that respect a specific symmetry on degree-five hypersurfaces and exact prescribed formulas in a large-volume regime.

What changes?

Numerical Bridgeland stability assigns phases and stability to complexes of sheaves, generalized bundles, using numerical data. For every smooth complex quintic threefold, the first manuscript reports conditions where tensoring by the hyperplane bundle followed by the spherical twist at the structure sheaf shifts phase by 2/5. For all smooth projective complex threefolds with trivial canonical bundle, the second reports stability at every sufficiently large volume with exact ordinary and square-root-Todd central charges, the formulas determining phases.

What does that help mathematicians do?

One large-volume threshold works across an open set of real twists and ample directions. Stable point sheaves give basic stable objects, while the support property on the full numerical Grothendieck group controls semistable numerical classes. Separately, every smooth projective complex threefold is claimed to satisfy a strong tilt inequality above a threshold uniform in the object and twist along a fixed polarization. This broader inequality is not a stability construction for all such threefolds.

Are there practical applications?

The immediate value is foundational. The quintic result would let researchers track stability under a specified geometric symmetry with an exact phase change. The large-volume result would provide the prescribed stability framework for studying sheaf complexes, rather than requiring a substitute central-charge formula.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

A Gepner stability condition on every smooth quintic threefold

September 24, 2026 22 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Prescribed large-volume charges on threefolds with trivial canonical bundle

September 24, 2026 49 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 10 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.