Algebraic and complex geometry · Result 055

Gepner symmetry and large-volume stability on threefolds

2 papers

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.

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▶Introduction: the problem and why it mattersin preparation

▶The presented solutionin preparation

Papers 2

A Gepner stability condition on every smooth quintic threefold

OpenAI · September 24, 2026

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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@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

OpenAI · September 24, 2026

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.

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@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}
}

Formal verification

No Lean formalization is listed for this result yet. The repository notes that unformalized results could have issues.