Existence of Bridgeland stability conditions on smooth projective varieties

About 11 years old · traced to 2015

Let $X$ be a smooth projective variety. A stability condition means a Bridgeland stability condition on $\operatorname{D^b}(X)$ , defined with respect to the numerical lattice $H^\ast_{\mathrm{alg}}(X)$ . Existence conjecture. The category $\operatorname{D^b}(X)$ admits stability conditions for any smooth projective variety $X$ .

This is presented as a difficult folk conjecture underlying the noncommutative minimal model program. The statement is intended to provide the stability-condition framework from which canonical semiorthogonal decompositions can arise; its general validity remains open.

References

Primary source

Daniel Halpern-Leistner, “The noncommutative minimal model program”, arXiv:2301.13168 (2024).

Additional references

2 papers in this index state this conjecture (2015–2023). The statement above is taken from the most recent of them; the others are arXiv:1504.01177.

derived-categories stability-conditions

Posted 1 month ago · edited 2 weeks ago

Progress summary

Refreshed 2 weeks ago

Claimed solved

A January 2026 paper claims to prove that every smooth projective variety has the required stability conditions, but the proof has not been independently checked.

The conjecture asks whether every smooth projective variety admits Bridgeland stability conditions on its derived category. It was framed as a difficult folk conjecture connected with the noncommutative minimal model program.

Known results

The standard construction was known in dimensions at most $2$ ; Bayer, Macrì, and Toda proved important threefold cases, including $\mathbb{P}^3$ .
Explicit examples included certain Calabi–Yau, Fano, abelian, and Kummer-type varieties.
In 2023, nonemptiness for a quintic threefold was still presented as open.

January 2026 claimed proof

On January 30, 2026, Chunyi Li submitted “A Remark on Stability Conditions on Smooth Projective Varieties,” claiming the full theorem over $\mathbb{C}$ . The strategy restricts conditions from projective space along closed embeddings and asserts the support property; the paper notes that its initial proof contained a mistake and presents a correction. No retrieved source supplies independent verification or a referee report.

Current status (as of September 2026): A general existence theorem is claimed by Li, but its proof and its precise numerical-lattice formulation remain independently unverified.

Sources

RemarkAI-assistedClaimed by OpenAI. The manuscript claims numerical Bridgeland stability conditions with exact ordinary and square-root-Todd large-volume charges on smooth complex projective threefolds with trivial canonical bundle, with support on the full numerical Grothendieck group.See full solutionHide full solution

Claimed by OpenAI.

This addresses the existence problem in the subclass of smooth complex projective threefolds with trivial canonical bundle. Theorem 1.1 allows open neighborhoods of any real twist and ample direction, with one sufficiently large volume threshold. The support property is on the full numerical Grothendieck group $K_{\mathrm{num}}(X)$ . The source also gives a separate large-volume tilt inequality. No identification with a larger homological-cohomology lattice, stability existence in all dimensions, or canonical semiorthogonal decomposition is inferred.

GitHub repository: https://github.com/openai/math

Manuscript: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Prescribed-large-volume-charges-on-threefolds-with-trivial-canonical-bundle-September-24-2026/paper.pdf

OpenAI-055-02-Prescribed-large-volume-charges-on-threefolds-with-trivial-canonical-bundle.pdf625,204 bytesOpen