View source for The hyperkähler SYZ conjecture

'''The hyperkähler SYZ conjecture''' asks whether, on a compact irreducible holomorphic symplectic (hyperkähler) manifold, every holomorphic line bundle whose first Chern class is nonzero, nef and isotropic for the Beauville–Bogomolov–Fujiki form is semiample, so that the sections of a power define a Lagrangian fibration. A companion question asks whether the base of a Lagrangian fibration is projective space. Two manuscripts in OpenAI's public mathematics collection ([https://github.com/openai/math github.com/openai/math]), dated 23 September 2026, claim both statements in every dimension and deformation type. This page summarizes the published claims and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper is not presented here as independent verification.

== Background ==
* As recounted in the introductions of the manuscripts:
** The strong form of the conjecture is Verbitsky's Conjecture 1.7; earlier nef-isotropic formulations appear in Hassett–Tschinkel and Sawon. Matsushita proved that fibrations of projective irreducible symplectic manifolds are half-dimensional and Lagrangian, with equidimensional fibres.
** Verbitsky proved nonvanishing for a nef isotropic bundle with a smooth semipositive metric, and Höring, Lazić and Lehn proved semiampleness in dimension four under a vanishing-Lelong-number current. Soldatenkov and Verbitsky showed that semiampleness at one nef point of a deformation component propagates to every nef point; with the constructions of Bayer–Macrì, Markman, Yoshioka, Mongardi–Rapagnetta and Mongardi–Onorati this settles the known deformation types.
** For the base, Hwang proved that a smooth base of a projective Lagrangian fibration is projective space, and Greb and Lehn extended this to compact Kähler total spaces; the conclusion is also known for the four known deformation types without a smoothness assumption (as recorded by Debarre, Huybrechts, Macrì and Voisin). In dimension four, Ou reduced the base to two cases and Huybrechts and Chenyang Xu excluded the second; Müller and Zheng Xu proved a local smoothness theorem for isolated quotient singularities.

== The original results ==
* '''Claim (SYZ).''' Theorem 1.1 of ''The strong hyperkähler SYZ conjecture'' states that if ''X'' is a compact irreducible holomorphic symplectic Kähler manifold of dimension 2''n'' and a holomorphic line bundle ''L'' has nonzero, nef, isotropic first Chern class, then ''L'' is semiample: some power ''L''<sup>''m''</sup> is the pullback of an ample bundle under a surjective map with connected fibres to a normal projective variety of dimension ''n'', all of whose fibre components are Lagrangian of dimension ''n''. The manuscript states that no restriction on the dimension, deformation type or second Betti number is imposed. Corollaries state that there are finitely many smooth deformation types in each dimension with ''b''<sub>2</sub> ≥ 5 (with the boundedness theorem of Engel, Filipazzi, Greer, Mauri and Svaldi) and, for projective ''X'', that the base is projective space.
* ''Method, as described:'' given the Soldatenkov–Verbitsky deformation theorem, it suffices to produce one semiample deformation of the prescribed class. The manuscript constructs a torus and rotates the complex structure, and uses the theorem of Greb, Lehn and Rollenske that a nonprojective hyperkähler manifold containing a holomorphic Lagrangian torus has a Lagrangian fibration.
* '''Claim (projective-space base).''' Theorem 1.1 of ''Projective-space bases of Lagrangian fibrations'' states that if ''f'' : ''X'' → ''B'' is a projective surjective map with connected fibres from a compact irreducible holomorphic symplectic Kähler manifold of dimension 2''n'' to a normal projective variety of dimension ''n'', with all fibre components ''n''-dimensional and Lagrangian, then ''B'' is isomorphic to '''P'''<sup>''n''</sup>. No smoothness of ''B'', bound on ''b''<sub>2</sub> or deformation type is assumed; the proof first shows that ''B'' is smooth and then applies Hwang's theorem.
* '''Sources''' (openai/math repository, commit <code>1581d4c</code>): [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/The-Strong-Hyperkahler-SYZ-Conjecture-September-23-2026 The strong hyperkähler SYZ conjecture] and [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026 Projective-space bases of Lagrangian fibrations].
* '''Formalization.''' The collection lists no Lean formalization for these results.

== Public follow-up work ==
{| class="wikitable"
! Date !! Work !! Relation !! Public claim !! Public status (checked 20 October 2026)
|-
| 2026-10-08 || P. Engel and M. Mauri, ''Hyperkähler SYZ conjecture'', [https://arxiv.org/abs/2610.12277v1 arXiv:2610.12277] || Parallel (independent) ||
* Theorem 1.1 states that every nef line bundle on a compact irreducible hyperkähler manifold is semiample; the paper notes that for a nontrivial isotropic bundle this is equivalent to the hyperkähler SYZ conjecture. Corollary 1.2 deduces that there are finitely many deformation classes of compact hyperkähler manifolds of fixed dimension with ''b''<sub>2</sub> ≥ 5. The paper names as its key new ingredient the metric SYZ conjecture, proved by Li and by Blum and Liu, which produces special Lagrangian tori in the large complex structure limit, followed by a hyperkähler rotation.
* Its AI disclosure states that a largely complete proof was human-written as a rough draft, that after the 6 October 2026 posting of manuscripts by OpenAI the authors used ChatGPT Astra to proofread it, and: "The structure of the proofs appears similar, but we have not yet had time to examine the OpenAI proof carefully."
| arXiv preprint, version 1 (7 pages).
|-
| 2026-10-16
| J. Jardine, Lattice24, referee reviews [https://github.com/JJardine919/openai-math-audit/blob/a06a8c6e77974d4c6021338a55a8e91b634f67f7/reviews/rank_446.json rank_446] (SYZ manuscript) and [https://github.com/JJardine919/openai-math-audit/blob/a06a8c6e77974d4c6021338a55a8e91b634f67f7/reviews/rank_220.json rank_220] (projective-base manuscript) in [https://github.com/JJardine919/openai-math-audit JJardine919/openai-math-audit] (archived at [https://doi.org/10.5281/zenodo.23422995 Zenodo 10.5281/zenodo.23422995])
| Review (AI-assisted audit)
| Reports no gap in either manuscript. For the SYZ manuscript, the reviewer states that it read every section; its two minor notes are that the return to ''X'' rests entirely on the Soldatenkov–Verbitsky fixed-class deformation theorem, used with a nonprojective Kähler endpoint, and on the projectivity of the base of Greb, Lehn and Rollenske's algebraic reduction, both of which it could not check against the sources (it saw no misapplication), and that the analysis on the limit space is assembled from several citations and stated somewhat loosely, with standard conclusions. For the projective-base manuscript, it states that it checked Sections 2–11 step by step, spot-checking parts of Sections 4 and 5, and that the companion cites this manuscript only in its final projective-base corollary, so there is no circularity.
| Part of an audit of the collection's manuscripts without Lean that the audit describes as complete. The reviews are published under CC BY 4.0.
|}

== Notes ==
* The partial results and the earlier classifications are due to the authors named in the Background section, as recounted by the manuscripts.
* The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.

[[Category:OpenAI results]]