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The strong hyperkähler SYZ conjecture
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Abstract
We prove the strong hyperkähler SYZ conjecture: every holomorphic line bundle with nonzero nef isotropic first Chern class on a compact irreducible holomorphic symplectic Kähler manifold is semiample.
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work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Follow-up works
- Philip Engel, Mirko Mauri, Hyperkähler SYZ conjecture (arXiv:2610.12277, 2026-10-08). Independent human-written proof of the same semiampleness statement and of the finiteness consequence for deformation classes with second Betti number at least five; its AI-disclosure section says "The structure of the proofs appears similar, but we have not yet had time to examine the OpenAI proof carefully." [family-level]
- Jie Fu, Bin Wang, Shihao Wang, Zhiwei Zheng, A Sharp Bound for the Second Betti Number of Hyperkähler Sixfolds (arXiv:2610.15803, 11 Oct 2026). Cites this preprint and Engel–Mauri in its introduction as recent proofs of SYZ statements and their finiteness consequences, and states that its own proofs do not use these SYZ results.
Comments and discussion
- OSVerify, The strong hyperkähler SYZ conjecture (automated review report, undated; reviewed manuscript dated 2026-09-23). The report flags the introduction's description of the Engel–Filipazzi–Greer–Mauri–Svaldi finiteness theorem as boundedness, while stating that Section 6 uses the needed finiteness conclusion. [AI-generated per source]
Related prior literature mentioned by others
- Philip Engel, Stefano Filipazzi, François Greer, Mirko Mauri and Roberto Svaldi, Boundedness of some fibered K-trivial varieties (arXiv:2507.00973, 2025-07-01). OSVerify's report singles out this paper's finiteness of deformation classes of primitive symplectic varieties admitting Lagrangian fibrations as the input to Corollary 1.2, distinguishing it from boundedness of a family.
- Daniel Huybrechts and Mirko Mauri, Lagrangian fibrations (arXiv:2108.10193v2, 2022-01-31; Milan Journal of Mathematics 90, 2022). Engel–Mauri identify Sections 3.5–3.6 of this survey as an antecedent for the degeneration, special Lagrangian torus and hyperkähler rotation strategy also pursued by the OpenAI manuscript.
Possibly relevant literature (found by the agent, not checked)
- Alessio Bottini, Emanuele Macrì, Paolo Stellari, Hyper-Kähler varieties: Lagrangian fibrations, atomic sheaves, and categories (arXiv:2603.23033, March 2026). A recent survey of compact hyper-Kähler varieties organised around Lagrangian fibrations; it gives background for the existence question that Theorem 1.1 addresses.
Digests and expositions
- AI Proved What?, Fibered by tori? Hyperkähler SYZ and projective-space bases (Episode 041, October 2026). This AI-assisted video and transcript explain both manuscripts' claims and proposed proof strategies, including the projectivity assumption in the base theorem and its dependence on the SYZ paper. [AI-generated per source] [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 041
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:The-Strong-Hyperkahler-SYZ-Conjecture-September-23-2026,
author = {{OpenAI}},
title = {{The strong hyperk\"ahler SYZ conjecture}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/The-Strong-Hyperkahler-SYZ-Conjecture-September-23-2026/main.pdf}{OAI:The-Strong-Hyperkahler-SYZ-Conjecture-September-23-2026}},
year = {2026}
}Family 041: Hyperkähler SYZ and projective-space bases
Proves the strong hyperkähler SYZ conjecture: every holomorphic line bundle with nonzero nef isotropic first Chern class on a compact irreducible holomorphic symplectic Kähler manifold is semiample. It also proves that every projective Lagrangian fibration with normal projective base has projective space as its base, in every dimension and deformation type.
Other papers in this family:
- Projective-space bases of Lagrangian fibrations · September 23, 2026