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Projective-space bases of Lagrangian fibrations
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Abstract
We prove that the normal projective base of a projective Lagrangian fibration from a compact irreducible holomorphic symplectic Kähler manifold is projective space. This resolves the projective-space base conjecture for such fibrations in every dimension and deformation type.
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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). A parallel preprint with an independent proof of the semiampleness statement that this paper imports from Theorem 1.1 of its companion preprint; its authors disclose a largely complete human-written draft preceding the release, later proofread with ChatGPT Astra, and that they had not yet carefully examined OpenAI's proof. [family-level]
Related prior literature mentioned by others
- Jun-Muk Hwang, Base manifolds for fibrations of projective irreducible symplectic manifolds (arXiv:0711.3224, 2007-11-20). The AI Proved What? family exposition identifies Hwang's projective-space theorem for smooth bases as the prior result that the manuscript seeks to extend to normal bases.
Possibly relevant literature (found by the agent, not checked)
- Benjamin Bakker, Christian Schnell, A Hodge-theoretic proof of Hwang's theorem on base manifolds of Lagrangian fibrations (arXiv:2311.08977, 2023). Gives a Hodge-theoretic proof of Hwang's smooth-base theorem, which the paper applies in its final step once it has shown that the base is smooth.
- Alessio Bottini, Emanuele Macrì, Paolo Stellari, Hyper-Kähler varieties: Lagrangian fibrations, atomic sheaves, and categories (arXiv:2603.23033, March 2026). A recent survey of hyper-Kähler varieties organised around Lagrangian fibrations; it gives background for the projective-space base conjecture 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:Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026,
author = {{OpenAI}},
title = {{Projective-space bases of Lagrangian fibrations}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026/main.pdf}{OAI:Projective-Space-Bases-of-Lagrangian-Fibrations-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:
- The strong hyperkähler SYZ conjecture · September 23, 2026