Result 041, Algebraic and complex geometry

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.

Proof

The bigger picture

Why it matters

The manuscripts claim that two central features of hyperkähler geometry are more rigid than they might appear: certain numerical conditions on line bundles guarantee geometric maps, and the bases of projective Lagrangian fibrations must be projective spaces.

What changes?

The SYZ manuscript reports that every holomorphic line bundle on a compact irreducible holomorphic symplectic Kähler manifold is semiample if its first Chern class is nonzero, nef and isotropic. A line bundle attaches a complex line to each point. Nef means its class is a limit of positive classes; isotropic means square zero for the manifold's canonical quadratic form. Semiample means some positive tensor power has global sections with no common zero, so these sections define a map.

What does that help mathematicians do?

The other manuscript reports that every projective Lagrangian fibration from such a manifold, with normal projective base, has projective space as its base, in every dimension and deformation type. Here the general fibers have half the manifold's complex dimension, and the symplectic form restricts to zero on them. This rules out other normal projective bases under those assumptions, letting researchers study these fibrations over a fixed, familiar kind of space rather than an unknown base.

Are there practical applications?

The immediate value is foundational. The semiampleness claim turns numerical information about a line bundle into the existence of a geometric map, while the base claim constrains the spaces that qualifying fibrations can map onto. Together they would sharpen the structural study of these manifolds without classifying their fibers or total spaces.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

Projective-space bases of Lagrangian fibrations

September 23, 2026 98 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

The strong hyperkähler SYZ conjecture

September 23, 2026 32 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.