Algebraic and complex geometry · Result 041

Hyperkähler SYZ and projective-space bases

2 papers

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.

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▶Introduction: the problem and why it mattersin preparation

▶The presented solutionin preparation

Papers 2

Projective-space bases of Lagrangian fibrations

OpenAI · September 23, 2026

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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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}
}

The strong hyperkähler SYZ conjecture

OpenAI · September 23, 2026

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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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}
}

Formal verification

No Lean formalization is listed for this result yet. The repository notes that unformalized results could have issues.