Strong hyperkähler SYZ semiampleness conjecture
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The strong hyperkähler SYZ conjecture asks whether every holomorphic line bundle with nonzero nef isotropic first Chern class on a compact irreducible holomorphic symplectic Kähler manifold is semiample. The resulting fibration should have a normal projective base of half the complex dimension and Lagrangian fibers.
Why it matters and what counts as progress
Why it matters: It predicts a geometric fibration from numerical positivity throughout all dimensions and deformation types.
What counts as progress: Prove semiampleness for every nonzero nef isotropic class, beyond special deformation types, or find an obstruction under the stated hypotheses.
How the problem developed
- 2010 · Partial result [1]
Frédéric Campana, Keiji Oguiso and Thomas Peternell Studied nef line bundles on non-algebraic hyperkähler manifolds and supplied evidence for the semiampleness question underlying the hyperkähler SYZ conjecture.
- 23 Sept 2026 · Proof claim [2]
OpenAI Released a manuscript claiming that every nonzero nef isotropic line bundle on a compact irreducible holomorphic symplectic Kähler manifold is semiample and induces a Lagrangian fibration.
- 8 Oct 2026 · Proof claim [3]
Philip Engel and Mirko Mauri Posted an independent manuscript claiming the hyperkähler SYZ semiampleness result for nontrivial nef isotropic line bundles.
AI contribution
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, according to the OpenAI manuscript. The resulting map has a normal projective base of half the complex dimension and Lagrangian fibers.
Sources
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