Hyperkähler SYZ and projective-space bases at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

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Hyperkähler SYZ and projective-space bases

at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

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If this game doesn't work on your computer, we can't help you. No Lean version yet. Some unformalized games could have issues!

expertly designed by an internal OpenAI model

Difficulty:	🧠🧠🧠🧠🧠	Ages:	13 - ∞
Skills:	shapes made of equations	Levels:	2
Category:	Algebraic and complex geometry	Lean version:	not yet
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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.

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LEVEL 1 Projective-space bases of Lagrangian fibrations

released 2026-09-23 | 3 theorems · 55 lemmas · 76 proofs · 52,020 words | PLAY LEVEL 1 » (pdf)

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.

LEVEL 2 The strong hyperkähler SYZ conjecture

released 2026-09-23 | 4 theorems · 9 lemmas · 12 proofs · 16,690 words | PLAY LEVEL 2 » (pdf)

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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