Hyperkähler SYZ and projective-space bases

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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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The paper states

Theorem 1.1

In The strong hyperkähler SYZ conjecture, 23 September 2026

Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.

Three other possible matches are listed below the card, among them Theorem 1.1 in Projective-space bases of Lagrangian fibrations and Corollary 1.2 in the same paper.

Let X be a compact irreducible holomorphic symplectic Kähler manifold of complex dimension 2n, where n\ge1. If a holomorphic line bundle L satisfies

c_1(L)\ne0,\qquad c_1(L)\in\overline{\mathcal K_X}, \qquad q_X(c_1(L))=0,

then L is semiample.

More precisely, there exist a positive integer m, a normal projective variety B, an ample line bundle A on B, and a surjective holomorphic map with connected fibers

f:X\longrightarrow B,\qquad L^m\simeq f^*A,

such that \dim_{\C}B=n. Every irreducible component of every fiber has complex dimension n, and \sigma vanishes on its smooth locus.

Three more possible matches

Theorem 1.1 in Projective-space bases of Lagrangian fibrations, 23 September 2026.
Corollary 1.2 in The strong hyperkähler SYZ conjecture, 23 September 2026.
Corollary 1.3 (Projective case) in The strong hyperkähler SYZ conjecture, 23 September 2026.

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Papers

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Projective-space bases of Lagrangian fibrations

23 September 2026, current edition.

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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.

The strong hyperkähler SYZ conjecture

23 September 2026, current edition.

Read it here 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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Author not stated. Source

Philip Engel, Mirko Mauri, Hyperkähler SYZ conjecture, 8 October 2026. Source
Engel and Mauri prove the hyperkaehler SYZ conjecture in a 7-page paper and say their proof structure appears similar to OpenAI’s, which they have not yet had time to examine.

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