Let X be a not necessarily projective hyperkähler manifold, and let L be a nef divisor on X such that q_X(L,L)=0, where q_X(·,·) denotes the Bogomolov--Beauville--Fujiki form on X. Non-vanishing conjecture. One has kappa(L)≥ 0. This conjecture concerns non-vanishing for nef isotropic divisors on hyperkähler manifolds and is examined as a comparison for the paper's method. The supplied text does not state that it has been resolved.

Non-vanishing conjecture for hyperkähler manifolds

About 5 years old · traced to 2021

Let $X$ be a not necessarily projective hyperkähler manifold, and let $L$ be a nef divisor on $X$ such that

q_X(L,L)=0,

where $q_X(\cdot,\cdot)$ denotes the Bogomolov--Beauville--Fujiki form on $X$ .

Non-vanishing conjecture. One has $\kappa(L)\geq 0$ .

This conjecture concerns non-vanishing for nef isotropic divisors on hyperkähler manifolds and is examined as a comparison for the paper's method. The supplied text does not state that it has been resolved.

References

Primary source

Haidong Liu and Shin-ichi Matsumura, “Strictly nef divisors on K-trivial fourfolds”, arXiv:2105.07259 (2021).

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RemarkAI-assistedClaimed by OpenAI. The manuscript claims semiampleness for holomorphic line bundles with nonzero nef isotropic first Chern class on compact irreducible holomorphic symplectic Kahler manifolds, giving nonvanishing in this subclass.See full solutionHide full solution

Claimed by OpenAI.

The source concerns compact irreducible holomorphic symplectic Kahler manifolds, without a projectivity assumption, and line bundles with nonzero nef first Chern class isotropic for the Beauville-Bogomolov-Fujiki form. Its claimed semiampleness implies a nonzero section of some positive tensor power, hence nonvanishing in this subclass of the problem. It does not assert that every hyperkahler manifold has such a line bundle, and the zero-class case is not covered by this statement.

GitHub repository: https://github.com/openai/math

Manuscript: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Strong-Hyperkahler-SYZ-Conjecture-September-23-2026/main.pdf

OpenAI-041-02-The-strong-hyperk-hler-SYZ-conjecture.pdf531,783 bytesOpen