Uniform Székelyhidi conjectures for complex Hessian equations on projective manifolds
partialconfidence 70%
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Precise statement
We prove a Nakai-Moishezon-type criterion for complex Hessian-type equations on projective manifolds whose associated degree-n polynomials are strongly strictly right-Noetherian. For strictly right-Noetherian polynomials of arbitrary degree, we prove a uniform Nakai-Moishezon-type criterion. This class includes the complex Hessian and Hessian quotient equations.
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What AI did
ChatGPT 5.6 Sol
The authors state that the proofs of the cone-inclusion lemmas in Section 3 are revised versions of arguments generated by ChatGPT 5.6 Sol. The authors subsequently checked and edited those arguments for mathematical clarity. ChatGPT 5.6 Sol also identified gaps in an earlier version of the manuscript and assisted with grammar correction. The Section 3 cone-inclusion lemmas are then used in Section 4 to prove the uniform Székelyhidi-type results.
Checked by this site on 17 August 2026 against the paper's LaTeX (arXiv:2608.03815, Chen-Nie-Xu). The declaration is verbatim as this entry describes: the Section 3 cone-inclusion lemma proofs are revised versions of arguments generated by ChatGPT 5.6 Sol, checked and edited by the authors, and the model was also used to find gaps in an earlier version - the v2 comment records that a gap in v1 (omega not a priori Kahler) was fixed, which is worth knowing when weighing a fresh preprint. The mathematics was not checked here. No independent review.
Chen, Nie and Xu prove a Nakai–Moishezon-type numerical criterion for a broad class of complex Hessian-type equations on compact projective manifolds. In particular, Corollary 1.3 gives a uniform version of Székelyhidi’s conjecture for complex Hessian quotient equations, while Corollary 1.5 proves the uniform version, formulated by Murakami, for complex k-Hessian equations. However, the results assume projectivity and a uniform numerical condition, whereas Székelyhidi’s original conjecture is formulated for arbitrary compact Kähler manifolds. Thus this should not be recorded as a full solution of the unrestricted original conjecture.
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argument (source-reported)
Source-reported tools: argument.
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What remains uncertain
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VibeMath has not independently audited the mathematical statement, proof, or novelty claim.