Uniform Székelyhidi conjectures for complex Hessian equations on projective manifolds
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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.
Context
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.
Szekelyhidi's Nakai-Moishezon-type criteria for complex Hessian equations are a standing target in Kahler geometry with a real partial-results literature. This settles a broad class (strictly right-Noetherian polynomials) on projective manifolds - substantial, specialist, partial.
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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.
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