Numerical criteria for projective complex Hessian equations
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When do projective uniform numerical conditions guarantee solvability of complex Hessian-type equations? The studied criterion uses right-Noetherian polynomials and inclusions of their associated cones.
Why it matters and what counts as progress
Why it matters: A numerical criterion connects geometric inequalities to analytic existence for this projective setting.
What counts as progress: A proved criterion under the stated projective and uniform hypotheses, including the cone-inclusion lemmas for adjoining a linear factor and perturbing f to f−εf′.
How the problem developed
- 2018 · Conjecture [1]
Gábor Székelyhidi Proposed a numerical criterion for solvability of Hessian quotient equations on compact Kähler manifolds.
- 4 Aug 2026 · Criterion [2]
Gao Chen, Sijie Nie and Yulun Xu Proved Nakai–Moishezon-type criteria for projective complex Hessian-type equations under right-Noetherian polynomial conditions; the result includes Hessian quotient equations.
AI contribution
A numerical criterion for complex Hessian type equations on projective manifolds
Chen, Nie and Xu’s cone-inclusion Lemmas 3.1 and 3.2 show that adjoining a linear factor preserves the required right-Noetherian structure and inclusion, while f−εf′ has a strengthened right-Noetherian property and inclusion. These lemmas feed a projective uniform numerical criterion for complex Hessian-type equations.
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