2025/12/08 by Pang, Kai, Sun, Haoyuan, Wang, Zhiwei +1
Mathematics · #Geometry and complex manifolds #Geometric Analysis and Curvature Flows #Algebraic Geometry and Number Theory
paper · doi:10.48550/arxiv.2512.07084
The aim of this paper is to further develop the theory of the degenerate complex Hessian equations on compact Hermitian manifolds. Building upon the generalization of the Bedford-Taylor pluripotential theory to complex Hessian equations by Kołodziej-Nguyen, we solve these equations in the (ω, m)-positive cone, (ω, m)-big classes and in nef classes, where ω is a reference Hermitian metric. These results are also new in the Kähler case. Moreover, we adapt our techniques to solve complex Monge-Ampère equations in nef classes with mild singularities. The solutions we obtain, in the compact Kähler case, coincide with those for the complex Monge-Ampère equations in the sense of the non-pluripolar product introduced by Boucksom-Eyssidieux-Guedj-Zeriahi. One of the key ingredients in the proof is the adaption, to the Hermitian setting, of a new a priori L^∞-estimate established by Guo-Phong-Tong and Guo-Phong-Tong-Wang.