Degenerate Complex Hessian type equations on compact Hermitian manifolds and Applications

Abstract: 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 $L^\infty$-estimate established by Guo-Phong-Tong and Guo-Phong-Tong-Wang.