Degenerate Complex Hessian type equations on compact Hermitian manifolds and Applications

TL;DR

This paper advances the theory of degenerate complex Hessian equations on compact Hermitian manifolds, solving these equations in various classes and extending techniques to complex Monge-Ampère equations with applications in Kähler geometry.

Contribution

It generalizes pluripotential theory to complex Hessian equations on Hermitian manifolds and solves these equations in new geometric contexts, including nef and big classes.

Findings

Solved degenerate complex Hessian equations in multiple classes on Hermitian manifolds.

Extended solutions to complex Monge-Ampère equations with mild singularities.

Adapted new a priori $L^ Infty$-estimates to Hermitian setting.

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 Ko\l odziej-Nguyen, we solve these equations in the $(\omega, m)$-positive cone, $(\omega, m)$-big classes and in nef classes, where $\omega$ is a reference Hermitian metric. These results are also new in the K\"ahler case. Moreover, we adapt our techniques to solve complex Monge-Amp\`ere equations in nef classes with mild singularities. The solutions we obtain, in the compact K\"ahler case, coincide with those for the complex Monge-Amp\`ere 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.