Hessian equations of Krylov type on K\"ahler manifolds

TL;DR

This paper studies Hessian equations on closed K"ahler manifolds, providing conditions for their solvability that extend previous results on Hessian and Hessian quotient equations.

Contribution

It offers a comprehensive criterion for solving Hessian equations of Krylov type on K"ahler manifolds, generalizing earlier work.

Findings

Derived necessary and sufficient conditions for solvability.

Unified framework for Hessian and Hessian quotient equations.

Extended classical results to complex K"ahler geometry.

Abstract

In this paper, we consider Hessian equations with its structure as a combination of elementary symmetric functions on closed K\"ahler manifolds. We provide a sufficient and necessary condition for the solvability of these equations, which generalize the results of Hessian equations and Hessian quotient equations.