Working on a compact Hermitian manifold, this paper proves second-order a priori estimates for admissible (chi-semi-convex) smooth solutions of the complex Hessian equation chi^k wedge omega^(n-k) = psi omega^n, in the general case where the right-hand side psi and the (1,1)-form chi both depend on the gradient of the unknown function u. The key tool is a modified concavity inequality for the underlying fully nonlinear operator, established under a semi-convexity assumption (adapting real-case ideas of Lu and of Zhang to the complex setting). Because such a priori estimates are the crucial step in running the continuity method, the result advances the solvability theory for gradient-dependent complex Hessian equations, which arise in geometric problems such as the J-flow and quaternionic geometry.
Working on a compact Hermitian manifold, this paper proves second-order a priori estimates for admissible (chi-semi-convex) smooth solutions of the complex Hessian equation chi^k wedge omega^(n-k) = …