Let (X, ω) be a compact Kähler manifold of dimension n and fix 1 ≤ m ≤ n. We prove that the total mass of the complex Hessian measure of ω-m-subharmonic functions is non-decreasing with respect to the singularity type. We then solve complex Hessian equations with prescribed singularity, and prove a Hodge index type inequality for positive currents.

Let (X, ω) be a compact Kähler manifold of dimension n and fix an integer m such that 1 ≤ m ≤ n. We characterize the relative full mass class E φ (X, ω, m). We also prove the integration by parts formula of Hessian type. Given a model potential φ, we study degenerate complex Hessian equations of the form (ω + dd c ϕ) m ∧ ω n−m = F(x, ϕ)ω n . Under some natrual conditions on F, this equation has a unique solution (up to a constant) which has the same singularity type as φ. These results unify some recent work in [DNL23][LN22][LN15][Lu13] etc.