Solution to Hessian type equation with prescribed singularities on compact Kähler manifolds

Abstract

Let (X,ω) be a compact Kähler manifold of dimension n and fix an integer m such that 1≤m≤n. We reformulate Darvas-Nezza-Lu's latest survey [12] into the Hessian setting. Namely, we characterize the relative full mass class Eϕ(X,ω,m) and prove the integration by parts formula of Hessian type. Given a model potential ϕ, we study degenerate complex Hessian equations of the form (ω+ddcφ)m∧ωn−m=F(x,φ)ωn. Under some natural conditions on F, we prove that this equation has a unique solution (up to a constant) which has the same singularity type as ϕ.