Solution to Hessian type equations with prescribed singularity on compact Kahler manifold

Let $(X,\omega )$ be a compact Kähler manifold of dimension $n$ and fix an integer $m$ such that $1\le m\le n$. We reformulate most relative pluripotential results of Darvas-DiNezza-Lu’s survey \cite{DNL23} to the Hessian setting. As an application, we use a slightly different method and give an characterization of finite energy range of the Hessian operator, which cannot be directly reformulated by \cite{DNL23}. Given a model potential $\phi$, we also study degenerate complex Hessian equations of the form $(\omega +d{d}^c\phi {)}^m\wedge {\omega }^{n-m}=F(x,\phi ){\omega }^n$. Under some natrual conditions on $F$, we prove that the solution of this type equation has the same singularity type as $\phi$.