Complex Hessian equations with prescribed singularity on compact Kähler manifolds

Abstract

Let $(X,\omega )$ be a compact Kähler manifold of dimension $n$ and fix $1\le m\le n$. We prove that the total mass of the complex Hessian measure of $\omega$-$m$-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.