Complex Hessian equations with prescribed singularity on compact K\"ahler manifolds

TL;DR

This paper studies complex Hessian equations on compact Kähler manifolds, demonstrating how the total mass of associated measures varies with singularity and solving equations with specified singularities, along with establishing a Hodge index inequality.

Contribution

It introduces a monotonicity property of the total mass of complex Hessian measures and provides solutions to Hessian equations with prescribed singularities, along with a new Hodge index inequality.

Findings

Total mass of complex Hessian measures is non-decreasing with singularity type.

Successfully solves complex Hessian equations with prescribed singularities.

Proves a Hodge index type inequality for positive currents.

Abstract

Let $(X,\omega)$ be a compact K\"ahler manifold of dimension $n$ and fix $1\leq m\leq n$. 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.