Degenerate complex Hessian equations on compact Kähler manifolds (Q2788636)

The work is devoted to \((w,m)\)-subharmonic functions (\((w,m)\)-sh) on a compact Kähler manifold \((X,w)\) of dimension \(n\), and their potential properties, here \(m\in\mathbb N\) such that \(1\leq m\leq n\). The main result of this paper is that every \((w,m)\)-sh function can be approximated from above by smooth \((w,m)\)-sh functions. Using this crucial regularization, the authors develop a pluripotential theory for complex Hessian equations on compact Kähler manifolds that generalizes the classical one studied by Guedj and Zeriahi for the case \(m=n\). Then using the variational approach inspired by Berman, Bouchsom, Guedj, and Zeriahi, the authors solve degenerate complex Hessian equations with right-hand sides vanishing on \(m\)-polar sets.