Solutions to degenerate complex Hessian equations

Abstract

Let (X,ω) be an n-dimensional compact Kähler manifold and fix an integer m such that 1⩽m⩽n. We study degenerate complex Hessian equations of the form (ω+ddcφ)m∧ωn−m=F(x,φ)ωn. Under some natural conditions on F, this equation has a unique continuous solution. When X is homogeneous and ω is invariant under the Lie group action, we further show that the solution is Hölder continuous.