Solutions to degenerate complex Hessian equations

Abstract

Let $(X,\omega )$ be an $n$-dimensional compact Kähler manifold. We 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 natural conditions on $F$, this equation has a unique continuous solution. When $(X,\omega )$ is rational homogeneous we further show that the solution is Hölder continuous.

We slightly modify the proof of Theorem 5.1 and we prove in Proposition 3.20 that the class ${\mathcal{P}}_m(X,\omega )$ is stable under decreasing sequences

References in corpus (2)

• Hölder continuous solutions to Monge-Ampère equations
• Capacities and Hessians in a class of m-subharmonic functions

Cited by in corpus (2)

• Viscosity solutions to complex Hessian equations
• Hölder continuous solutions to complex Hessian equations