Degenerate Complex Monge–Ampère Equations on Some Compact Hermitian Manifolds

Abstract

Let X be a compact complex manifold which admits a hermitian metric satisfying a curvature condition introduced by Guan–Li. Given a semipositive form \(\theta \) with positive volume, we define the Monge–Ampère operator for unbounded \(\theta \)-psh functions and prove that it is continuous with respect to convergence in capacity. We then develop pluripotential tools to study degenerate complex Monge–Ampère equations in this context, extending recent results of Tosatti–Weinkove, Kolodziej–Nguyen, Guedj–Lu and many others who treat bounded solutions.