Plurisubharmonic functions in calibrated geometry and q-convexity

Let (M,ω) be a Kähler manifold. An integrable function φ on M is called ω-plurisubharmonic if the current ddφ ∧ ω is positive. We prove that φ is ωplurisubharmonic if and only if φ is subharmonic on all q-dimensional complex subvarieties. We prove that a ωplurisubharmonic function is q-convex, and admits a local approximation by smooth, ω-plurisubharmonic functions. For any closed subvariety Z ⊂ M , dimC Z 6 q − 1, there exists a strictly ω-plurisubharmonic function in a neighbourhood of Z (this result is known for q-convex functions). This theorem is used to give a new proof of Sibony’s lemma on integrability of positive closed (p, p)-forms which are integrable outside of a complex subvariety of codimension > p+ 1.