Plurisubharmonic functions in calibrated geometry and q-convexity

Abstract

Let $(M,\omega)$ be a Kahler manifold. An integrable function on M is called $\omega^q$-plurisubharmonic if it is subharmonic on all q-dimensional complex subvarieties. We prove that a smooth $\omega^q$-plurisubharmonic function is q-convex. A continuous $\omega^q$-plurisubharmonic function admits a local approximation by smooth, $\omega^q$-plurisubharmonic functions. For any closed subvariety $Z\subset M$, $\dim Z < q$, there exists a strictly $\omega^q$-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.