Plurisubharmonic functions in calibrated geometry and q-convexity

Abstract

Let (M, ω) be a Kähler manifold. An integrable function ${\varphi}$ on M is called ω ^q-plurisubharmonic if the current ${dd^c\varphi\wedge \omega^{q-1}}$ is positive. We prove that ${\varphi}$ is ω ^q-plurisubharmonic if and only if ${\varphi}$ is subharmonic on all q-dimensional complex subvarieties. We prove that a ω ^q-plurisubharmonic function is q-convex, and admits a local approximation by smooth, ω ^q-plurisubharmonic functions. For any closed subvariety ${Z\subset M}$ , ${\dim_\mathbb{C} Z\leq q-1}$ , there exists a strictly ω ^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 + 1.