Plurisubharmonic functions in calibrated geometry and q-convexity

TL;DR

This paper explores the properties of ta^q-plurisubharmonic functions on Ke4hler manifolds, establishing their q-convexity, approximation capabilities, and applications to Sibony's lemma, advancing understanding in complex geometry.

Contribution

It proves that smooth ta^q-plurisubharmonic functions are q-convex and provides approximation results, offering a new proof of Sibony's lemma.

Findings

Smooth ta^q-plurisubharmonic functions are q-convex

Continuous ta^q-plurisubharmonic functions can be approximated locally by smooth ones

Existence of strictly ta^q-plurisubharmonic functions near subvarieties

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.