Pluricomplex charge at weak singularities

Abstract

Let $u$ be a plurisubharmonic function, defined on a neighbourhood of a point $x,$ such that the complex Monge-Ampère operator is well-defined on $u.$ Suppose also that $u$ has a weak singularity, in the sense that the Lelong number of $u$ at $x$ vanish. Is it true that the residual mass of the measure $(d{d}^{c}u{)}^n$ vanish at $x$? To our knowledge there is no known example that falsifies the posed question. In this paper some partial results are obtained. We find that for a significant subset of plurisubharmonic functions with well defined Monge-Ampère mass vanishing Lelong number does implies vanishing residual mass of the Monge-Ampère measure.

References in corpus (2)

• Singularities of plurisubharmonic functions and positive closed currents
• Convergence in capacity

Cited by in corpus (3)

• An Inequality for Mixed Monge-Ampère measures
• Higher Lelong numbers and convex geometry
• Open problems in pluripotential theory