
Mathematics 2005
Pluricomplex charge at weak singularitiesAbstract: Let $u$ be a plurisubharmonic function, defined on a neighbourhood of a point $x,$ such that the complex MongeAmp\`ere operator is welldefined 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 $(dd^c{u})^n$ 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 MongeAmp\`ere mass vanishing Lelong number does implies vanishing residual mass of the MongeAmp\`ere measure.
