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Newton numbers and residual measures of plurisubharmonic functions

Alexander Rashkovskii — 2000

Annales Polonici Mathematici

We study the masses charged by ${(d d^{c} u)}^{n}$ at isolated singularity points of plurisubharmonic functions u. This is done by means of the local indicators of plurisubharmonic functions introduced in [15]. As a consequence, bounds for the masses are obtained in terms of the directional Lelong numbers of u, and the notion of the Newton number for a holomorphic mapping is extended to arbitrary plurisubharmonic functions. We also describe the local indicator of u as the logarithmic tangent to u.

Extreme plurisubharmonic singularities

Alexander Rashkovskii — 2012

Annales Polonici Mathematici

A plurisubharmonic singularity is extreme if it cannot be represented as the sum of non-homothetic singularities. A complete characterization of such singularities is given for the case of homogeneous singularities (in particular, those determined by generic holomorphic mappings) in terms of decomposability of certain convex sets in ℝⁿ. Another class of extreme singularities is presented by means of a notion of relative type.

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