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Valuative analysis of planar plurisubharmonic functions
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
2005 (English)In: Inventiones Mathematicae, ISSN 0020-9910, E-ISSN 1432-1297, Vol. 162, no 2, 271-311 p.Article in journal (Refereed) Published
Abstract [en]

We show that valuations on the ring R of holomorphic germs in dimension 2 may be naturally evaluated on plurisubharmonic functions, giving rise to generalized Lelong numbers in the sense of Demailly. Any plurisubharmonic function thus defines a real-valued function on the set V of valuations on R and - by way of a natural Laplace operator defined in terms of the tree structure on V - a positive measure on V. This measure contains a great deal of information on the singularity at the origin. Under mild regularity assumptions, it yields an exact formula for the mixed Monge-Ampere mass of two plurisubharmonic functions. As a consequence, any generalized Lelong number can be interpreted as an average of valuations. Using our machinery we also show that the singularity of any positive closed ( 1, 1) current T can be attenuated in the following sense: there exists a finite composition of blowups such that the pull-back of T decomposes into two parts, the first associated to a divisor with normal crossing support, the second having small Lelong numbers.

Place, publisher, year, edition, pages
2005. Vol. 162, no 2, 271-311 p.
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URN: urn:nbn:se:kth:diva-38160DOI: 10.1007/s00222-005-0443-2ISI: 000232406700002ScopusID: 2-s2.0-26444477317OAI: diva2:436037
QC 20110822Available from: 2011-08-22 Created: 2011-08-22 Last updated: 2011-08-22Bibliographically approved

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