Newton numbers and residual measures of plurisubharmonic functions

Alexander Rashkovskii

Annales Polonici Mathematici (2000)

  • Volume: 75, Issue: 3, page 213-231
  • ISSN: 0066-2216

Abstract

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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.

How to cite

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Rashkovskii, Alexander. "Newton numbers and residual measures of plurisubharmonic functions." Annales Polonici Mathematici 75.3 (2000): 213-231. <http://eudml.org/doc/208396>.

@article{Rashkovskii2000,
abstract = {We study the masses charged by $(dd^cu)^n$ 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.},
author = {Rashkovskii, Alexander},
journal = {Annales Polonici Mathematici},
keywords = {Monge-Ampère operator; local indicator; directional Lelong number; plurisubharmonic function; Newton polyhedron},
language = {eng},
number = {3},
pages = {213-231},
title = {Newton numbers and residual measures of plurisubharmonic functions},
url = {http://eudml.org/doc/208396},
volume = {75},
year = {2000},
}

TY - JOUR
AU - Rashkovskii, Alexander
TI - Newton numbers and residual measures of plurisubharmonic functions
JO - Annales Polonici Mathematici
PY - 2000
VL - 75
IS - 3
SP - 213
EP - 231
AB - We study the masses charged by $(dd^cu)^n$ 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.
LA - eng
KW - Monge-Ampère operator; local indicator; directional Lelong number; plurisubharmonic function; Newton polyhedron
UR - http://eudml.org/doc/208396
ER -

References

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  1. [1] L. A. Aĭzenberg and Yu. P. Yuzhakov, Integral Representations and Residues in Multidimensional Complex Analysis, Nauka, Novosibirsk, 1979 (in Russian); English transl.: AMS, Providence, RI, 1983. 
  2. [2] J.-P. Demailly, Monge-Ampère operators, Lelong numbers and intersection theory, in: Complex Analysis and Geometry, V. Ancona and A. Silva (eds.), Plenum Press, New York, 1993, 115-193. Zbl0792.32006
  3. [3] L. Hörmander, Notions of Convexity, Progr. Math. 127, Birkhäuser, 1994. 
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  7. [7] C. O. Kiselman, Attenuating the singularities of plurisubharmonic functions, Ann. Polon. Math. 60 (1994), 173-197. Zbl0827.32016
  8. [8] M. Klimek, Pluripotential Theory, Oxford Univ. Press, London, 1991. 
  9. [9] A. G. Kouchnirenko, Newton polyhedron and the number of solutions of a system of k equations with k indeterminates, Uspekhi Mat. Nauk 30 (1975), no. 2, 266-267 (in Russian). 
  10. [10] A. G. Kouchnirenko, Polyèdres de Newton et nombres de Milnor, Invent. Math. 32 (1976), 1-31. 
  11. [11] S. Lang, Fundamentals of Diophantine Geometry, Springer, New York, 1983. Zbl0528.14013
  12. [12] P. Lelong, Plurisubharmonic Functions and Positive Differential Forms, Gordon and Breach, New York, and Dunod, Paris, 1969. Zbl0195.11604
  13. [13] P. Lelong, Remarks on pointwise multiplicities, Linear Topol. Spaces Complex Anal. 3 (1997), 112-119. Zbl0923.32003
  14. [14] P. Lelong and L. Gruman, Entire Functions of Several Complex Variables, Springer, Berlin, 1986. Zbl0583.32001
  15. [15] P. Lelong and A. Rashkovskii, Local indicators for plurisubharmonic functions, J. Math. Pures Appl. 78 (1999), 233-247. Zbl0933.32049
  16. [16] J. Rauch and B. A. Taylor, The Dirichlet problem for the multidimensional Monge-Ampère equation, Rocky Mountain J. Math. 7 (1977), 345-364. Zbl0367.35025
  17. [17] Y. Xing, Continuity of the complex Monge-Ampère operator, Proc. Amer. Math. Soc. 124 (1996), 457-467. Zbl0849.31010

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