Green functions, Segre numbers, and King’s formula

Mats Andersson[1]; Elizabeth Wulcan[1]

  • [1] Department of Mathematics Chalmers University of Technology and the University of Gothenburg S-412 96 Gothenburg SWEDEN

Annales de l’institut Fourier (2014)

  • Volume: 64, Issue: 6, page 2639-2657
  • ISSN: 0373-0956

Abstract

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Let 𝒥 be a coherent ideal sheaf on a complex manifold X with zero set Z , and let G be a plurisubharmonic function such that G = log | f | + 𝒪 ( 1 ) locally at Z , where f is a tuple of holomorphic functions that defines 𝒥 . We give a meaning to the Monge-Ampère products ( d d c G ) k for k = 0 , 1 , 2 , ... , and prove that the Lelong numbers of the currents M k 𝒥 : = 1 Z ( d d c G ) k at x coincide with the so-called Segre numbers of J at x , introduced independently by Tworzewski, Gaffney-Gassler, and Achilles-Manaresi. More generally, we show that M k 𝒥 satisfy a certain generalization of the classical King formula.

How to cite

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Andersson, Mats, and Wulcan, Elizabeth. "Green functions, Segre numbers, and King’s formula." Annales de l’institut Fourier 64.6 (2014): 2639-2657. <http://eudml.org/doc/275584>.

@article{Andersson2014,
abstract = {Let $\mathcal\{J\}$ be a coherent ideal sheaf on a complex manifold $X$ with zero set $Z$, and let $G$ be a plurisubharmonic function such that $G=\log |f|+\mathcal\{O\}(1)$ locally at $Z$, where $f$ is a tuple of holomorphic functions that defines $\mathcal\{J\}$. We give a meaning to the Monge-Ampère products $(dd^c G)^k$ for $k=0,1,2,\ldots $, and prove that the Lelong numbers of the currents $M_k^\{\mathcal\{J\}\}:=\{\bf 1\}_Z(dd^c G)^k$ at $x$ coincide with the so-called Segre numbers of $J$ at $x$, introduced independently by Tworzewski, Gaffney-Gassler, and Achilles-Manaresi. More generally, we show that $M_k^\{\mathcal\{J\}\}$ satisfy a certain generalization of the classical King formula.},
affiliation = {Department of Mathematics Chalmers University of Technology and the University of Gothenburg S-412 96 Gothenburg SWEDEN; Department of Mathematics Chalmers University of Technology and the University of Gothenburg S-412 96 Gothenburg SWEDEN},
author = {Andersson, Mats, Wulcan, Elizabeth},
journal = {Annales de l’institut Fourier},
keywords = {Green function; Segre numbers; Monge-Ampère products; King’s formula; King's formula},
language = {eng},
number = {6},
pages = {2639-2657},
publisher = {Association des Annales de l’institut Fourier},
title = {Green functions, Segre numbers, and King’s formula},
url = {http://eudml.org/doc/275584},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Andersson, Mats
AU - Wulcan, Elizabeth
TI - Green functions, Segre numbers, and King’s formula
JO - Annales de l’institut Fourier
PY - 2014
PB - Association des Annales de l’institut Fourier
VL - 64
IS - 6
SP - 2639
EP - 2657
AB - Let $\mathcal{J}$ be a coherent ideal sheaf on a complex manifold $X$ with zero set $Z$, and let $G$ be a plurisubharmonic function such that $G=\log |f|+\mathcal{O}(1)$ locally at $Z$, where $f$ is a tuple of holomorphic functions that defines $\mathcal{J}$. We give a meaning to the Monge-Ampère products $(dd^c G)^k$ for $k=0,1,2,\ldots $, and prove that the Lelong numbers of the currents $M_k^{\mathcal{J}}:={\bf 1}_Z(dd^c G)^k$ at $x$ coincide with the so-called Segre numbers of $J$ at $x$, introduced independently by Tworzewski, Gaffney-Gassler, and Achilles-Manaresi. More generally, we show that $M_k^{\mathcal{J}}$ satisfy a certain generalization of the classical King formula.
LA - eng
KW - Green function; Segre numbers; Monge-Ampère products; King’s formula; King's formula
UR - http://eudml.org/doc/275584
ER -

References

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