Green functions, Segre numbers,  and King’s formula

Andersson, Mats; Wulcan, Elizabeth

doi:10.5802/aif.2922

Green functions, Segre numbers, and King’s formula
[Fonctions de Green, nombres de Segre, et formule de King]

Andersson, Mats ¹ ; Wulcan, Elizabeth ¹

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

Annales de l'Institut Fourier, Tome 64 (2014) no. 6, pp. 2639-2657.

Résumé
Abstract

Soit $𝒥$ un faisceau cohérent d’ideaux sur un variété complexe lisse $X$ , et soit $Z$ la variété de $𝒥$ . Soit $G$ une fonction plurisousharmonique telle que $G = log | f | + 𝒪 (1)$ localement sur $Z$ , où $f$ est un $n$ -uple de fonctions holomorphes qui définit $𝒥$ . Nous donnons un sens au produit de Monge-Ampère ${(d d^{c} G)}^{k}$ pour $k = 0, 1, 2, ...$ , et nous montrons que les nombres de Lelong des courants $M_{k}^{𝒥} : = 1_{Z} {(d d^{c} G)}^{k}$ en $x$ coïncident avec les nombres de Segre de $𝒥$ en $x$ , introduits indépendemment par Tworzewski, Gaffney-Gassler et Achilles-Manaresi. Plus généralement, nous montrons que les $M_{k}^{𝒥}$ satisfont une certaine généralisation de la formule de King.

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.

MR Zbl

DOI : 10.5802/aif.2922

Classification : 32U35, 32U25, 32U40, 32B30, 14B05
Keywords: Green function, Segre numbers, Monge-Ampère products, King’s formula
Mot clés : Fonctions de Green, nombres de Segre, produits de Monge-Ampère, formule de King

Affiliations des auteurs :

Andersson, Mats ¹ ; Wulcan, Elizabeth ¹

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

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     author = {Andersson, Mats and Wulcan, Elizabeth},
     title = {Green functions, {Segre} numbers,  and {King{\textquoteright}s} formula},
     journal = {Annales de l'Institut Fourier},
     pages = {2639--2657},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {64},
     number = {6},
     year = {2014},
     doi = {10.5802/aif.2922},
     mrnumber = {3331176},
     zbl = {06387349},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2922/}
}

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Andersson, Mats; Wulcan, Elizabeth. Green functions, Segre numbers,  and King’s formula. Annales de l'Institut Fourier, Tome 64 (2014) no. 6, pp. 2639-2657. doi : 10.5802/aif.2922. https://aif.centre-mersenne.org/articles/10.5802/aif.2922/

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