Green functions, Segre numbers, and King’s formula

[Fonctions de Green, nombres de Segre, et formule de King]

Andersson, Mats ; Wulcan, Elizabeth

Annales de l'Institut Fourier, Tome 64 (2014), p. 2639-2657 / Harvested from Numdam

Résumé

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

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.

Détail

Publié le : 2014-01-01

MR 3331176 | Zbl 06387349

DOI : https://doi.org/10.5802/aif.2922
Classification:  32U35,  32U25,  32U40,  32B30,  14B05
Mots clés: Fonctions de Green, nombres de Segre, produits de Monge-Ampère, formule de King

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