Green functions, Segre numbers, and King’s formula

Andersson, Mats; Wulcan, Elizabeth

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

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 $G=log\left|f\right|+𝒪\left(1\right)$ localement sur $Z$, où $f$ est un $n$-uple de fonctions holomorphes qui définit $𝒥$. Nous donnons un sens au produit de Monge-Ampère ${\left(d{d}^{c}G\right)}^k$ pour $k=0,1,2,...$, et nous montrons que les nombres de Lelong des courants $M_k^{𝒥}:={\mathbf{1}}_Z{\left(d{d}^{c}G\right)}^k$ 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 $G=log\left|f\right|+𝒪\left(1\right)$ locally at $Z$, where $f$ is a tuple of holomorphic functions that defines $𝒥$. We give a meaning to the Monge-Ampère products ${\left(d{d}^{c}G\right)}^k$ for $k=0,1,2,...$, and prove that the Lelong numbers of the currents $M_k^{𝒥}:={\mathbf{1}}_Z{\left(d{d}^{c}G\right)}^k$ 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.

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

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