Green functions, Segre numbers, and King’s formula
Annales de l'Institut Fourier, Volume 64 (2014) no. 6, p. 2639-2657

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 (dd c G) k for k=0,1,2,..., and prove that the Lelong numbers of the currents M k 𝒥 :=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 𝒥 satisfy a certain generalization of the classical King formula.

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 (dd c G) k pour k=0,1,2,..., et nous montrons que les nombres de Lelong des courants M k 𝒥 :=1 Z (dd 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.

DOI : https://doi.org/10.5802/aif.2922
Classification:  32U35,  32U25,  32U40,  32B30,  14B05
Keywords: Green function, Segre numbers, Monge-Ampère products, King’s formula
@article{AIF_2014__64_6_2639_0,
     author = {Andersson, Mats and Wulcan, Elizabeth},
     title = {Green functions, Segre numbers,  and King's formula},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {64},
     number = {6},
     year = {2014},
     pages = {2639-2657},
     doi = {10.5802/aif.2922},
     mrnumber = {3331176},
     zbl = {06387349},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2014__64_6_2639_0}
}
Andersson, Mats; Wulcan, Elizabeth. Green functions, Segre numbers,  and King’s formula. Annales de l'Institut Fourier, Volume 64 (2014) no. 6, pp. 2639-2657. doi : 10.5802/aif.2922. http://www.numdam.org/item/AIF_2014__64_6_2639_0/

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