Green functions, Segre numbers, and King’s formula. (Fonctions de Green, nombres de Segre, et formule de King.) (English. French summary) Zbl 1322.32026
Authors’ 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 \((d d^c G)^k\) for \(k = 0, 1, 2, \dots\), and prove that the Lelong numbers of the currents \(M_k^{\mathcal{J}} : = 1_Z(d d^c G)^k\) at \(x\) coincide with the so-called Segre numbers of \(\mathcal J\) 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.
Reviewer: Jean Ruppenthal (Wuppertal)
MSC:
| 32U35 | Plurisubharmonic extremal functions, pluricomplex Green functions |
| 32U25 | Lelong numbers |
| 32U40 | Currents |
| 32B15 | Analytic subsets of affine space |
| 14B05 | Singularities in algebraic geometry |
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