A geometric proof of the Lelong-Poincaré fórmula
DOI:
https://doi.org/10.4067/S0716-09172013000100001Keywords:
Complex analytic manifolds, Analytic sets, Local parametrization theorem, Integration currents, Branching coverings.Abstract
We propose a geometric proof of the fundamental Lelong-Poincaré formula : ddc log |ƒ| = [ƒ = 0] where f is any nonzero holomorphic function defined on a complex analytic manifold V and [ƒ = 0] is the integration current on the divisor of the zeroes of ƒ.References
[1] Barlet D. : Le théorème d’intégration sur un ensemble analytique complexe de P. Lelong. Séminaire de géométrie analytique. Institut Elie Cartan 5, pp. 1-11, (1982).
[2] Barlet D. : Espace analytique réduit des cycles analytiques complexes compacts d’un espace analytique complexe de dimension finie. Fonctions de plusieurs variables complexes II. Lecture Notes in Mathematics 482, pp. 1-159.
[3] Barlet D. : Convexité de l’espace des cycles. Bull. Soc. Math. France 106, pp. 373-397, (1978).
[4] Demailly J.-P. : Courants positifs et théorie de l’intersection. Gaz. Math. 53, pp. 131-159, (1992).
[5] Gunning P., Rossi H. : Analytic functions of several complex variables. AMS Chelsea Publishing, Providence, RI, (2009).
[6] Lelong P. : Intégration sur un ensemble analytique complexe. Bull. Soc. Math. France, 85, pp. 239-262, (1957).
[7] Lelong P. : Fonctions plurisousharmoniques et formes différentielles positives. Dunod, Paris, Gordon & Breach, New-York, (1968).
[2] Barlet D. : Espace analytique réduit des cycles analytiques complexes compacts d’un espace analytique complexe de dimension finie. Fonctions de plusieurs variables complexes II. Lecture Notes in Mathematics 482, pp. 1-159.
[3] Barlet D. : Convexité de l’espace des cycles. Bull. Soc. Math. France 106, pp. 373-397, (1978).
[4] Demailly J.-P. : Courants positifs et théorie de l’intersection. Gaz. Math. 53, pp. 131-159, (1992).
[5] Gunning P., Rossi H. : Analytic functions of several complex variables. AMS Chelsea Publishing, Providence, RI, (2009).
[6] Lelong P. : Intégration sur un ensemble analytique complexe. Bull. Soc. Math. France, 85, pp. 239-262, (1957).
[7] Lelong P. : Fonctions plurisousharmoniques et formes différentielles positives. Dunod, Paris, Gordon & Breach, New-York, (1968).
Published
2013-06-23
How to Cite
[1]
M. El Amrani and A. Jeddi, “A geometric proof of the Lelong-Poincaré fórmula”, Proyecciones (Antofagasta, On line), vol. 32, no. 1, pp. 1-13, Jun. 2013.
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