A geometric proof of the Lelong-Poincaré formula

M El Amrani,A Jeddi

doi:10.4067/s0716-09172013000100001

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.

Highlights

Since the Lelong-Poincare formula plays a crucial role in complex analytic geometry, notably in intersection theory, it is a natural aim to look for a geometric proof of this fundamental formula
Recall that the integration current exists on any analytic set
We prove the formula (LP ) for differential forms which are locally given by φ = ρ(t, z) dt ∧ dt where ρ is a smooth function with compact support in V, and dt ∧ dt = dz1 ∧ . . . dzn−1 ∧ dz1 ∧ . . . ∧ dzn−1

Summary

Introduction

Since the Lelong-Poincare formula plays a crucial role in complex analytic geometry, notably in intersection theory (see [4]), it is a natural aim to look for a geometric proof of this fundamental formula. Theorem Let V be a connected complex analytic manifold of dimension n and let f : V → C be a holomorphic nonzero function. We denote by D(V ) the set of compactly supported differential forms of class C∞ in V .

Results

Conclusion