# Décroissance rapide de la distribution fλ

Research paper by D. Barlet, H.-M. Maire

Indexed on: 01 Feb '98Published on: 01 Feb '98Published in: Compositio Mathematica

#### Abstract

Let X be an open subset of $$\mathbb{C}$$n and (f1, ...,fp): X → $$\mathbb{C}$$p be a holomorphic mapping. We prove that if (x0,0, λ0) ∈ T* × $$\mathbb{C}$$p does not belong to the characteristic variety of the $$\mathcal{D}$$X [λ]-module $$\mathcal{D}$$X[λ]fλ, then there exists a conic neighborhood V × Γ of (x0, λ0) such the function $$(\lambda_{\text{1}} , \ldots ,\lambda_p) \mapsto \int {\left| {f_1} \right|} ^{\lambda_{\text{1}}}\ldots \left| {f_1} \right|^{\lambda p} \omega$$ is rapidely decreasing in | Im λ | for λ ∈ Γ with Re λ bounded, for any (n,n)-form ω of class C∞ with compact support in V. The following partial converse of this result is also established: if $$s \mapsto \int {_{f = s}} {\text{}}\varphi {\text{is of class}}C^\infty{\text{in}}\mathbb{C}^p$$ for all (n,n)-forms ϕ of class C∞ with compact support in X, then $${\text{d}}f_1\wedge\cdots\wedge {\text{d}}f_p (x) \ne 0,\forall x \in X$$.