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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 \({\text{d}}f_1\wedge\cdots\wedge {\text{d}}f_p (x) \ne 0,\forall x \in X\).