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Développements asymptotiques, transformation de Mellin complexe et intégration sur les fibres. (Asymptotic expansions, complex Mellin transformation and integration on fibers). (French) Zbl 0649.32008
Sémin. d’analyse P. Lelong - P. Dolbeault - H. Skoda, Paris 1985/86, Lect. Notes Math. 1295, 11-23 (1987).
[For the entire collection see Zbl 0623.00006.]
Let X be an analytic complex manifold of dimension \(n+1\) and f a holomorphic function on X, such that \(f^{-1}(s)\) is a submanifold for \(s\neq 0\); if u is an (n,n)-differential form with compact support, it was proved by the first author [Invent. Math. 68, 129-174 (1982; Zbl 0508.32003)] that the fiber integral \(F(s)=\int_{f^{-1}(s)}u\) has an asymptotic expansion of the form \[ F(s)\sim \sum a^ i_{jkm}| s|^{2r_ i} s^ j \bar s^ k(\log | s|)^ m\quad (s\to 0), \] with \(r_ i\in {\mathbb{Q}}\), \(0\leq r_ 1<...<r_ p<1\), j,k,m\(\in {\mathbb{N}}\), \(m\leq n.\)
This paper gives a simplified proof of this result, using relations between the poles of the meromorphic continuation of the “complex Mellin transform” \[ M_ qF(t)=\int_{C}| z|^{2t} z^ q \bar z^{-q} F(z,\bar z)dz d\bar z \] and the asymptotic expansion of F.
Reviewer: G.Roos

MSC:
32C30 Integration on analytic sets and spaces, currents
32Sxx Complex singularities
32S45 Modifications; resolution of singularities (complex-analytic aspects)