zbMATH — the first resource for mathematics

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

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