zbMATH — the first resource for mathematics

Contribution effective de la monodromie aux développements asymptotiques. (French) Zbl 0542.32003
Let $$f:({\mathbb{C}}^{n+1},0)\to({\mathbb{C}},0)$$ be a germ of a non constant holomorphic function. Using the Bernstein-Sato polynomial of f it is easy to show that the distribution $$| f|^{2\lambda}$$, defined near 0 for $$Re(\lambda)>0$$ admits a meromorphic extension to $${\mathbb{C}}$$. We prove that if the monodromy of f has a jordan block of size (k,k) for the eigenvalue $$e^{-i\pi r}$$, the meromorphic extension of $$| f|^{2\lambda}$$ has a pole of order at least k at $$-r/2-p$$ for $$p\in {\mathbb{N}}$$ big enough. Using Malgrange’s results linking monodromy and Bernstein-Sato polynomial, we can then deduce that each root of this polynomial produces a sequence of poles for the meromorphic extension of $$| f|^{2\lambda}.$$
Further results of this kind are obtained in the author’s following articles: ”Contribution du cup-produit de la fibre de Milnor aux pôles de $$| f|^{2\lambda}$$”, Ann. Inst. Fourier 34, No. 4, 75-107 (1984; Zbl 0525.32007); ”Monodromie et pôles du prolongement méromorphe de $$\int_{X}| f|^{2\lambda}\square$$”, Inst. E. Cartan, Univ. Nancy I (to appear).

MSC:
 32S05 Local complex singularities 32B10 Germs of analytic sets, local parametrization 32C30 Integration on analytic sets and spaces, currents 32Sxx Complex singularities 32C35 Analytic sheaves and cohomology groups
Full Text:
References:
 [1] D. BARLET , Développements asymptotiques des fonctions obtenues par intégration sur les fibres (Inv. Math., vol. 68, 1982 ). MR 84a:32021 | Zbl 0508.32003 · Zbl 0508.32003 [2] P. DELIGNE , Equations différentielles à points singuliers réguliers (Lecture Notes n^\circ 163, Springer-Verlag, 1970 ). MR 54 #5232 | Zbl 0244.14004 · Zbl 0244.14004 [3] M. KASHIWARA , B-functions and Holonomic Systems (Inv. Math., vol. 38-1, 1976 ). MR 55 #3309 | Zbl 0354.35082 · Zbl 0354.35082 [4] B. MALGRANGE , Polynôme de Bernstein-Sato , (Publication de l’IRMA de Strasbourg, vol. 28, RCP 25, 1980 ). [5] B. MALGRANGE , Polynôme de Bernstein-Sato et cohomologie évanescente [Colloque d’Analyse et Topologie sur les espaces singuliers, Luminy, 1981 (à paraître)]. [6] J. MILNOR , Singular Points of Complex Hypersurfaces (Ann. of Math. Studies 61, Princeton, 1968 ). MR 39 #969 | Zbl 0184.48405 · Zbl 0184.48405
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.