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Filtration asymptotique et pôles de \(\int _{X}| f| ^{2\lambda}\square\). (Asymptotic filtration and poles of \(\int _{X}| f| ^{2\lambda}\square)\). (French) Zbl 0728.32018
Théorie de Hodge, Actes Colloq., Luminy/Fr. 1987, Astérisque 179-180, 13-37 (1989).
[For the entire collection see Zbl 0695.00012.]
Following Varchenko’s idea [A. N. Varchenko, Math. USSR, Izv. 18, 469-512 (1982; Zbl 0489.14003)], the author introduces an asymptotic filtration \(F^{\bullet}\) on the cohomology of the Milnor fiber of a germ of holomorphic mapping f: (\({\mathbb{C}}^{n+1},0)\to ({\mathbb{C}},0)\) without assumption on the singularity of \(\{f=0\}\). The filtration is defined by using relative de Rahm cohomology of \(f:X\to {\mathbb{C}}\) over \(Y=f^{-1}(0)\) and is not used a resolution of singularities.
In the fundamental proposition in § 2 the author gives a relation between nilpotent logarithm N of the unipotent part of the monodromy, the filtrations \(F^{\bullet}\) and \(\bar F^{\bullet}\) and the poles of the meromorphic continuation of the distribution \(\int_{X}| f|^{2\lambda}\square\). As an application of the proposition, the author gives estimates of the integral shift of the roots of the Bernstein-Sato polynomial of f in terms of the action of N on the filtration \(F^{\bullet}\).
Reviewer: K.Ueno (Kyoto)

MSC:
32S40 Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects)
14F40 de Rham cohomology and algebraic geometry
32S55 Milnor fibration; relations with knot theory
14J17 Singularities of surfaces or higher-dimensional varieties
32S25 Complex surface and hypersurface singularities