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Asymptotics of integrals over vanishing cycles and the Newton polyhedron. (English. Russian original) Zbl 0595.32012
Sov. Math., Dokl. 32, 122-127 (1985); translation from Dokl. Akad. Nauk SSSR 283, 521-525 (1985).
Let f: (\({\mathbb{C}}^ n,0)\to ({\mathbb{C}},0)\) be the germ of a holomorphic function at an isolated critical point and let \(\Omega^ p\) be the space of germs at \(0\in {\mathbb{C}}^ n\) of holomorphic differential p-forms. Now let \(\Omega_ f:=\Omega^ n/df\wedge \Omega^{n-1}\) and let \({\bar \Omega}{}_ f:=\Omega^ n/df\wedge d\Omega^{n-2}\). The authors establish a relationship between the Newton filtration in \(\Omega_ f\) (resp. \({\bar \Omega}{}_ f)\) and its Hodge filtration. As a consequence they prove a conjecture due to Steenbrink regarding the relation between the spectrum of the Newton filtration and the one of the mixed Hodge- Steenbrink structure; this conjecture has been settled down also by M. Saito by different techniques.
Reviewer: Vo Van Tan

32S05 Local complex singularities
14B05 Singularities in algebraic geometry
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
32Sxx Complex singularities
32J25 Transcendental methods of algebraic geometry (complex-analytic aspects)
32C30 Integration on analytic sets and spaces, currents