# zbMATH — the first resource for mathematics

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

##### MSC:
 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