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On the Milnor fibrations of weighted homogeneous polynomials. (English) Zbl 0726.14002
“Let \(w=(w_ 0,...,w_ n)\) be a set of integer positive weights and let S be the polynomial ring \({\mathbb{C}}[x_ 0,...,x_ n]\) graded by the conditions \(\deg (x_ i)=w_ i\). Let f be a weighted homogeneous polynomial of degree \( N.\) The Milnor fibration of f is the locally trivial fibration \(f:\;{\mathbb{C}}^{n+1}\setminus f^{-1}(0)\to {\mathbb{C}}\setminus \{0\}\) with typical fiber F: \(f^{-1}(1).''\)
Spectral sequence techniques are used to arrive at a formula for the cohomology groups of F. In the case of isolated singularities more is known about F [see for example E. Brieskorn, Manuscr. Math. 2, 103- 161 (1970; Zbl 0186.261)]. The results in this work apply to more general f with (possibly) nonisolated singularities. There is a good number of interesting examples of explicit calculations using the spectral sequences introduced which illustrate why they should be useful in many situations. - A theorem is stated relating a filtration induced by one of the spectral sequences introduced and the Hodge filtration, and it allows one to extend a result of P. A. Griffiths [Ann. Math., II. Ser. 90, 460-495; 496-541 (1969; Zbl 0215.081)] on reductions of rational n-forms on (complex) projective spaces with a nonsingular polar locus to the singular case.
The author points out in a note that the theorem is correct (although the proof given is flawed), consequence of a more general result in work done with P. Deligne [“Hodge and order of the pole filtrations for singular hypersurfaces” (preprint); see also Ann. Sci. Éc. Norm. Supér., IV. Sér. 23, No.4, 645-656 (1990)].

MSC:
14B05 Singularities in algebraic geometry
14D99 Families, fibrations in algebraic geometry
32S99 Complex singularities
57R45 Singularities of differentiable mappings in differential topology
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