By F. A. Bogomolov, A. N. Landia (auth.), H. Kurke, J. H. M. Steenbrink (eds.)

The convention on Algebraic Geometry, held in Berlin 9-15 March 1988, was once organised by means of the Sektion Mathematik of the Humboldt-Universitat. The establishing committee consisted of H. Kurke, W. Kleinert, G. Pfister and M. Roczen. The convention is one in a sequence organised by means of the Humboldt-Universitat at normal durations of 2 or 3 years, with the aim of supplying a gathering position for mathematicians from japanese and western nations. the current quantity comprises embellishments of a part of the lectures awarded on the convention and a few articles on similar matters. All papers have been topic to the common refereeing technique of Compositio Mathematica, and H. Kurke acted as a visitor editor of this magazine. The papers concentrate on real topics in algebraic geometry and singularity conception, equivalent to vector bundles, arithmetical algebraic geometry, intersection thought, moduli and Hodge concept. we're thankful to all those that, by way of their hospitality, their presence on the Con ference, their help or their written contributions, have made this convention to a hit. The editors Compositio Mathematica seventy six: viii, 1990.

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**Example text**

Is the Milnor lattice of the singularity l = 0 (resp. g~ = 0). When t varies, these Milnor lattices are constant and hence the morphism cpt has to be constant too. Hence H"(V) = H"(V1) and so we can assume from the beginning that gN -1, gN satisfy the condition (ii). Let cp: L1 C:L--+L be the lattice morphism in this case. 2). But we may think of gr as being a even smaller deformation (of order r- 2 ) of the 46 Alexan dru Dimca the germ g' = gN - 1 • r - 1 + 9 N, which is a small deformation of 9 N.

If we define F'HO(F)a = FSHO(F) n HO(F)a it follows that F'HO(F) = tBaF'HO(F)a. And one has a similar result for the Hodge filtration F~. 2), since the isomorphism HO(U) ~HO(F)o c HO(F) is clearly compatible with both filtrations. 19) and the next two facts. 20) it follows that On the Milnor fibrations of weighted homogeneous polynomials 31 Note also that the filtration po on HO(F) is very close to the filtrations considered by Scherk and Steenbrink in the isolated singularity case in [21]. 7) COROLLARY.

Let Z denote the singular locus of V. 8) and the associated morphisms A moment thought shows that Grpp is a quasi-isomorphism for s < O. A computation using an E 2 -spectral sequence shows that H'(Grp(ni>n(* V))) = H'(Gr}Ao). Assume from now on that Z is a finite set, namely Z = {a 1 , ••• , ap }. Note that On the Milnor fibrations of weighted homogeneous polynomials 35 the singularity (V, ai) is precisely the transversal singularity of f along the line C*" a i as defined in the beginning of this section.