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 way of the Sektion Mathematik of the Humboldt-Universitat. The setting up committee consisted of H. Kurke, W. Kleinert, G. Pfister and M. Roczen. The convention is one in a chain organised through the Humboldt-Universitat at common periods of 2 or 3 years, with the aim of offering a gathering position for mathematicians from japanese and western nations. the current quantity includes 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 usual refereeing process of Compositio Mathematica, and H. Kurke acted as a visitor editor of this magazine. The papers specialize in real issues in algebraic geometry and singularity thought, reminiscent of vector bundles, arithmetical algebraic geometry, intersection idea, moduli and Hodge thought. we're thankful to all those that, through their hospitality, their presence on the Con ference, their help or their written contributions, have made this convention to successful. The editors Compositio Mathematica seventy six: viii, 1990.

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Ii) below! 19) PROPOSITION. (i) Assume that a polynomial X(V) = X(V') + (_1)n L dimM(ga)-W X(F) = X(F') + (_1)n+1 I' as above exists. Then aeZ L L aeZ j= 1,N dimM(ga)-N-w+ j. 7). Then X(V) = Xm(V(w, N)) + (_1)n L dim M(gat N- w X(F) = Xm(F(w, N)) + (_1)n+1 L L aeZ aeZ j= 1,N dimM(ga)(m-1)N-w+j for all m large enough. When all the singularities ga are weighted homogeneous, it is enough to take m ~ n + 1. Proof On a formal level, note that the formulas in (i) are a special case of the formulas in (ii), obtained by taking m divisible by all ka = 1Gal, a E Z.

Since every V contains an open neighborhood ~ corresponding by IX to Zll {(x, a) E L x (0, 1): a < g(r(x))} for some continuous function g, and H·(~\Z, F) is independent of 9 by the hypothesis on F, we get the assertion. D When Z is compact we expect that an analytic link is a topological link. The argument is similar to the proof of the Thorn isotopy theorem, but also the limit of integral curves must be controlled using a good partition of unity. This problem will be treated elsewhere. 7. Assume that the stratifications of the objects in D~(U) and D~(Z) are algebraic.

Then multiplication by df becomes a map of degree 0 and one has P(t) = L (-1t+ l-k p(Qk)(t) k=O,n+ 1 where p(Qk) is the Poincare series of Qk with respect to this new grading [22]. Then it is obvious that