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Betti numbers of hypersurfaces and defects of linear systems. (English) Zbl 0729.14017
Let \(f(x_ 0,...,x_ n)\) be a weighted homogeneous polynomial in \(n+1\) variables, and let V be the hypersurface defined by \(f=0\) in the associated weighted projective space \({\mathbb{P}}\). One assumes that the singular locus \(\Sigma\) of V is isolated. It is important to calculate the Betti numbers \(b_ j(V)\), especially \(b_ n(V)\) and \(b_{n-1}(V)\). In the earlier works of H. Clemens [Adv. Math. 47, 107-230 (1983; Zbl 0509.14045)], H. Esnault [Invent. Math. 68, 477-496 (1982; Zbl 0489.14009)], C. Schoen [Math. Ann. 270, 17-27 (1985; Zbl 0533.14002)], and J. Werner [Bonn. Math. Schr. 186 (1987; Zbl 0657.14021)], some special cases were studied and it turned out that \(b_ n(V)\) depends, in an intricate manner, on the positions of the singularities of V. - In the paper under review, the author gives a general account of these facts through the study of the cohomology of the complement \(U={\mathbb{P}}\setminus V\). He uses the differential forms with poles along V, following the work of P. A. Griffiths [Ann. Math., II. Ser. 90, 460-495, 496-541 (1969; Zbl 0215.081)]. By a detailed study of the mixed Hodge structures on \(H^ n(U)\) and \(H^ n_{\Sigma}(V)\), the computation of \(b_ n(V)\) (or, more precisely, its primitive part) is reduced to the calculation of the number of homogeneous polynomials of certain degree with some condition imposed by the singularities.
Reviewer: E.Horikawa (Tokyo)

MSC:
14F45 Topological properties in algebraic geometry
14J17 Singularities of surfaces or higher-dimensional varieties
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14C20 Divisors, linear systems, invertible sheaves
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