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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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