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Hodge numbers of hypersurfaces. (English) Zbl 0879.14018
Let \(V\) be a hypersurface with isolated singularities in the complex projective space \(\mathbb{P}^{n+1}\). Then there are two possibly nontrivial primitive cohomology groups of \(V\), namely \(H^n_0 (V)\) and \(H_0^{n+1} (V)\), each of which carries a canonical mixed Hodge structure. It is the (mixed) Hodge numbers of \(H^n_0(V)\) that the author compute in terms of some data connected with the singularity of \(V\).
As applications, he gives certain formulas which express its Hodges numbers, when \(V\) is a \(\mathbb{Q}\)-manifold, and when all the singularities are weighted homogeneous. Furthermore, he obtains
bounds for the Euler characteristic of the real points of \(V\), and restrictions on the possible configurations of singularities.

MSC:
14J70 Hypersurfaces and algebraic geometry
14J17 Singularities of surfaces or higher-dimensional varieties
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
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