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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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##### References:
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