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Defect and Hodge numbers of hypersurfaces. (English) Zbl 1144.14033
The Hodge numbers of double solids [C. H. Clemens, Adv. Math. 47, 107–230 (1983; Zbl 0509.14045)] and the middle Betti numbers of hypersurfaces having only isolated singularities in weighted projective spaces [the reviewer, “Singularities and topology of hypersurfaces.” Universitext. New York etc.: Springer-Verlag. xvi (1992; Zbl 0753.57001)] depend on a subtle invariant, the defect. This number reflects the position of the singularities with respect to some linear systems. The author extends these results to the case of hypersurfaces \(Y\) with A-D-E singularities in a complex normal Cohen-Macaulay 4-fold \(X\) satisfying some vanishing properties of Bott-type.
As a main example, \(Y\) can be an ample hypersurface in a complete simplicial toric 4-fold \(X\). The study of the relation between the Hodge numbers of a Kähler small resolution and a big one, and the computation of the Hodge numbers of some Calabi-Yau manifolds obtained as Kähler small resolutions complete the paper.

MSC:
14J30 \(3\)-folds
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14Q10 Computational aspects of algebraic surfaces
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