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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
##### Keywords:
defect; Hodge numbers; Calabi-Yau manifolds; hypersurfaces
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##### References:
 [1] DOI: 10.1007/s00209-004-0699-z · Zbl 1073.14051 [2] DOI: 10.1002/mana.200410463 · Zbl 1120.14026 [3] DOI: 10.1215/S0012-7094-94-07509-1 · Zbl 0851.14021 [4] DOI: 10.1016/0040-9383(95)00051-8 · Zbl 0864.14022 [5] DOI: 10.1007/BF01361440 · Zbl 0145.09402 [6] Clemens C. H., Math. 47 pp 107– (1983) [7] DOI: 10.1080/00927879908826685 · Zbl 0958.14032 [8] DOI: 10.1007/s002290170030 · Zbl 0983.14017 [9] DOI: 10.1007/s002090100396 · Zbl 1004.14007 [10] DOI: 10.1002/mana.200310388 · Zbl 1101.14006 [11] DOI: 10.1515/advg.2001.023 · Zbl 1092.32015 [12] DOI: 10.1070/IM1987v029n02ABEH000970 · Zbl 0669.14012 [13] Dimca A., MR1 pp 32058– (1941) [14] Dolgachev I., Math. 34 pp 14060– (1982) [15] Durfee A. H., Enseign. Math. 25 pp 131– (2) [16] Esnault H., MR1 pp 14017– (1939) [17] DOI: 10.1007/BF01458602 · Zbl 0576.14013 [18] Kawamata Y., Stud. Pure Math. 283 pp 14015– (1987) [19] Labs O., J. London Math. Soc. 74 (2) pp 607– (2007) [20] Materov E. N., Mosc. Math. J. 2 pp 161– (2002) [21] DOI: 10.1007/BF01456083 · Zbl 0521.14013 [22] Reid M., Stud. Pure Math. 131 pp 14010– (1983) [23] Treger R., Math. 592 pp 14003– (1979) [24] DOI: 10.1016/0040-9383(93)90054-Y · Zbl 0801.14015 [25] Varchenko A. N., Dokl. 27 pp 735– (1983)
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