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Global smoothing of Calabi-Yau threefolds. (English) Zbl 0861.14036
Let \(Z\) be a Calabi-Yau threefold, i.e. a projective threefold with only rational singularities and with \(K_Z \sim 0\). Assume that \(Z\) admits only isolated rational hypersurface singularities. Then \(Z\) can be deformed to a Calabi-Yau threefold \(Z'\) with only ordinary double points; \(Z'\) is smooth if \(Z\) is \(\mathbb{Q}\)-factorial. Furthermore if \(X\) is a normal projective threefold with only isolated rational hypersurface singularities and \(h^2(X, {\mathcal O}_X) =0\), then the rank of the abelian group of Weil divisors of \(X\) modulo the Cartier ones is expressed in terms of the Betti numbers for the singular cohomology of \(Z\). The smoothing problems here considered are faced by two different ways: one using the vanishing theorems by Guillèn, Navarro Aznar and Puerto; the other using an invariant introduced by the first author for an isolated rational singularity.

MSC:
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
14J30 \(3\)-folds
32J25 Transcendental methods of algebraic geometry (complex-analytic aspects)
14B05 Singularities in algebraic geometry
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References:
[1] [AC] A’Campo, N.: Le nombre de Lefschetz d’une monodromie. Indag. Math. (N.S.)8, 113-118 (1973) · Zbl 0276.14004
[2] [Be] Beauville, A.: Variétés Kähleriennes dont la première classe de Chern est nulle. J. Differ. Geom.18, 755-782 (1983) · Zbl 0537.53056
[3] [C1] Clemens, H.: Double solids, Adv. Math.47, 107-230 (1983) · Zbl 0509.14045 · doi:10.1016/0001-8708(83)90025-7
[4] [Di] Dimca, A.: Betti numbers of hypersurfaces and defects of linear systems. Duke Math. J.60, 285-298 (1990) · Zbl 0729.14017 · doi:10.1215/S0012-7094-90-06010-7
[5] [Fr] Friedman, R.: Simultaneous resolution of threefold double points. Math. Ann.274, 671-689 (1986) · Zbl 0576.14013 · doi:10.1007/BF01458602
[6] [G-H] Green, P.S., Hübsh, T.: Connecting moduli spaces of Calabi-yau threefolds. Comm. Math. Phys.119, 431-441 (1988) · Zbl 0684.53077 · doi:10.1007/BF01218081
[7] [H] Hirzebruch, F.: Some examples of threefolds with trivial canonical bundle. M.P.I. preprint, no. 85-58, Bonn (1985)
[8] [Ka 1] Kawamata, Y.: Crepant blowing-up of 3-dimensional canonical singularities and its application of degenerations of surfaces. Ann. Math.127(2), 93-163 (1988) · Zbl 0651.14005 · doi:10.2307/1971417
[9] [Ka 2] Kawamata, Y.: Minimal models and the Kodaira dimension of algebraic fiber spaces. J. Reine Angew. Math.363, 1-46 (1986)
[10] [Ka 3] Kawamata, Y.: Abundance theorem for minimal threefolds. preprint (1991) · Zbl 0777.14011
[11] [Ka 4] Kawamata, Y.: Unobstructed deformations, a remark on a paper of Z. Ran. J. Algebraic Geom.1, 183-190 (1992) · Zbl 0818.14004
[12] [K-M-M] Kawamata, Y., Matsuda, K., Matsuki, K.: Introduction to the minimal model problem. Adv. Stu. Pure Math.10, 283-360 (1987), Kinokuniya and North-Holland · Zbl 0672.14006
[13] [K-M] Kollár, J., Mori, S.: Classification of three-dimensional flips. J. Am. Math. Soc.5(3), 533-703 · Zbl 0773.14004
[14] [Ko] Kollár, J.: Shafarevich maps and plurigenera of algebraic varieties. Invent. Math.113, 177-215 (1993) · Zbl 0819.14006 · doi:10.1007/BF01244307
[15] [Lo] Looijenga, E.J.N.: Isolated singular points on complete intersections. London Math. Soc. Lect. Note. Ser.77, Cambridge University Press 1984 · Zbl 0552.14002
[16] [Mo] Mori, S.: Flip theorem and the existence of minimal threefolds. J. Am. Math. Soc.1, 117-253 (1988) · Zbl 0649.14023
[17] [Na] Namikawa, Y.: On deformations of Calabi-Yau threefolds with terminal singularities. Topology33(3), 429-446 (1994) · Zbl 0813.14004 · doi:10.1016/0040-9383(94)90021-3
[18] [Ra] Ran, Z.: Deformations of Calabi-Yau Kleinfolds. In: Essays on Mirror manifolds · Zbl 0827.32021
[19] [Re 1] Reid, M.: Minimal models of canonical threefolds. Adv. Stud. Pure Math.1, 131-180 (1983) Kinokuniya, North-Holland
[20] [Re 2] Reid, M.: The moduli space of threefolds withK=0 may nevertheless be irreducible. Math. Ann.278, 329-334 (1987) · Zbl 0649.14021 · doi:10.1007/BF01458074
[21] [S-S] Scherk, J., Steenbrink, J.H.M.: On the mixed Hodge structure on the cohomology of the Milnor fibre. Math. Ann.271, 641-665 (1985) · Zbl 0618.14002 · doi:10.1007/BF01456138
[22] [Sch] Schlessinger, M.: Rigidity of quotient singularities. Invent. Math.14, 17-26 (1971) · Zbl 0232.14005 · doi:10.1007/BF01418741
[23] [St 1] Steenbrink, J.H.M.: Mixed Hodge structure on the vanishing cohomology, In: Real and complex singularities, Oslo 1976. P. Holm ed. pp. 525-563, Sijthoff-Noordhoff, Alphen a/d Rijn 1977
[24] [St 2] Steenbrink, J.H.M.: Mixed Hodge structures associated with isolated singularities. Proc. Symp. Pure Math.40 Part 2, 513-536 (1983) · Zbl 0515.14003
[25] [St 3] Steenbrink, J.H.M.: Vanishing theorems on singular spaces. Astérisque130, 330-341 (1985) · Zbl 0582.32039
[26] [Te] Teissier, B.: The hunting of invariants in the geometry of discriminants. In: Real and complex singularities, Oslo 1976. P. Holm ed. pp. 565-677, Sijthoff-Noordhoff, Alphen a/d Rijn 1977
[27] [W] Werner, J.: Kleine Auflösungen spezieller dreidimensionaler Varietäten. Bonner Math. Schriften Nr.186 (1987) · Zbl 0657.14021
[28] [Wi] Wilson, P.M.H.: Calabi-Yau manifolds with large Picard number. Invent. Math.98, 139-155 (1989) · Zbl 0688.14032 · doi:10.1007/BF01388848
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