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Small resolutions and non-liftable Calabi-Yau threefolds. (English) Zbl 1177.14081
The authors construct further examples of Calabi–Yau threefolds at small primes that do not lift to characteristic zero. The starting point are families of Calabi–Yau threefolds in mixed characteristic whose generic fiber is smooth and whose special fiber is rigid with only nodes as singularities. The key observation is that, although the special fiber \(X\) admits a lifting, any small resolution \(Y\) of the special fiber does not admit a lifting.
Using coverings of projective \(3\)-space branched along suitable arrangements of planes, the authors construct projective nonliftable Calabi–Yau threefolds for \(p=3,5\). Taking fiber products of elliptic surfaces, they construct further nonliftable examples for certain primes up to \(p=9001\). Here, however, small resolutions \(Y\rightarrow X\) seem to exist only as algebraic spaces.
[See also M. Hirokado, Tohoku Math. J., II. Ser. 51, No.4, 479–487 (1999; Zbl 0969.14028); S. Schröer, Compos. Math. 140, No. 6, 1579–1592 (2004; Zbl 1074.14037); M. Hirokado, H. Ito, N. Saito, Manuscr. Math. 125, No. 3, 325–343 (2008; Zbl 1155.14031); C. Schoen, Compos. Math. 145, No. 1, 89–111 (2009; Zbl 1163.14006)].

MSC:
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
14B12 Local deformation theory, Artin approximation, etc.
14J17 Singularities of surfaces or higher-dimensional varieties
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References:
[1] Artin M.: Algebraic construction of Brieskorn’s resolutions. J. Algebra 29, 330–348 (1974) · Zbl 0292.14013
[2] Atiyah M.: On analytic surfaces with double points. Proc. R. Soc. Lond. Ser. A 247, 237–244 (1958) · Zbl 0135.21301
[3] Beauville A.: Les familles stables de courbes elliptiques sur P 1 admettant quatre fibres singulières. (French) [The stable families of elliptic curves on P 1 with four singular fibers]. C. R. Acad. Sci. Paris Sér. I Math. 294(19), 657–660 (1982) · Zbl 0504.14016
[4] Brieskorn E.: Über die Auflösung gewisser Singularitäten von holomorphen Abbildungen. Math. Ann. 166, 76–102 (1966) · Zbl 0145.09402
[5] Cynk S., Meyer C.: Geometry and arithmetic of certain Double octic Calabi-Yau manifolds. Can. Math. Bull. 48(2), 180–194 (2005) · Zbl 1079.14049
[6] Cynk S., van Straten D.: Infinitesimal deformations of double covers of smooth algebraic varieties. Math. Nachr. 279(7), 716–726 (2006) · Zbl 1101.14006
[7] Ekedahl, T.: On non-liftable Calabi-Yau threefolds (preprint). math.AG/0306435
[8] Friedman R.: Simultaneous resolution of threefold double points. Math. Ann. 274(4), 671–689 (1986) · Zbl 0576.14013
[9] van der Geer G.: On the height of Calabi-Yau varieties in positive characteristic. Doc. Math. 8, 97–113 (2003) · Zbl 1074.14524
[10] Grothendieck, A.: Géométrie formelle et géometrie algébrique. Sem. Bourbaki, Exposé 182 (1959)
[11] Grothendieck, A.: Revetements Étales et Group Fondamental (SGA1). Séminaire de géométire algébrique du Bois Marie 1960–1961. Lecture Notes in Mathematics, vol. 224. Springer, Berlin (1971)
[12] Hirokado M.: A non-liftable Calabi-Yau threefold in characteristic 3. Tohoku Math. J. (2) 51(4), 479–487 (1999) · Zbl 0969.14028
[13] Hirokado M., Ito H., Saito N.: Calabi-Yau threefolds arising from fiber products of rational quasi-elliptic surfaces, I. Arkiv Mat. 45(2), 279–296 (2007) · Zbl 1156.14033
[14] Hirokado M., Ito H., Saito N.: Calabi-Yau threefolds arising from fiber products of rational quasi-elliptic surfaces, II. Manuscripta Math. 125(3), 325–343 (2008) · Zbl 1155.14031
[15] Hirzebruch F.: Hilberts modular group of the field \({\mathbb{Q}(\sqrt{5})}\) and the cubic diagonal surface of Clebsch and Klein. Russ. Math. Surv. 31(5), 96–110 (1976) · Zbl 0356.14010
[16] Illusie, L.: Complex Cotangent et déformations. Lecture Notes in Mathematics, vol. 239. Springer, Berlin (1971) · Zbl 0224.13014
[17] Illusie, L.: Grothendieck’s existence theorem in formal geometry (with a letter of J-P. Serre). In: Fantechi, B., Goettsche, L., Illusie, L., Kleiman, S., Nitsure, N., Vistoli, A. (eds.) Advanced School in Basic Algebraic Geometry, ICTP Trieste, 2003, Fundamental Algebraic Geometry, Grothendieck’s FGA explained. Mathematical Surveys and Monographs, vol. 123. American Mathemcatical Society, Providence (2005)
[18] Kodaira K.: On stability of compact submanifolds of complex manifolds. Am. J. Math. 85, 79–94 (1963) · Zbl 0173.33101
[19] Schoen C.: On fiber products of rational elliptic surfaces with Section. Math. Z. 197, 177–199 (1988) · Zbl 0631.14032
[20] Schoen, C.: Desingularized fiber products of semi-stable elliptic surfaces with vanishing third Betti number (preprint). arXiv:0804.1078 · Zbl 1163.14006
[21] Schröer S.: Some Calabi-Yau threefolds with obstructed deformations over the Witt vectors. Compos. Math. 140(6), 1579–1592 (2004) · Zbl 1074.14037
[22] Schröer S.: The T 1-lifting theorem in positive characteristic. J. Algebraic Geom. 12(4), 699–714 (2003) · Zbl 1079.14505
[23] Schütt M.: New examples of modular rigid Calabi-Yau threefolds. Collect. Math. 55(2), 219–228 (2004) · Zbl 1062.14050
[24] Wahl J.: Simultaneous resolution and discriminantal loci. Duke Math. J. 46(2), 341–375 (1979) · Zbl 0472.14002
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