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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)].

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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