Arkiv för Matematik

Non-liftable Calabi–Yau spaces

Sławomir Cynk and Matthias Schütt

Full-text: Open access


We construct many new non-liftable three-dimensional Calabi–Yau spaces in positive characteristic. The technique relies on lifting a nodal model to a smooth rigid Calabi–Yau space over some number field as introduced by one of us jointily with D. van Straten.


Funding from MNiSW under grant no N N201 388834 and DFG under grant Schu 2266/2-2 is gratefully acknowledged.

Article information

Ark. Mat., Volume 50, Number 1 (2012), 23-40.

Received: 2 December 2009
First available in Project Euclid: 31 January 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

2010 © Institut Mittag-Leffler


Cynk, Sławomir; Schütt, Matthias. Non-liftable Calabi–Yau spaces. Ark. Mat. 50 (2012), no. 1, 23--40. doi:10.1007/s11512-010-0130-4.

Export citation


  • Artin, M., Algebraic Spaces, Yale University Press, New Haven–London, 1971.
  • Artin, M., Algebraic construction of Brieskorn’s resolutions, J. Algebra 29 (1974), 330–348.
  • Beauville, A., Les familles stables de courbes elliptiques sur ℙ1 admettant quatre fibres singuliéres, C. R. Acad. Sci. Paris Sér. I Math. 294 (1982), 657–660.
  • Cynk, S. and van Straten, D., Small resolutions and non-liftable Calabi–Yau threefolds, Manuscripta Math. 130 (2009), 233–249.
  • Elkies, N. D. and Schütt, M., Modular forms and K3 surfaces. Preprint, 2008.
  • Herfurtner, S., Elliptic surfaces with four singular fibres, Math. Ann. 291 (1991), 319–342.
  • Hirokado, M., A non-liftable Calabi–Yau threefold in characteristic 3, Tohoku Math. J. 51 (1999), 479–487.
  • Hulek, K. and Verrill, H., On the modularity of Calabi–Yau threefolds containing elliptic ruled surfaces, in Mirror Symmetry V, AMS/IP Studies in Advanced Mathematics 34, pp. 19–34, Amer. Math. Soc., Providence, RI, 2006.
  • Kapustka, G. and Kapustka, M., Fiber products of elliptic surfaces with section and associated Kummer fibrations, Internat. J. Math. 20 (2009), 401–426.
  • Kapustka, M., Correspondences between modular Calabi–Yau fiber products, Manuscripta Math. 130 (2009), 121–135.
  • Khare, C. and Wintenberger, J.-P., Serre’s modularity conjecture. I, II, Invent. Math. 178 (2009), 485–504, 505–586.
  • Lang, W., Extremal rational elliptic surfaces in characteristic p. I: Beauville surfaces, Math. Z. 207 (1991), 429–437.
  • Lang, W., Extremal rational elliptic surfaces in characteristic p. II: Surfaces with three or fewer singular fibers, Ark. Mat. 32 (1994), 423–448.
  • Schoen, C., On fiber products of rational elliptic surfaces with section, Math. Z. 197 (1988), 177–199.
  • Schoen, C., Desingularized fiber products of semi-stable elliptic surfaces with vanishing third Betti number, Compos. Math. 145 (2009), 89–111.
  • Schröer, S., Some Calabi–Yau threefolds with obstructed deformations over the Witt vectors, Compos. Math. 140 (2004), 1579–1592.
  • Schütt, M., New examples of modular rigid Calabi–Yau threefolds, Collect. Math. 55 (2004), 219–228.
  • Schütt, M., On the modularity of three Calabi–Yau threefolds with bad reduction at 11, Canad. Math. Bull. 46 (2006), 296–312.
  • Schütt, M. and Schweizer, A., On the uniqueness of elliptic K3 surfaces with maximal singular fibre. Preprint, 2007.
  • Shioda, T., On the Mordell–Weil lattices, Comment. Math. Univ. St. Pauli 39 (1990), 211–240.