d-Semistable Calabi–Yau threefolds of type III

  • Nam-Hoon Lee


We develop some methods to construct normal crossing varieties whose dual complexes are two-dimensional, which are smoothable to Calabi–Yau threefolds. We calculate topological invariants of smoothed Calabi–Yau threefolds and show that several of them are new examples.

Mathematics Subject Classification

14J32 14D05 32G20 


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The author is very thankful to the referee for making several valuable suggestions for the initial draft of this note. This work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2017R1D1A2B03029525) and Hongik University Research Fund.


  1. 1.
    Altman, R., Gray, J., He, Y.-H., Jejjala, V., Nelson, B.D.: A Calabi–Yau database: threefolds constructed from the Kreuzer–Skarke list. J. High Energy Phys. (2015).
  2. 2.
    Donagi, R., Katz, S., Wijnholt, M.: Weak coupling, degeneration and log Calabi–Yau spaces. Pure Appl. Math. Q. 9(4), 655–738 (2013)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Doran, C.F., Harder, A., Thompson, A.: Mirror symmetry, Tyurin degenerations and fibrations on Calabi–Yau manifolds. arXiv:1601.08110v2
  4. 4.
    Friedman, R.: Global smoothings of varieties with normal crossings. Ann. Math. (2) 118(1), 75–114 (1983)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Fujita, K.: Simple normal crossing Fano varieties and log Fano manifolds. Nagoya Math. J. 214, 95–123 (2014)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Kawamata, Y., Namikawa, Y.: Logarithmic deformations of normal crossing varieties and smoothing of degenerate Calabi–Yau varieties. Invent. Math. 118(3), 395–409 (1994)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Kreuzer, M., Skarke, H.: Complete classification of reflexive polyhedra in four dimensions. Adv. Theor. Math. Phys. 4(6), 12091230 (2000)MathSciNetCrossRefGoogle Scholar
  8. 8.
  9. 9.
    Kulikov, V.S.: Degenerations of \(K3\) surfaces and Enriques surfaces. Izv. Akad. Nauk SSSR Ser. Mat. 41(5), 1008-1042, 1199 (1977)MathSciNetGoogle Scholar
  10. 10.
    Lee, N.-H.: Constructive Calabi–Yau manifolds. Ph.D. thesis, University of Michigan (2006). Accessed 4 Dec 2018
  11. 11.
    Lee, N.-H.: Calabi–Yau construction by smoothing normal crossing varieties. Int. J. Math. 21(6), 701–725 (2010)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Lee, N.-H.: Mirror pairs of Calabi–Yau threefolds from mirror pairs of quasi-Fano threefolds. arXiv:1708.02489
  13. 13.
    Lee, N.-H.: 6518 mirror pairs of Calabi–Yau threefolds: appendix to “Mirror pairs of Calabi–Yau theefolds from mirror pairs of quasi-Fano threefolds”. Accessed 4 Dec 2018
  14. 14.
    Persson, U.: On degenerations of algebraic surfaces. Memoirs of the American Mathematical Society 11, no. 189 (1977)Google Scholar
  15. 15.
    Persson, U., Pinkham, H.: Degeneration of surfaces with trivial canonical bundle. Ann. Math. (2) 113(1), 45–66 (1981)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Tyurin, A.N.: Fano versus Calabi–Yau. In: The Fano Conference, pp. 701–734. Universitá di Torino, Turin (2004)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics EducationHongik UniversityMapo-gu, SeoulKorea
  2. 2.School of MathematicsKorea Institute for Advanced StudyDongdaemun-gu, SeoulSouth Korea

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