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Mathematische Annalen

, Volume 365, Issue 1–2, pp 155–172 | Cite as

Derived equivalent Calabi–Yau threefolds from cubic fourfolds

  • John R. CalabreseEmail author
  • Richard P. Thomas
Article

Abstract

We describe pretty examples of derived equivalences and autoequivalences of Calabi-Yau threefolds arising from pencils of cubic fourfolds. The cubic fourfolds are chosen to be special, so they each have an associated K3 surface. Thus a pencil gives rise to two different Calabi-Yau threefolds: the associated pencil of K3 surfaces, and the baselocus of the original pencil—the intersection of two cubic fourfolds. They both have crepant resolutions which are derived equivalent.

Keywords

Complete Intersection Double Cover Exceptional Divisor Crepant Resolution Linear Subsystem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgments

This paper is an extended exercise in Kuznetsov’s beautiful ideas about derived categories of cubic fourfolds [10], fractional Calabi–Yau categories [8, Section 4], and HPD [9]. Our debt to him is clear. We would like to warmly thank Nick Addington and Ed Segal for useful discussions. In addition, JC is grateful to Roland Abuaf, Michele Bolognesi, Enrica Floris, J\(\otimes \)rgen Rennemo and Brian Lehmann for helpful conversations on topics related to this paper. Finally, we would like to thank the referee for helpful comments. RT was partially supported by EPSRC programme Grant EP/G06170X/1 and JC was partially supported by NSF RTG Grant 1148609.

References

  1. 1.
    Addington, N.: The derived category of the intersection of four quadrics. arXiv:0904.1764
  2. 2.
    Addington, N.: Spinor sheaves and complete intersections of quadrics. Ph.D., University of Wisconsin-Madison (2009). http://www.math.duke.edu/~adding
  3. 3.
    Auel, A., Bernardara, M., Bolognesi, M.: Fibrations in complete intersections of quadrics, Clifford algebras, derived categories, and rationality problems. J. Math. Pures Appl. 102, 249–291 (2014). arXiv:1109.6938 MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ballard, M., Deliu, D., Favero, D., Isik, M.U., Katzarkov, L.: Homological projective duality via variation of geometric invariant theory quotients. arXiv:1306.3957
  5. 5.
    Bernardara, M.: A semiorthogonal decomposition for Brauer Severi schemes. Math. Nachr. 282, 1406–1413 (2009). arXiv:math.AG/0511497
  6. 6.
    Carocci, F., Turcinovic, Z.: Homological projective duality and blowups (preprint)Google Scholar
  7. 7.
    Harris, J., Tu, L.: On symmetric and skew-symmetric determinantal varieties. Topology 23, 71–84 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Kuznetsov, A.: Derived category of a cubic threefold and the variety \({V}_{14}\). Proc. Steklov Inst. Math. 246, 171–194 (2004). arXiv:math.AG/0303037
  9. 9.
    Kuznetsov, A.: Homological projective duality. Pub. Math. I.H.E.S. 105, 157–220 (2007). arXiv:math.AG/0507292
  10. 10.
    Kuznetsov, A.: Derived categories of cubic fourfolds. Prog. Math. 282, 219–243 (2010). arXiv:0808.3351 MathSciNetzbMATHGoogle Scholar
  11. 11.
    Kuznetsov, A.: Scheme of lines on a family of 2-dimensional quadrics: geometry and derived category. Math. Z. 276, 655–672 (2014). arXiv:1011.4146 MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kuznetsov, A.: Base change for semiorthogonal decompositions. Compos. Math. 147, 852–876 (2011). arXiv:0711.1734 MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kuznetsov, A.: Derived categories of quadric fibrations and intersections of quadrics. Adv. Math. 218, 1340–1369 (2008). arXiv:math.AG/0510670
  14. 14.
    Orlov, D.: Projective bundles, monoidal transformations, and derived categories of coherent sheaves. Izv. Ross. Akad. Nauk Ser. Mat. 56, 852–862 (1992). [translation in Russian Acad. Sci. Izv. Math. 41, 133–141 (1993)]Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Rice University MS136HoustonUSA
  2. 2.Department of MathematicsImperial College LondonLondonUK

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