Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



The Pfaffian-Grassmannian derived equivalence

Authors: Lev Borisov and Andrei Caldararu
Journal: J. Algebraic Geom. 18 (2009), 201-222
Published electronically: March 17, 2008
MathSciNet review: 2475813
Full-text PDF

Abstract | References | Additional Information

Abstract: We argue that there exists a derived equivalence between Calabi-Yau threefolds obtained by taking dual hyperplane sections (of the appropriate codimension) of the Grassmannian $ \mathbf{G}(2,7)$ and the Pfaffian $ \mathbf{Pf}(7)$. The existence of such an equivalence has been conjectured by physicists for almost ten years, as the two families of Calabi-Yau threefolds are believed to have the same mirror. It is the first example of a derived equivalence between non-birational Calabi-Yau threefolds.

References [Enhancements On Off] (What's this?)

  • 1. Tom Bridgeland, Equivalences of triangulated categories and Fourier-Mukai transforms, Bull. London Math. Soc. 31 (1999), no. 1, 25–34. MR 1651025,
  • 2. David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960
  • 3. Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157
  • 4. Hori, K., Tong, D., Aspects of non-abelian gauge dynamics in two-dimensional $ \mathscr{N}=(2,2)$ theories, preprint, hep-th/0609032.
  • 5. János Kollár and Shigefumi Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998. With the collaboration of C. H. Clemens and A. Corti; Translated from the 1998 Japanese original. MR 1658959
  • 6. Kuznetsov, A., Homological Projective Duality, preprint, math.AG/0507292.
  • 7. Kuznetsov, A., Exceptional collections for Grassmannians of isotropic lines, preprint, math.AG/0512013.
  • 8. Kuznetsov, A., Lefschetz exceptional collections and categorical resolutions of singularities, preprint, math.AG/0609240.
  • 9. Einar Andreas Rødland, The Pfaffian Calabi-Yau, its mirror, and their link to the Grassmannian 𝐺(2,7), Compositio Math. 122 (2000), no. 2, 135–149. MR 1775415,

Additional Information

Lev Borisov
Affiliation: Mathematics Department, University of Wisconsin–Madison, 480 Lincoln Drive, Madison, Wisconsin 53706–1388

Andrei Caldararu
Affiliation: Mathematics Department, University of Wisconsin–Madison, 480 Lincoln Drive, Madison, Wisconsin 53706–1388

Received by editor(s): December 13, 2006
Received by editor(s) in revised form: June 9, 2007
Published electronically: March 17, 2008
Additional Notes: This material is based upon work supported by the National Science Foundation under Grants No. DMS-0456801 and DMS-0556042

American Mathematical Society