Journal of Algebraic Geometry

Author(s): Lev Borisov; Andrei Caldararu
Journal: J. Algebraic Geom. 18 (2009), 201-222.
Posted: March 17, 2008
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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.

1.: Bridgeland, T., Equivalences of triangulated categories and Fourier-Mukai transforms, Bull. London Math. Soc. 31 (1999), no. 1, 25-34 (also preprint, math.AG/9809114). MR 1651025 (99k:18014)
2.: Eisenbud, D., Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics Vol. 150, Springer-Verlag (1995). MR 1322960 (97a:13001)
3.: Hartshorne, R., Algebraic Geometry, Graduate Texts in Mathematics Vol. 52, Springer-Verlag (1977). MR 0463157 (57:3116)
4.: Hori, K., Tong, D., Aspects of non-abelian gauge dynamics in two-dimensional $\mathscr{N}=(2,2)$ theories, preprint, hep-th/0609032.
5.: Kollár, J., Mori, S., Birational Geometry of Algebraic Varieties, Cambridge Tracts in Mathematics Vol. 134, Cambridge University Press (1998). MR 1658959 (2000b:14018)
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.: Rødland, E., A., The Pfaffian Calabi-Yau, its Mirror, and their link to the Grassmannian , Compositio Math. 122 (2000), no. 2, 135-149 (also preprint, math.AG/9801092). MR 1775415 (2001h:14051)

Lev Borisov
Affiliation: Mathematics Department, University of Wisconsin--Madison, 480 Lincoln Drive, Madison, Wisconsin 53706--1388
Email: borisov@math.wisc.edu

Andrei Caldararu
Affiliation: Mathematics Department, University of Wisconsin--Madison, 480 Lincoln Drive, Madison, Wisconsin 53706--1388
Email: andreic@math.wisc.edu

PII: S 1056-3911(08)00496-7
Received by editor(s): December 13, 2006
Received by editor(s) in revised form: June 9, 2007
Posted: 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

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