The Pfaffian-Grassmannian derived equivalence
Authors:
Lev Borisov and Andrei Căldăraru
Journal:
J. Algebraic Geom. 18 (2009), 201-222
DOI:
https://doi.org/10.1090/S1056-3911-08-00496-7
Published electronically:
March 17, 2008
MathSciNet review:
2475813
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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
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- Kuznetsov, A., Homological Projective Duality, preprint, math.AG/0507292.
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References
- 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)
- Eisenbud, D., Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics Vol. 150, Springer-Verlag (1995). MR 1322960 (97a:13001)
- Hartshorne, R., Algebraic Geometry, Graduate Texts in Mathematics Vol. 52, Springer-Verlag (1977). MR 0463157 (57:3116)
- Hori, K., Tong, D., Aspects of non-abelian gauge dynamics in two-dimensional $\mathscr {N}=(2,2)$ theories, preprint, hep-th/0609032.
- Kollár, J., Mori, S., Birational Geometry of Algebraic Varieties, Cambridge Tracts in Mathematics Vol. 134, Cambridge University Press (1998). MR 1658959 (2000b:14018)
- Kuznetsov, A., Homological Projective Duality, preprint, math.AG/0507292.
- Kuznetsov, A., Exceptional collections for Grassmannians of isotropic lines, preprint, math.AG/0512013.
- Kuznetsov, A., Lefschetz exceptional collections and categorical resolutions of singularities, preprint, math.AG/0609240.
- Rødland, E., A., The Pfaffian Calabi–Yau, its Mirror, and their link to the Grassmannian $G(2,7)$, Compositio Math. 122 (2000), no. 2, 135–149 (also preprint, math.AG/9801092). MR 1775415 (2001h:14051)
Additional Information
Lev Borisov
Affiliation:
Mathematics Department, University of Wisconsin–Madison, 480 Lincoln Drive, Madison, Wisconsin 53706–1388
Email:
borisov@math.wisc.edu
Andrei Căldăraru
Affiliation:
Mathematics Department, University of Wisconsin–Madison, 480 Lincoln Drive, Madison, Wisconsin 53706–1388
Email:
andreic@math.wisc.edu
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
