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A geometric approach to Orlov’s theorem

Part of: Homological algebra (Co)homology theory

Published online by Cambridge University Press:  10 July 2012

Ian Shipman
Ian Shipman*
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA (email: ishipman@MIT.edu)
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Abstract

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A famous theorem of D. Orlov describes the derived bounded category of coherent sheaves on projective hypersurfaces in terms of an algebraic construction called graded matrix factorizations. In this article, I implement a proposal of E. Segal to prove Orlov’s theorem in the Calabi–Yau setting using a globalization of the category of graded matrix factorizations (graded D-branes). Let X⊂ℙ be a projective hypersurface. Segal has already established an equivalence between Orlov’s category of graded matrix factorizations and the category of graded D-branes on the canonical bundle Kℙ to ℙ. To complete the picture, I give an equivalence between the homotopy category of graded D-branes on Kℙ and Dbcoh(X). This can be achieved directly, as well as by deforming Kℙ to the normal bundle of X⊂Kℙ and invoking a global version of Knörrer periodicity. We also discuss an equivalence between graded D-branes on a general smooth quasiprojective variety and on the formal neighborhood of the singular locus of the zero fiber of the potential.

Keywords

matrix factorizationsD-branesbrane transport

MSC classification

Secondary: 14F05: Sheaves, derived categories of sheaves and related constructions 18G35: Chain complexes
Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 5 , September 2012 , pp. 1365 - 1389
Copyright
Copyright © Foundation Compositio Mathematica 2012

References

[BP10]Baranovsky, V. and Pecharich, J., On equivalences of derived and singular categories, Cent. Eur. J. Math. 8 (2010), 1–14.Google Scholar
[BN93]Bökstedt, M. and Neeman, A., Homotopy limits in triangulated categories, Compositio Math. 86 (1993), 209–234.Google Scholar
[BB03]Bondal, A. and van den Bergh, M., Generators and representability of functors in commutative and noncommutative geometry, Mosc. Math. J. 3 (2003), 1–36.Google Scholar
[BW]Burke, J. and Walker, M., Matrix factorizations over quasiprojective schemes, in preparation.Google Scholar
[Eis80]Eisenbud, D., Homological algebra on a complete intersection, with an application to group representations, Trans. Amer. Math. Soc. 260 (1980), 35–64.Google Scholar
[HW05]Hori, K. and Walcher, J., F-term equations near Gepner points, J. High Energy Phys. (2005), 008 (electronic) 23 pp.Google Scholar
[Isi10]Isik, M. U., Equivalence of the derived category of a variety with a singularity category, Preprint (2010), arXiv:1011.1484.Google Scholar
[Kno87]Knörrer, H., Cohen-Macaulay modules on hypersurface singularities. I, Invent. Math. 88 (1987), 153–164.Google Scholar
[LP11]Lin, K. and Pomerleano, D., Global matrix factorizations, Preprint (2011), arXiv:1101.5847.Google Scholar
[Nee92]Neeman, A., The connection between the K-theory localization theorem of Thomason, Trobaugh and Yao and the smashing subcategories of Bousfield and Ravenel, Ann. Sci. École Norm. Sup. (4) 25 (1992), 547–566.Google Scholar
[Orl06]Orlov, D. O., Triangulated categories of singularities, and equivalences between Landau-Ginzburg models, Mat. Sb. 197 (2006), 117–132.Google Scholar
[Orl09a]Orlov, D., Formal completions and idempotent completions of triangulated categories of singularities, Preprint (2009), arXiv:0901.1859.Google Scholar
[Orl09b]Orlov, D., Derived categories of coherent sheaves and triangulated categories of singularities, in Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Volume II, Progress in Mathematics, vol. 270 (Birkhäuser, Boston, MA, 2009), 503–531.Google Scholar
[Orl11]Orlov, D., Matrix factorizations for nonaffine LG-models, Preprint (2011), arXiv:1101.4051.Google Scholar
[Pos09]Positselski, L., Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence, Preprint (2009), arXiv:0905.2621.Google Scholar
[Pos11]Positselski, L., Coherent analogs of matrix factorizations and relative categories of singularities, Preprint (2011), arXiv:1102.0261.Google Scholar
[Seg09]Segal, Ed., Equivalences between GIT quotients of Landau-Ginzburg B-models, Preprint (2009), arXiv:0910.5534.Google Scholar
[Sum74]Sumihiro, H., Equivariant completion, J. Math. Kyoto Univ. 14 (1974), 1–28.Google Scholar
[Wit93]Witten, E., Phases of N=2 theories in two dimensions, Nuclear Phys. B 403 (1993), 159–222.Google Scholar