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A geometric approach to Orlov’s theorem

Part of: (Co)homology theory Homological algebra

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

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