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A geometric approach to Orlov’s theorem

  • Ian Shipman (a1)

Abstract

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 K to ℙ. To complete the picture, I give an equivalence between the homotopy category of graded D-branes on K and Dbcoh(X). This can be achieved directly, as well as by deforming K to the normal bundle of XK 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.

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References

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A geometric approach to Orlov’s theorem

  • Ian Shipman (a1)

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