• DOI:10.1112/S0010437X12000255
• Corpus ID: 55411253

A geometric approach to Orlov’s theorem

@article{Shipman2012AGA,
  title={A geometric approach to Orlov’s theorem},
  author={Ian Shipman},
  journal={Compositio Mathematica},
  year={2012},
  volume={148},
  pages={1365 - 1389}
}

• I. Shipman
• Published 23 December 2010
• Mathematics
• Compositio Mathematica

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 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.