A Geometric Approach to Orlov's Theorem

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 be a projective hypersurface. Already, Segal has established an equivalence between Orlov's category of graded matrix factorizations and the category of graded D-branes on the canonical bundle K of the ambient projective space. To complete the picture, I give an equivalence between the homotopy category of graded D-branes on K and Dcoh(X). This can be achieved directly and by deforming K to the normal bundle of X, embedded in K and invoking a global version of Knörrer periodicity. We also discuss an equivalence between graded D-branes on a general smooth quasi-projective variety and on the formal neighborhood of the singular locus of the zero fiber of the potential.

27 pages. Extensively revised from previous version. Final version to appear in Compositio Mathematica

References in corpus (1)

• Matrix factorizations for nonaffine LG-models

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