A Geometric Approach to Orlov's Theorem

Details A Geometric Approach to Orlov's Theorem A Geometric Approach to Orlov's Theorem

2010-12-23

arxiv.org/abs/1012.5282v3

Mathematics

Algebraic Geometry

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

Scientific paper

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 Kn\"{o}rrer 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.

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