A Geometric Approach to Orlov's Theorem

TL;DR

This paper provides a geometric proof of Orlov's theorem relating derived categories of coherent sheaves on hypersurfaces to graded matrix factorizations, using a globalization approach involving graded D-branes and Kn"{o}rrer periodicity.

Contribution

It introduces a geometric method to prove Orlov's theorem in the Calabi-Yau setting by connecting graded D-branes, matrix factorizations, and derived categories through global equivalences.

Findings

Established an equivalence between graded D-branes on the canonical bundle and Dcoh(X).

Provided a direct and deformation-based proof of the equivalence between graded D-branes and derived categories.

Extended the framework to general smooth quasi-projective varieties and their singular loci.

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 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.