A geometric approach to Orlov’s theorem

By Ian Shipman. Compositio Mathematica, 2010

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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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/50a3baafd05918fbc1e5166cbaef65824e148455

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Citations in the corpus

Autoequivalences of derived categories via geometric invariant theory (2013, unknown): “sms are obtained by a certain localization procedure. The maps in the above diagram induce functors D(X s) = D(X s;0) ˇ /D(Vj Xs ;0) i /D(V;W) whose composite i ˇ is an equivalence. (For details, see [17,21].) Suppose that V is a smooth quasiprojective variety with an action of a reductive algebraic”
Towards Mirror Symmetry for Varieties of General Type (2012, background): “ave an equivalence5 (2.3) Db(S) ∼=Db(X,w,Zk) where Zk indicates a Zk-grading given by the (C∗)k-action on ˇw−1(0) induced from the split vector bundle. The hypersurface case is also treated in [Is10],[Sh11]. This equivalence is called renormalization ﬂow in the physics literature. We assumed in §1 t”
The derived category of a GIT quotient (2012, background): “In addition, if we introduce an auxiliary Gm action on X with respect to which W ∈ Γ(OX) has weight 2, then the categories B(X/G×Gm,W ) are often referred to as graded categories of singularities [25].”