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 ⊂ 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 K P to P. To complete the picture, I give an equivalence between the homotopy category of graded D-branes on K P and D b coh(X). This can be achieved directly, as well as by deforming K P to the normal bundle of X ⊂ K P 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.

Proof of a conjecture of Batyrev and Nill

“…The statement without G is exactly that in [Isi13]. Using the equivalent category of factorizations which we avoid in this paper for technical simplicity, the result is also in [Shi12].…”

On Clifford double mirrors of toric complete intersections

“…Finally, we observe that D b (coh[V − /C * ], C) is equivalent to the derived category of the complete intersection D b (Y ) by the work of Isik [Isik13] and Shipman [Shi12].…”

“…Importantly, we can relate the above defined category D B (K, c) to the derived category of a Calabi-Yau complete intersection for any choice of a decomposition deg ∨ = t 1 + · · · + t k . Specifically, there is the following result, due to multiple authors, see [FK14,Isik13,Shi12].…”

Fractional Calabi�Yau categories from Landau�Ginzburg models

“…The following theorem is originally due to Isik [Isi13] and Shipman [Shi12] and due to Hirano [Hir16] in the G-equivariant case, which is the case we will use. Theorem 3.5 (Proposition 4.8 of [Hir16]).…”