A geometric approach to Orlov's theorem

作者：Shipman Ian^*

来源：Compositio Mathematica, 2012, 148(5): 1365-1389.

DOI：10.1112/S0010437X12000255

• matrix factorizations
• D-branes
• brane transport

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 subset of 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 subset of K-P and invoking a global version of Knorrer 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.

• 出版日期2012-9