Triangulated categories of singularities and D-branes in Landau-Ginzburg models

In spite of physics terms in the title, this paper is purely mathematical. Its purpose is to introduce triangulated categories related to singularities of algebraic varieties and establish a connection of these categories with D-branes in Landau-Ginzburg models It seems that two different types of categories can be associated with singularities (or singularities of maps). Categories of the first type are connected with vanishing cycles and closely related to the categories which were introduced in [24] for symplectic Picard-Lefschetz pencils. Categories of the second type are purely algebraic and come from derived categories of coherent sheaves. Categories of this type will be central in this work. An important notion here is the concept of a perfect complex, which was introduced in [3]. A perfect complex is a complex of sheaves which locally is quasi-isomorphic to a bounded complex of locally free sheaves of finite type (a good reference is [25]). To any algebraic variety X one can attach the bounded derived category of coherent sheaves D(coh(X)) . This category admits a triangulated structure. The derived category of coherent sheaves has a triangulated subcategory Perf(X) formed by perfect complexes. If the variety X is smooth then any coherent sheaf has a finite resolution of locally free sheaves of finite type and the subcategory of perfect complexes coincides with the whole of D(coh(X)). But for singular varieties this property is not fulfilled. We introduce a notion of triangulated category of singularities DSg(X) as the quotient of the triangulated category D(coh(X)) by the full triangulated subcategory of perfect complexes Perf(X) . The category DSg(X) reflects the properties of the singularities of X and ”does not depend on all of X ”. For example we prove that it is invariant with respect to a localization in Zariski topology (Proposition 1.14). The category DSg(X) has good properties when X is Gorenstein. In this case, if the locus of singularities is complete then all Hom’s between objects are finite-dimensional vector spaces (Corollary 1.24). The investigation of such categories is inspired by the Homological Mirror Symmetry Conjecture ([21]). Works on topological string theory are mainly concerned with the case of N=2 superconformal sigmamodels with a Calabi-Yau target space. In this case the field theory has two topologically twisted versions: Amodels and B-models. The corresponding D-branes are called A-branes and B-branes. The mirror symmetry should interchange these two classes of D-branes. From the mathematical point of view the category of B-branes on a Calabi-Yau is the derived category of coherent sheaves on it ([21],[7]). As a candidate for a category of A-branes on Calabi-Yau manifolds so-called Fukaya category has been proposed. Its objects