Enhanced Triangulated Categories

Bondal, Alexey; Kapranov, Mikhail

doi:10.1070/sm1991v070n01abeh001253

Cited by 311 publications

(449 citation statements)

References 13 publications

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“…Denote by mod-B the category of finitely generated right modules over B. There is a theorem according to which if D is an enhanced triangulated category in the sense of Bondal and Kapranov [5], then it is equivalent to the bounded derived category…”

Section: Definition 24 the Algebra Of A Strong Exceptional Collectionmentioning

confidence: 99%

Mirror symmetry for Del Pezzo surfaces: Vanishing cycles and coherent sheaves

Auroux¹,

2006

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Abstract. We study homological mirror symmetry for Del Pezzo surfaces and their mirror LandauGinzburg models. In particular, we show that the derived category of coherent sheaves on a Del Pezzo surface X k obtained by blowing up CP 2 at k points is equivalent to the derived category of vanishing cycles of a certain elliptic fibration W k : M k → C with k +3 singular fibers, equipped with a suitable symplectic form. Moreover, we also show that this mirror correspondence between derived categories can be extended to noncommutative deformations of X k , and give an explicit correspondence between the deformation parameters for X k and the cohomology class [B + iω] ∈ H 2 (M k , C).

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Section: Definition 24 the Algebra Of A Strong Exceptional Collectionmentioning

confidence: 99%

Mirror symmetry for Del Pezzo surfaces: Vanishing cycles and coherent sheaves

Auroux¹,

2006

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“…Finally, 0 mod L tr is the only point in E G . Therefore, π is ramified over 3 points with multiplicities (2,3,6).…”

Section: 2mentioning

confidence: 99%

“…, E n ) is strong if Hom a (E i , E j ) = 0 for a = 0 and all i, j. If an exceptional collection is full and strong then E = ⊕ n i=1 E i is a tilting object in A, i.e., the functor X → R Hom(E, X) gives an equivalence between A and D b (mod −A) (provided A satisfies some natural finiteness assumptions and is framed, see [6]). …”

Section: 2mentioning

confidence: 99%

Holomorphic bundles on 2-dimensional noncommutative toric orbifolds

Polishchuk¹

Noncommutative Geometry and Number Theory

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Abstract. We define the notion of a holomorphic bundle on the noncommutative toric orbifold T θ /G associated with an action of a finite cyclic group G on an irrational rotation algebra. We prove that the category of such holomorphic bundles is abelian and its derived category is equivalent to the derived category of modules over a finite-dimensional algebra Λ. As an application we finish the computation of K 0 -groups of the crossed product algebras describing the above orbifolds initiated in [17], [28], [29], [12] and [13]. Also, we describe a torsion pair in the category of Λ-modules, such that the tilting with respect to this torsion pair gives the category of holomorphic bundles on T θ /G.

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“…Given a triangulated category (more precisely, an "enhanced triangulated category" [3]), one can study its deformations. Information about deformations is encoded in the Hochschild cohomology of the category in question.…”

Section: Mirror Symmetry From a Physical Viewpointmentioning

confidence: 99%

“…vanishing) on some scalar fields which appear in the Lagrangian. 3 A classical D-brane is simply a choice of such a boundary condition.…”

Section: Mirror Symmetry From a Physical Viewpointmentioning

confidence: 99%

Lectures on mirror symmetry, derived categories, andD-branes

Kapustin¹,

Orlov²

2004

Russ. Math. Surv.

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Abstract. This paper is an introduction to Homological Mirror Symmetry, derived categories, and topological D-branes aimed mainly at a mathematical audience. In the paper we explain the physicists' viewpoint of the Mirror Phenomenon, its relation to derived categories, and the reason why it is necessary to enlarge the Fukaya category with coisotropic A-branes; we discuss how to extend the definition of Floer homology to such objects and describe mirror symmetry for flat tori. The paper consists of four lectures which were given at the Institute for Pure and Applied Mathematics (Los Angeles), March 2003, as part of a program on Symplectic Geometry and Physics. Mirror Symmetry From a Physical ViewpointThe goal of the first lecture is to explain the physicists' viewpoint of the Mirror Phenomenon and its interpretation in mathematical terms proposed by Maxim Kontsevich in his 1994 talk at the International Congress of Mathematicians [25]. Another approach to Mirror Symmetry was proposed by A. Strominger, S-T. Yau, and E. Zaslow [41], but we will not discuss it here.From the physical point of view, Mirror Symmetry is a relation on the set of 2d conformal field theories with N = 2 supersymmetry. A 2d conformal field theory is a rather complicated algebraic object whose definition will be sketched in a moment. Thus Mirror Symmetry originates in the realm of algebra. Geometry will appear later, when we specialize to a particular class of N = 2 superconformal field theories related to Calabi-Yau manifolds.Let us start with 2d conformal field theory. The data needed to specify a 2d CFT consist of an infinite-dimensional vector space V (the space of states), three special elements in V (the vacuum vector |vac , and two more elements L andL ), and a linear map Y from V to the space of "formal fractional power series in z,z with coefficients in End(V ) " ( Y is called the state-operator correspondence). The precise definition of what a "formal fractional power series" means can be found in [19]; to keep things simple, one can pretend that Y takes values in the space of Laurent series in z,z with coefficients in End(V ), although such a definition is not sufficient for applications to Mirror Symmetry. These data must satisfy a number of axioms whose precise form can be found in [19]. Roughly speaking, they are (i) Y (|vac ) = id V .

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Enhanced Triangulated Categories

Cited by 311 publications

References 13 publications

Mirror symmetry for Del Pezzo surfaces: Vanishing cycles and coherent sheaves

Mirror symmetry for Del Pezzo surfaces: Vanishing cycles and coherent sheaves

Holomorphic bundles on 2-dimensional noncommutative toric orbifolds

Lectures on mirror symmetry, derived categories, andD-branes

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