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439
The Orbifold Chow Ring of Toric DeligneMumford Stacks
 JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY
, 2004
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On Gorenstein projective, injective and flat dimensions  a functorial description with applications
, 2004
"... For a large class of rings, including all those encountered in algebraic geometry, we establish the conjectured Moritalike equivalence between the full subcategory of complexes of finite Gorenstein flat dimension and that of complexes of finite Gorenstein injective dimension. This functorial descr ..."
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Cited by 53 (18 self)
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For a large class of rings, including all those encountered in algebraic geometry, we establish the conjectured Moritalike equivalence between the full subcategory of complexes of finite Gorenstein flat dimension and that of complexes of finite Gorenstein injective dimension. This functorial description meets the expectations and delivers a series of new results, which allows us to establish a wellrounded theory for Gorenstein dimensions. For any pair of adjoint functors, C
Computation Of Superpotentials For Dbranes
, 2004
"... We present a general method for the computation of treelevel superpotentials for the worldvolume theory of Btype Dbranes. This includes quiver gauge theories in the case that the Dbrane is marginally stable. The technique involves analyzing the A∞structure inherent in the derived category of co ..."
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Cited by 39 (2 self)
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We present a general method for the computation of treelevel superpotentials for the worldvolume theory of Btype Dbranes. This includes quiver gauge theories in the case that the Dbrane is marginally stable. The technique involves analyzing the A∞structure inherent in the derived category of coherent sheaves. This effectively gives a practical method of computing correlation functions in holomorphic Chern–Simons theory. As an example, we give a more rigorous proof of previous results concerning
Dbranes on CalabiYau manifolds
, 2004
"... In this review we study BPS Dbranes on Calabi–Yau threefolds. Such Dbranes naturally divide into two sets called Abranes and Bbranes which are most easily understood from topological field theory. The main aim of this paper is to provide a selfcontained guide to the derived category approach to ..."
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Cited by 35 (7 self)
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In this review we study BPS Dbranes on Calabi–Yau threefolds. Such Dbranes naturally divide into two sets called Abranes and Bbranes which are most easily understood from topological field theory. The main aim of this paper is to provide a selfcontained guide to the derived category approach to Bbranes and the idea of Πstability. We argue that this mathematical machinery is hard to avoid for a proper understanding of Bbranes. Abranes and Bbranes are related in a very complicated and interesting way which ties in with the “homological mirror symmetry ” conjecture of Kontsevich. We motivate and exploit this form of mirror symmetry. The examples of the quintic 3fold, flops and orbifolds are discussed at some length. In the latter
Syzygies of abelian varieties
 J. Amer. Math. Soc
"... This is the first in a series of papers meant to introduce a notion of regularity on abelian varieties and more general irregular varieties. This notion, called Mukai regularity, is based on Mukai’s concept of Fourier transform, and in a very particular form (called Theta regularity) it parallels an ..."
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Cited by 34 (8 self)
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This is the first in a series of papers meant to introduce a notion of regularity on abelian varieties and more general irregular varieties. This notion, called Mukai regularity, is based on Mukai’s concept of Fourier transform, and in a very particular form (called Theta regularity) it parallels and strengthens the usual CastelnuovoMumford
Derived categories and zerobrane stability
 JHEP
"... We define a particular class of topological field theories associated to open strings and prove the resulting Dbranes and open strings form the bounded derived category of coherent sheaves. This derivation is a variant of some ideas proposed recently by Douglas. We then argue that any 0brane on an ..."
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Cited by 30 (1 self)
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We define a particular class of topological field theories associated to open strings and prove the resulting Dbranes and open strings form the bounded derived category of coherent sheaves. This derivation is a variant of some ideas proposed recently by Douglas. We then argue that any 0brane on any Calabi–Yau threefold must become unstable along some path in the Kähler moduli space. As a byproduct of this analysis we see how the derived category can be invariant under a birational transformation The idea that a Dbrane is simply some subspace of the target space where open strings are allowed to end is clearly too simple. Even at zero string coupling, we are faced with carefully analyzing the nonlinear σmodel of maps from the string worldsheet into the target space. It is wellknown that the nonlinear σmodel modifies the usual rules of classical geometry
Ktheory and derived equivalences
 Duke Math. J
"... Abstract. We show that if two rings have equivalent derived categories then they have the same algebraic Ktheory. Similar results are given for Gtheory, and for a large class of abelian categories. Contents ..."
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Cited by 29 (6 self)
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Abstract. We show that if two rings have equivalent derived categories then they have the same algebraic Ktheory. Similar results are given for Gtheory, and for a large class of abelian categories. Contents
The heterotic string, the tangent bundle, and derived categories
 Adv. Theor. Math. Phys
, 1998
"... We consider the compactification of the E8×E8 heterotic string on a K3 surface with “the spin connection embedded in the gauge group ” and the dual picture in the type IIA string (or Ftheory) on a Calabi–Yau threefold X. It turns out that the same X arises also as dual to a heterotic compactificati ..."
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Cited by 27 (3 self)
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We consider the compactification of the E8×E8 heterotic string on a K3 surface with “the spin connection embedded in the gauge group ” and the dual picture in the type IIA string (or Ftheory) on a Calabi–Yau threefold X. It turns out that the same X arises also as dual to a heterotic compactification on 24 pointlike instantons. X is necessarily singular, and we see that this singularity allows the RamondRamond moduli on X to split into distinct components, one containing the (dual of the heterotic) tangent bundle, while another component contains the pointlike instantons. As a practical application we derive the result that a heterotic string compactified on the tangent bundle of a K3 with ADE singularities acquires nonperturbatively enhanced gauge symmetry in just the same fashion as a type IIA string on a singular K3 surface. On a more philosophical level we discuss how it appears to be natural to say that the heterotic string is compactified using an object in the derived category of coherent sheaves. This is necessary to properly extend the notion of Tduality to the heterotic
Invariance and localization for cyclic homology of DG algebras
 J. PURE APPL. ALGEBRA
, 1998
"... We show that two flat differential graded algebras whose derived categories are equivalent by a derived functor have isomorphic cyclic homology. In particular, ‘ordinary ’ algebras over a field which are derived equivalent [48] share their cyclic homology, and iterated tilting [19] [3] preserves cyc ..."
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Cited by 26 (6 self)
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We show that two flat differential graded algebras whose derived categories are equivalent by a derived functor have isomorphic cyclic homology. In particular, ‘ordinary ’ algebras over a field which are derived equivalent [48] share their cyclic homology, and iterated tilting [19] [3] preserves cyclic homology. This completes results of Rickard’s [48] and Happel’s [18]. It also extends well known results on preservation of cyclic homology under Morita equivalence [10], [39], [25], [26], [41], [42]. We then show that under suitable flatness hypotheses, an exact sequence of derived categories of DG algebras yields a long exact sequence in cyclic homology. This may be viewed as an analogue of ThomasonTrobaugh’s [51] and Yao’s [58] localization theorems in Ktheory (cf. also [55]).