Flops and derived categories

By T. Bridgeland. 2000

387 citations recorded by the source corpus.

This paper contains some applications of Fourier-Mukai techniques to the birational geometry of threefolds. In particular, we prove that birational Calabi-Yau threefolds have equivalent derived categories. To do this we show how flops arise naturally as moduli spaces of perverse coherent sheaves.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6419b179768731c0916b315ffdd7438cf22e88b5

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Three-dimensional flops and noncommutative rings (2002, background): “!) subcategory of coh(Y ) consisting of objects E such that Rf∗E = 0 (this is a deviation of the notation used by Bridgeland).”
Derived categories of coherent sheaves (2006, background): “Full details of the proof can be found in [11].”
D-Equivalence and K-Equivalence (2002, background): “erived categories of coherent orbifold sheaves D(X) and D(Y) are equivalent as triangulated categories. Proof. The assertion is already proved in the case where K X is a Cartier divisor by Bridgeland [4] and Chen [8] (see also [23]). Indeed, it is proved that the structure sheaf O Z of the ﬁber prod”
Dirichlet branes, homological mirror symmetry, and stability (2002, background): “re is that this equivalence holds for any birationally equivalent Mand M′. Indeed, this is generally believed to be true (for arbitrary smooth projective varieties) and to some extent has been proven [30, 31]. One can also deﬁne and work with boundary conditions at various other “exactly solvable” p”
Deformations and Fourier-Mukai transforms (2005, background): “flops. Recently the relationship between derived categories and birational geometry has been developed. For example see [ 4 ], [7], [10].”
Curve counting theories via stable objects Ⅱ : DT/ncDT flop formula (2009, methodology): “effectively use an equivalence of bounded derived categories of coherent sheaves by Bridgeland [4], Φ: D(Coh(X)) ∼ −→ D(Coh(X)).”
Phases Of N=2 Theories In 1+1 Dimensions With Boundary (2008, background): “The equivalence for OCP2(−3) and C3/Z3 is a special case of the McKay correspondence by Bridgeland-King-Reid [4], as already mentioned in Section 8.4.”
D-branes, categories and N=1 supersymmetry (2000, background): “W e would like to thank T.Banks,N.Berkovits,F.Bogom olov,T.Bridgeland,I. Brunner,P.Deligne,D.-E.Diaconescu, B.Fiol,A.Kapustin,S.Katz,A.Klem m ,M .”
Projectivity and birational geometry of Bridgeland moduli spaces (2012, neutral): “…A t, − (cid:10) does not depend on t ; it is then straightforward to check that O X and all T are also σ t, − (cid:10) -stable for all t ∈ [ 12 , 2]: indeed, the imaginary part of Z t, − (cid:10) ( w ) for any Mukai vector w is of the form t · const, and the real part is of the form const…”
Combinatorial constructions of derived equivalences (2016, background): “rror symmetry and originates in the work of Bondal and Orlov [BO]. In the “global” case where R = k is the ground ﬁeld (always assumed to have characteristic 0), the conjecture is settled for 3-folds [Br], as well as for toric varieties [Kaw], and some K-equivalences arising from variation of geomet”
Dimer models and the special McKay correspondence (2009, methodology): “A G ] 1 M ay 2 00 9 Dimer models and the special McKay correspondence Akira Ishii and Kazushi Ueda”
Derived categories of small toric Calabi-Yau 3-folds and counting invariants (2008, background): “B04], [VdB]). In such a case, the Abelian category of modules of the noncommutative crepant resolution corresponds to the Abelian category of perverse coherent sheaves in the sense of Tom Bridgeland ([Bri02]). Recently, Balazs Szendroi proposed to study counting invariants of ideals of such noncommu”
Curve counting via stable pairs in the derived category (2007, background): “Amongst other technical 3Earlier examples of well-behaved components of the moduli space of complexes in the derived category can be found in [9, 24].”
Fomin-Zelevinsky mutation and tilting modules over Calabi-Yau algebras (2006, methodology): “By using the method of Fourier-Mukai transformations, Bridgeland [ Bri1 ] proved a conjecture of Bondal and Orlov [BO], which states that any birational 3-dimensional Calabi-Yau varieties are derived equivalent.”
Mirror symmetry and projective geometry of Reye congruences I (2011, background): “Picard numbers of X˜ i and X˜0 are two. Thus all of X˜ i(i = 0,1,2) are smooth Calabi-Yau 3-folds which are birational to each other. In particular, they have the equivalent derived categories due to [Br]. Note that X˜ 0,X˜1 and X˜2 are all complete intersections of ﬁve (1,1)-divisors, and thus in t”
Derived categories of toric varieties III (2006, background): “The derived McKay correspondence was already proved for finite subgroups of SL(2,C), SL(3,C) (Bridgeland-King-Reid [6]), and Sp(2n,C) (Bezrukavnikov-Kaledin [2]).”
Stability conditions and extremal contractions (2012, methodology): “When f(D) is a curve, Van den Bergh [14] shows that X is derived equivalent to a certain coherent OY -algebras A , and the perverse heart in [10] corresponds to the category of right A -modules.”