Flops and Equivalences of Derived Categories for Threefolds with Only Terminal Gorenstein Singularities

By Jiun C. Chen. 2002

87 citations recorded by the source corpus.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/35d863c2e8df7464fe66bd2fe5d2fc1caaa7f982

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In Geometry and Topology

1. Yukinobu Toda
2. T. Bridgeland
3. Michel van den Bergh
4. M. Wemyss
5. Osamu Iyama
6. G. Moore
7. Emanuele Macrì
8. P. Aspinwall

In Mathematical Physics

1. Yukinobu Toda
2. T. Bridgeland
3. Michel van den Bergh
4. G. Moore
5. Emanuele Macrì
6. Osamu Iyama
7. M. Wemyss
8. P. Aspinwall

Citations in the corpus

• Dirichlet Branes and Mirror Symmetry (2009, background): “For recent progress on special cases and versions for singular varieties, look in [93, 301].”
• Maximal modifications and Auslander–Reiten duality for non-isolated singularities (2010, background): “ings Aand Bare said to be derived equivalent if D(ModEndR(N)) ≈ D(ModEndR(M)) [R89]. Theorem 1.14 now immediately gives the following, which we view as the noncommutative analogue of a result of Chen [C02]. Corollary 1.15. Let Rbe a 3-dimensional Gorenstein equi-codimensional normal domain with a MM”
• Relative integral functors for singular fibrations and singular partners (2006, methodology): “The results in [13] can be generalized to Q-Gorenstein teminal threefolds (see [1, 18, 19]).”
• Flops and clusters in the homological minimal model programme (2014, background): “For Flop, the inverse of the Bridgeland–Chen flop functor, this is well–known; see e.g. [DW1, 7.16] or [T3, Appendix B].”
• Derived categories of coherent sheaves (2006, background): “In this way Chen [16] was able to extend Theorem 3.1 so as to allow Gorenstein terminal singularities.”
• Derived categories of sheaves: a skimming (2005, background): “ that Db coh (X −) ∼= Db coh (X +) and they were led to conjecture that this would be true in the case of an arbitrary flop. Their conjecture was proven by Bridgeland [6] and later generalized by Chen [10] and Kawamata [18]. There are some indications that ideas from derived category theory could pro”
• ℙ-objects and autoequivalences of derived categories (2005, unknown): “ines the Pn-twist PE and the spherical twist Tj∗E, i.e. one has the following commutative diagram Db(X) PE j∗ //Db(X) T j∗E Db(X) j∗ //D b(X). Proof. This is an application of Chen’s lemma (see [4] or [5, Ch. 11]). One simply has to show that there exists an object L on X ×C X with ℓ∗L ∼= Pj ∗E an”
• Handbook of Tilting Theory: Fourier-Mukai transforms (2004, methodology): “dgeland’s proof the second author produced a mildly different proof of the result [77]. Some of the ingredients in this new proof were adapted to the case of stacks by Kawamata. We should also mention [23] which uses a different method to extend Bridgeland’s result to singular spaces. Let us give some”
• Derived categories of toric varieties III (2006, background): “The derived equivalence in the case of flops (case (c) with KX +B = KY +C) was proved for general (nontoric) case under the additional assumptions as follows: if dimX = 3 andX is smooth by Bridgeland ([5]), if dimX = 3 andX has only Gorenstein terminal singularities by Chen ([8]) and Van den Bergh (”
• Singular Derived Categories of Q-factorial terminalizations and Maximal Modification Algebras (2011, unknown): “stein singularity, or a cAn singularity in §5). Then Conjecture 1.8 is true. Proof. The Q-factorial terminalizations are connected by a finite sequence of flops [K89], so they are derived equivalent by [C02]. By [V04a] there is a derived equivalence between Y and some algebra EndR(M) with EndR(M) ∈ CM”
• Moduli of complexes on a proper morphism (2005, background): “Recent work ([8], [9], [10]) has indicated the usefulness of constructing moduli spaces of certain types of objects in the derived category D(X) of a variety X .”