A note on derived McKay correspondence

Chen, Jiun-Cheng; Tseng, Hsian-Hua

doi:10.4310/mrl.2008.v15.n3.a4

Cited by 17 publications

(19 citation statements)

References 13 publications

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“…Let us recall that the authors of [24] define a natural bijective correspondence between conjugacy classes of junior elements of G \ {1} (here, G is a finite subgroup of SL n (C)) and exceptional prime divisors of a minimal model of C n /G (if f : Y → C n /G is a crepant resolution, then Y is a minimal model of C n /G). This result has been interpreted, and extended in several directions, using derived categories in [9] and [13].…”

Section: Mckay Correspondence For Complexified Bianchi Orbifoldsmentioning

confidence: 70%

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On Ruan’s cohomological crepant resolution conjecture for the complexified Bianchi orbifolds

Perroni

Rahm

2019

Algebr. Geom. Topol.

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We give formulae for the Chen-Ruan orbifold cohomology for the orbifolds given by a Bianchi group acting on complex hyperbolic 3-space. The Bianchi groups are the arithmetic groups PSL 2 (O), where O is the ring of integers in an imaginary quadratic number field. The underlying real orbifolds which help us in our study, given by the action of a Bianchi group on real hyperbolic 3-space (which is a model for its classifying space for proper actions), have applications in physics.We then prove that, for any such orbifold, its Chen-Ruan orbifold cohomology ring is isomorphic to the usual cohomology ring of any crepant resolution of its coarse moduli space. By vanishing of the quantum corrections, we show that this result fits in with Ruan's Cohomological Crepant Resolution Conjecture. 55N32, Orbifold cohomology arXiv:1109.5923v4 [math.KT] 24 Aug 2018Let Γ be a discrete group acting properly discontinuously, hence with finite stabilizers, by bi-holomorphisms on a complex manifold Y . For any element g ∈ Γ, denote by C Γ (g) the centralizer of g in Γ.Denote by Y g the subspace of Y consisting of the fixed points of g.Definition 1 ([11]) Let T ⊂ Γ be a set of representatives of the conjugacy classes of elements of finite order in Γ. Then the Chen-Ruan orbifold cohomology vector space of [Y/Γ] is:The grading on this vector space is reviewed in Equation (1) below.This definition is slightly different from, but equivalent to, the original one in [11]. We can verify this fact using arguments analogous to those used by Fantechi and Göttsche [16] in the case of a finite group Γ acting on Y . The additional argument needed when considering some element g in Γ of infinite order is the following. As the action of Γ on Y is properly discontinuous, g does not admit any fixed point in Y . Thus, H * Y g /C Γ (g); Q = H * ∅; Q = 0. For another proof, see [1].

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Section: Mckay Correspondence For Complexified Bianchi Orbifoldsmentioning

confidence: 70%

“…Therefore, it admits a crepant resolution (see e.g. [44], [13]). Then we prove the following result.…”

Section: Statement Of the Resultsmentioning

confidence: 99%

On Ruan’s cohomological crepant resolution conjecture for the complexified Bianchi orbifolds

Perroni

Rahm

2019

Algebr. Geom. Topol.

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“…In fact, in the case where X = [V/G], it amounts to requiring that the canonical bundle of V be G-equivariantly locally trivial. This condition seems to be missing in [CT08].…”

Section: The Equivalence Between Per(y/x) and Coh(x)mentioning

confidence: 96%

“…Concretely, we relate counting invariants of X and Y. The proof employs a derived equivalence between X and Y, which is a "global" version of the McKay correspondence of Bridgeland-King-Reid [BKR01,CT08]. We prove that the image of the heart Coh(X) under this equivalence is Bridgeland's category of perverse coherent sheaves Per(Y/X) [Bri02].…”

Section: Introductionmentioning

confidence: 99%

On the crepant resolution conjecture for Donaldson-Thomas invariants

Calabrese

2015

J. Algebraic Geom.

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We prove a comparison formula for curve-counting invariants in the setting of the McKay correspondence, related to the crepant resolution conjecture for Donaldson-Thomas invariants. The conjecture is concerned with comparing the invariants of a (hard Lefschetz) Calabi-Yau orbifold of dimension three with those of a specific crepant resolution of its coarse moduli space. We prove the conjecture for point classes and give a conditional proof for general curve classes. We also prove a variant of the formula for curve classes. Along the way we identify the image of the standard heart of the orbifold under the Bridgeland-King-Reid equivalence.

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“…This result is known as the categorical McKay correspondence. I refer to [7] for some discussions on the global version of the last part of this conjecture. We will prove a very special case of it in the context of cyclic groups acting by dilations on finite dimensional vector spaces.…”

Section: In Particular If Z Is a Crepant Resolution Of V /G Then Thmentioning

confidence: 99%

Categorical crepant resolutions for quotient singularities

Abuaf

2015

Math. Z.

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A note on derived McKay correspondence

Abstract: Abstract. We obtain a global version and a twisted version (in the sense of [BP05]) of the main theorem of [BKR01].

Cited by 17 publications

References 13 publications

On Ruan’s cohomological crepant resolution conjecture for the complexified Bianchi orbifolds

On Ruan’s cohomological crepant resolution conjecture for the complexified Bianchi orbifolds

On the crepant resolution conjecture for Donaldson-Thomas invariants

Categorical crepant resolutions for quotient singularities

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