Minimal resolutions of Gorenstein orbifolds in dimension three

Roan, Shi-shyr

doi:10.1016/0040-9383(95)00018-6

Cited by 84 publications

(82 citation statements)

References 11 publications

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“…In the classical McKay case, the minimal resolution is crepant, but in higher dimensions crepant resolutions do not necessarily exist and, even when they do, they are not usually unique. However, it is now known that crepant resolutions of Gorenstein quotient singularities do exist in dimension n = 3, through a case by case analysis of the local linear actions by Ito, Markushevich and Roan (see Roan [21] and references given there). In dimension ≥ 4, even such quotient singularities only have crepant resolutions in rather special cases.…”

Section: Introductionmentioning

confidence: 99%

The McKay correspondence as an equivalence of derived categories

Bridgeland

King

Reid

2001

J. Amer. Math. Soc.

462

584

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Let G G be a finite group of automorphisms of a nonsingular three-dimensional complex variety M M , whose canonical bundle ω M \omega _M is locally trivial as a G G -sheaf. We prove that the Hilbert scheme Y = G Y = G - Hilb ⁡ M \operatorname {Hilb}M parametrising G G -clusters in M M is a crepant resolution of X = M / G X=M/G and that there is a derived equivalence (Fourier–Mukai transform) between coherent sheaves on Y Y and coherent 𝐺-sheaves on M M . This identifies the K theory of Y Y with the equivariant K theory of M M , and thus generalises the classical McKay correspondence. Some higher-dimensional extensions are possible.

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Section: Introductionmentioning

confidence: 99%

The McKay correspondence as an equivalence of derived categories

Bridgeland

King

Reid

2001

J. Amer. Math. Soc.

462

584

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“…Note that this theorem, besides establishing the expected derived equivalence, also produces a specific crepant resolution of W . For n = 3 this was done earlier by a case by case analysis (see [64] and the references therein).…”

Section: Non-commutative Rings In Algebraic Geometrymentioning

confidence: 99%

Fourier-Mukai transforms

Hille¹,

Bergh²

2007

Handbook of Tilting Theory

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“…Our main tool will be the string theoretic formula [14]: Here M g,h denotes the common fixed point set of g, h.…”

Section: Computation Of the Euler Numbermentioning

confidence: 99%

On Siegel three-folds with a projective Calabi–Yau model

Freitag

Manni

2011

Communications in Number Theory and Physics

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In two recent papers we described some Siegel modular threefolds which admit a weak Calabi-Yau model. Not all of them admit a projective model. In fact, Bert van Geemen, in a private communication, pointed out a significative example which cannot admit a projective model. A weak Calabi-Yau threefold is projective if, and only if, it admits a Kaehler metric. The purpose of this paper is to exhibit criteria for the projectivity, to treat several examples, and to compute their Hodge numbers. Some of these Calabi-Yau manifolds seem to be new. We obtain a rigid Calabi-Yau manifold with Euler number 4. IntroductionIn the papers [9, 10] we described some Siegel modular three-folds which admit a weak Calabi-Yau model. 1 Not all of them admit a projective model. In fact, Bert van Geemen, in a private communication, pointed out a significative example which cannot admit a projective model. His comment was a starting motivation for this paper. We mention that a weak CalabiYau three-fold is projective if, and only if, it admits a Kaehler metric. The purpose of this paper is to exhibit criteria for the projectivity, to treat several examples and to compute their Hodge numbers. Some of these CalabiYau manifolds seem to be new. For example, we obtain a rigid Calabi-Yau manifold with Euler number 4.

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Minimal resolutions of Gorenstein orbifolds in dimension three

Cited by 84 publications

References 11 publications

The McKay correspondence as an equivalence of derived categories

The McKay correspondence as an equivalence of derived categories

Fourier-Mukai transforms

On Siegel three-folds with a projective Calabi–Yau model

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