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Crepant resolutions of \(\mathbb{C}^n/A_1(n)\) and flops of \(n\)-folds for \(n=4,5\). (English) Zbl 1061.14052
Yui, Noriko (ed.) et al., Calabi-Yau varieties and mirror symmetry. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3355-3/hbk). Fields Inst. Commun. 38, 27-41 (2003).
The paper under review is intended to describe the explicit constructions of crepant resolutions of higher-dimensional orbifolds with Gorenstein quotient singularities, that is, the algebraic (or analytic) varieties such that the analytic type of each singular point is described as \({\mathbb C}^n / G\), where \(G\) is a (non-trivial) finite subgroup of the special linear group \(\text{SL}_n ({\mathbb C})\). For the 2-dimensional case, finite subgroups \(G\) of \(\text{SL}_2 ({\mathbb C})\) were classically classified into ADE series. It is well known that \({\mathbb C}^2 / G\) are always hypersurface singularities, and the minimal resolutions of them give the desired crepant resolutions. For the 3-dimensional case, the required crepant resolutions were found for all finite subgroups \(G \subset \text{SL}_3 ({\mathbb C})\) by virtue of Y. Ito [Proc. Japan Acad., Ser. A 70, 131–136 (1994; Zbl 0831.14006)], D. Markushevich [Math. Ann. 308, 279–289 (1997; Zbl 0899.14016)], S. S. Roan [Int. J. Math. 5, 523–536 (1994; Zbl 0856.14005)] and S. S. Roan [Topology 35, 489–508 (1996; Zbl 0872.14034)], which are depending on the classical result on the classification of finite subgroups of \(\text{SL}_3 ({\mathbb C})\) due to Miller-Blichfeldt-Dickson [Y. A. Miller, H. F. Blichfeldt and L. E. Dickson, “Theory and application of finite groups”. New York, Wiley (1915; JFM 45.0255.12)]. But, for such a higher-dimensional case, the non-uniqueness of crepant resolutions happens due to the existence of certain kinds of codimension 2 birational operations known as flops. In order to understand higher-dimensional crepant resolutions qualitatively, the development has resulted in the theory of \(G\)-Hilbert schemes \(\text{Hilb}^G ({\mathbb C}^n)\) associated to the quotient singularities \({\mathbb C}^n / G\) as in Y. Ito and I. Nakamura [Proc. Japan Acad.,Ser. A 72, 135–138 (1996; Zbl 0881.14002)]: the crepant resolutions of \({\mathbb C}^n / G\) would be related to \(G\)-Hilbert schemes \(\text{Hilb}^G ({\mathbb C}^n)\) of \(G\)-stable 0-dimensional subschemes of \({\mathbb C}^n\) of length equal to the order \(| G | \) of \(G\). As a result, the structure of \(\text{Hilb}^G ({\mathbb C}^n)\) now yields that \(\text{Hilb}^G ({\mathbb C}^3)\) is a toric crepant resolution of \({\mathbb C}^3 / G\) for every finite abelian subgroup \(G \subset \text{SL}_3 ({\mathbb C})\) by virtue of T. Bridgeland, A. King and M. Reid [J. Am. Math. Soc. 14, 535–554 (2001; Zbl 0966.14028)], Y. Ito and H. Nakajima [Topology 39, 1155–1191 (2000; Zbl 0995.14001)] and I. Nakamura [J. Alg. Geom. 10, 757–779 (2001; Zbl 1104.14003)]. Whereas, for the cases \(n \geq 4\), there are very few results concerning the crepant resolutions of \({\mathbb C}^n / G\) and the structure of \(\text{Hilb}^G ({\mathbb C})\) for finite subgroups \(G \subset \text{SL}_n ({\mathbb C})\). The authors restricted themselves to the case where \(G\) is the subgroup \(A_r (n)\) of \(\text{SL}_n({\mathbb C})\) consisting of all diagonal matrices of order \(r+1\). In an earlier paper [Int.. J. Math. Math. Sci. 26, 649–669 (2001; Zbl 1065.14018)], the authors studied the case of \(n=4\) and \(G=A_r (4)\), and obtained crepant resolutions of \({\mathbb C}^4 / A_r (4)\) through the detailed investigations of the structure of \(\text{Hilb}^{A_r (4)} ({\mathbb C}^4)\). In the present paper, the authors study the case of \(n=4, 5\) and \(r=1\). More precisely, for \(n=4\) and \(r=1\), they describe the toric variety structure of \(\text{Hilb}^{A_1 (4)} ({\mathbb C}^4)\) which is NOT crepant. Then, blowing-down the divisor \({\mathbb P}^1 \times {\mathbb P}^1 \times {\mathbb P}^1\) on \(\text{Hilb}^{A_1 (4)} ({\mathbb C}^4)\) onto \({\mathbb P}^1 \times {\mathbb P}^1\) in different ways, they obtain three different toric crepant resolutions of \({\mathbb C}^4 / A_1(4)\). These three crepant resolutions are related to each other by 4-fold flops. For the case \(n=5\) and \(r=1\) also, as in the 4-dimensional case just above, they describe the toric variety structure of \(\text{Hilb}^{A_1(5)} ({\mathbb C}^5)\) which is NOT crepant, and obtain twelve mutually different crepant resolutions of \({\mathbb C}^5 / A_1(5)\), all of which are dominated by \(\text{Hilb}^{A_1(5)}({\mathbb C}^5)\). These twelve crepant resolutions are related to each other by 5-fold flops.
For the entire collection see [Zbl 1022.00014].

14M17 Homogeneous spaces and generalizations
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
14C05 Parametrization (Chow and Hilbert schemes)
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