Resolution of \(\mathbb{C}^3/H_{168}\)
Let \(M_0\) be a smooth complex threefold with trivial canonical bundle \(\omega_{M_0}\) acted on by a finite group \(G\) of automorphisms acting trivially on \(\omega_{M_0}\). A conjecture of Dixon-Harvey-Vafa-Witten says that there always exists a desingularization \(M_0/G\) with trivial canonical bundle, and predicts its Euler number. This conjecture has a local form from which it follows: If \(G\) is a finite subgroup of \(SL(3,{\mathbb{C}})\), there exists a crepant (i.e. with trivial canonical bundle) desingularization of \({\mathbb{C}}^3/G\), and its Euler number is the number of conjugacy classes of \(G\). The conjecture is now completely solved: There is a list of all finite subgroups of \(SL(3,{\mathbb{C}})\) due to Miller, Blichfeldt and Dickson, and a case-by-case analysis, of which this paper is part, shows that there always exists a crepant resolution, constructed by an explicit sequence of blow-ups. The formula for the Euler number follows from these explicit constructions; it also has an independent general proof by the generalization of the MacKay correspondence to dimension \(3\) by \textit{Y. Ito} and \textit{M. Reid}, in: Higher-dimensional complex varieties, Proc. Int. Conf., Trento 1994, 221-240 (1996; Zbl 0894.14024). The construction of a crepant resolution of \({\mathbb{C}}^3/H_{168}\) is pretty straightforward: One makes \(4\) successive blow-ups of singular curves until all the singular points disappear. The fact that the resolution obtained in this way is crepant follows from a remark of Reid (1979).
scientific article; zbMATH DE number 1057239