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Minimal resolutions of Gorenstein orbifolds in dimension three. (English) Zbl 0872.14034
A Gorenstein orbifold (G.o.) is a variety \(X\) with at most Gorenstein quotient singularities, i.e. obtained as quotients of \(\mathbb{C}^n\) by finite subgroups of \(SL_n (\mathbb{C})\). Let \(X= M/G\) be a G.o. Then, as introduced in the string theory, the orbifold Euler number \(\chi (M,G)\) is an invariant of the pair \((M,G)\) which should define the corrected Euler characteristic of \(X=M/G\). The problem to justify this definition is reduced to the question about the existence of a minimal resolution \(\widetilde X\) of the quotient \(X\) such that the Euler characteristic \(\chi (\widetilde X) =\) the orbifold number \(\chi (M,G)\). For \(n=2\) the problem is solved by F. Hirzebruch and T. Höfer [Math. Ann. 286, No. 1-3, 255-260 (1990; Zbl 0679.14006)], and for \(n=3\) in many special cases by D. Markushevich, Y. Ito, the author, and others [see Y. Ito, J. Math. Sci., Tokyo 2, No. 2, 419-440 (1995; Zbl 0869.14002); D. G. Markushevich, M. A. Olshanetskij and A. M. Perelomov, Commun. Math. Phys. 111, 247-274 (1987; Zbl 0628.53065); S.-S. Roan, J. Differ. Geom. 30, No. 2, 523-537 (1989; Zbl 0661.14031), Int. J. Math. 5, No. 4, 523-536 (1994; Zbl 0856.14005); S. S. Roan and S. Yau, Acta Math. Sin., New Ser. 3, 256-288 (1987; Zbl 0649.14024)].
In the paper under review the author solves completely the problem, by constructing minimal resolutions also for the quotients by these finite subgroups of \(SL_3 (\mathbb{C})\) for which the expected identity has not been verified earlier.
Reviewer: A.Iliev (Sofia)

14J30 \(3\)-folds
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
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