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Six-dimensional exceptional quotient singularities. (English) Zbl 1281.14004

One calls a strictly log canonical (lc) pair \((X,B)\) exceptional if there exists only one exceptional divisor of discrepancy \(-1\) w.r.t. \((X,B)\). Similarly, replacing \((X,B)\) by \((X,cB)\), where \(c\) is the log canonical threshold, one introduces the same exceptionality notion for Kawamata log terminal (klt) pairs \((X,B)\). Finally, taking a singularity germ \((V\ni O)\) in place of \(X\), one calls the singularity exceptional if the pair \((X,B)\) is so for any effective \(\mathbb{Q}\)-Cartier divisor \(B\) on \(V\) (near \(O\)).
All these notions were introduced in [V. V. Shokurov, Russ. Acad. Sci., Izv., Math. 40, No.1, 95–202 (1993); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 56, No.1, 105–201, Appendix 201–203 (1992; Zbl 0785.14023)] (see also V. V. Shokurov [J. Math. Sci., New York 102, No. 2, 3876–3932 (2000; Zbl 1177.14078)], Yu. G. Prokhorov [Lectures on complements on log surfaces. MSJ Memoirs 10. Tokyo: Mathematical Society of Japan. (2001; Zbl 1037.14003)] for foundations of the theory) along the proof of existence of \(3\)-dimensional flip. In particular, it was shown that the surface canonical singularity is exceptional iff it is of type \(E_6,E_7\) or \(E_8\). In higher dimensions an important characterization of exceptional quotient singularities has been found in [D. Markushevich and Yu. G. Prokhorov, Am. J. Math. 121, No. 6, 1179–1189 (1999; Zbl 0958.14003)]. Namely, let \(G \subset \mathrm{GL}_{n+1}(\mathbb{C})\) be a finite group without quasi-reflections and with induced action on \(\mathbb{P}^n\) (we will be identifying \(G\) with its image in \(\mathrm{PGL}_{n+1}(\mathbb{C})\)). Then the quotient singularity \(\mathbb{C}^{n+1}/ G\) is exceptional iff for any \(G\)-invariant \(\mathbb{Q}\)-divisor \(D\) on \(\mathbb{P}^n\), satisfying \(D\sim_{\mathbb{Q}}-K_{\mathbb{P}^n}\), the pair \((\mathbb{P}^n,D)\) is klt.
The authors of the paper under review restate the latter criterion in terms of the \(G\)-invariant log canonical threshold \(\mathrm{lct}(\mathbb{P}^n,G)\) (see G. Tian and S.-T. Yau [Commun. Math. Phys. 112, No. 1, 175–203 (1987; Zbl 0631.53052)]) by saying that \(\mathbb{C}^{n+1}/ G\) is exceptional if \(\mathrm{lct}(\mathbb{P}^n,G)>1\) (see Corollary 1.1 in the text for the precise statement). This allows one extend the study of exceptional quotient singularities (elaborated up to dimension \(5\)) to higher dimensions. The main result (Theorem 1.4) of the paper claims that for \(n=5\) the quotient singularity \(\mathbb{C}^6/ G\) is exceptional iff \(\mathrm{lct}(\mathbb{P}^5,G)\geq 7/6\) and \(G\) is either a Hall-Janko group or \(6.A_7\) (note that \(G\) must be primitive due to Markushevich and Prokhorov [Zbl 0958.14003]). The authors also prove (Theorem 1.5) that the are no \(7\)-dimensional quotient exceptional singularities.
The proofs of both of the results proceed, essentially, by reducing to a small number of primitive groups \(G\) (see Theorems 3.2, 5.1), and after that (assuming that \(\mathrm{lct}(\mathbb{P}^n,G)\) is less than \((n+2)/(n+1)\)) one finds a \(G\)-invariant divisor \(B\) as above such that the pair \((X\lambda B)\) is strictly lc for some \(\lambda < (n+2)/(n+1)\). Then one applies the crucial Lemma 2.3 to estimate the degree and dimension of the linear span of the minimal (\(G\)-invariant and irreducible) lc center of \((X,B)\) to get contradiction.

MSC:

14B05 Singularities in algebraic geometry
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