On Vorontsov’s Theorem on K3 surfaces with non-symplectic group actions

Oguiso, Keiji; Zhang, De-Qi

doi:10.1090/S0002-9939-00-05427-7

On Vorontsov’s Theorem on K3 surfaces with non-symplectic group actions
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by Keiji Oguiso and De-Qi Zhang PDF

Proc. Amer. Math. Soc. 128 (2000), 1571-1580 Request permission

Abstract:

We shall give a proof for Vorontsov’s Theorem and apply this to classify log Enriques surfaces with large prime canonical index.

References

W. Barth, C. Peters, and A. Van de Ven, Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 4, Springer-Verlag, Berlin, 1984. MR 749574, DOI 10.1007/978-3-642-96754-2
Raimund Blache, The structure of l.c. surfaces of Kodaira dimension zero. I, J. Algebraic Geom. 4 (1995), no. 1, 137–179. MR 1299007
Yujiro Kawamata, Elementary contractions of algebraic $3$-folds, Ann. of Math. (2) 119 (1984), no. 1, 95–110. MR 736561, DOI 10.2307/2006964
K. Kodaira, On compact analytic surfaces. II, III, Ann. of Math. (2) 77 (1963), 563–626; ibid. 78 (1963), 1–40. MR 0184257, DOI 10.2307/1970500
Shigeyuki Kond\B{o}, Automorphisms of algebraic $K3$ surfaces which act trivially on Picard groups, J. Math. Soc. Japan 44 (1992), no. 1, 75–98. MR 1139659, DOI 10.2969/jmsj/04410075
J. Myron Masley and Hugh L. Montgomery, Cyclotomic fields with unique factorization, J. Reine Angew. Math. 286(287) (1976), 248–256. MR 429824
N. Machida and K. Oguiso, On $K3$ surfaces admitting finite non-symplectic group actions, J. Math. Sci. Univ. Tokyo 5 (1998), no. 2, 273–297.
André Néron, Modèles minimaux des variétés abéliennes sur les corps locaux et globaux, Inst. Hautes Études Sci. Publ. Math. 21 (1964), 128 (French). MR 179172, DOI 10.1007/bf02684271
V. V. Nikulin, Finite groups of automorphisms of Kählerian $K3$ surfaces, Trudy Moskov. Mat. Obshch. 38 (1979), 75–137 (Russian). MR 544937
V. V. Nikulin, Factor groups of the automorphism group of hyperbolic forms by the subgroups generated by $2-$reflections, J. Soviet Math. 22 (1983), 1401-1475.
V. V. Nikulin, Integer symmetric bilinear forms and some of their geometric applications, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 1, 111–177, 238 (Russian). MR 525944
Keiji Oguiso, On algebraic fiber space structures on a Calabi-Yau $3$-fold, Internat. J. Math. 4 (1993), no. 3, 439–465. With an appendix by Noboru Nakayama. MR 1228584, DOI 10.1142/S0129167X93000248
Keiji Oguiso and De-Qi Zhang, On the most algebraic $K3$ surfaces and the most extremal log Enriques surfaces, Amer. J. Math. 118 (1996), no. 6, 1277–1297. MR 1420924
K. Oguiso and D.-Q. Zhang, On extremal log Enriques surfaces, II,, Tohoku Math. J. 50 (1998), 419 - 436.
K. Oguiso and D.-Q. Zhang, On the complete classification of extremal log Enriques surfaces, Math. Z. to appear.
I. I. Pjateckiĭ-Šapiro and I. R. Šafarevič, Torelli’s theorem for algebraic surfaces of type $\textrm {K}3$, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 530–572 (Russian). MR 0284440
A. N. Rudakov and I. R. Shafarevich, Surfaces of type $K3$ over fields of finite characteristic, Current problems in mathematics, Vol. 18, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow, 1981, pp. 115–207 (Russian). MR 633161
Tetsuji Shioda, On the Mordell-Weil lattices, Comment. Math. Univ. St. Paul. 39 (1990), no. 2, 211–240. MR 1081832
Kenji Ueno, A remark on automorphisms of Enriques surfaces, J. Fac. Sci. Univ. Tokyo Sect. I A Math. 23 (1976), no. 1, 149–165. MR 0404268
S. P. Vorontsov, Automorphisms of even lattices arising in connection with automorphisms of algebraic $K3$-surfaces, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 2 (1983), 19–21 (Russian, with English summary). MR 697215
De-Qi Zhang, Logarithmic Enriques surfaces, J. Math. Kyoto Univ. 31 (1991), no. 2, 419–466. MR 1121173, DOI 10.1215/kjm/1250519795
De-Qi Zhang, Logarithmic Enriques surfaces. II, J. Math. Kyoto Univ. 33 (1993), no. 2, 357–397. MR 1231749, DOI 10.1215/kjm/1250519265