×

zbMATH — the first resource for mathematics

Orders of automorphisms of \(K3\) surfaces. (English) Zbl 1375.14125
Summary: We determine all orders of automorphisms of complex \(K3\) surfaces and of \(K3\) surfaces in characteristic \(p > 3\). In particular, 66 is the maximum finite order in each characteristic \(p \neq 2, 3\). As a consequence, we give a bound for the orders of finite groups acting on \(K3\) surfaces in characteristic \(p > 7\).

MSC:
14J28 \(K3\) surfaces and Enriques surfaces
14J50 Automorphisms of surfaces and higher-dimensional varieties
14J27 Elliptic surfaces, elliptic or Calabi-Yau fibrations
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Artin, M., Coverings of the rational double points in characteristic p, (Complex Analysis and Algebraic Geometry, (1977), Iwanami Shoten Tokyo), 11-22 · Zbl 0358.14008
[2] Cossec, F.; Dolgachev, I., Enriques surfaces I, (1989), Birkhäuser · Zbl 0569.14016
[3] Cox, D.; Zucker, S., Intersection numbers of sections of elliptic surfaces, Invent. Math., 53, 1-44, (1979) · Zbl 0444.14004
[4] Deligne, P., Relèvement des surfaces K3 en caractéristique nulle, (Giraud, J.; Illusie, L.; Raynaud, M., Surface Algébrique, Lecture Notes in Math., vol. 868, (1981), Springer), 58-79
[5] Deligne, P.; Lusztig, G., Representations of reductive groups over finite fields, Ann. of Math. (2), 103, 1, 103-161, (1976) · Zbl 0336.20029
[6] Dolgachev, I.; Keum, J., Wild p-cyclic actions on K3 surfaces, J. Algebraic Geom., 10, 101-131, (2001) · Zbl 1044.14015
[7] Dolgachev, I.; Keum, J., Finite groups of symplectic automorphisms of K3 surfaces in positive characteristic, Ann. of Math., 169, 269-313, (2009) · Zbl 1187.14047
[8] Dolgachev, I.; Keum, J., K3 surfaces with a symplectic automorphism of order 11, J. Eur. Math. Soc. (JEMS), 11, 799-818, (2009) · Zbl 1185.14035
[9] Hwang, D.; Keum, J., The maximum number of singular points on rational homology projective planes, J. Algebraic Geom., 20, 495-523, (2011) · Zbl 1231.14032
[10] Illusie, L., Report on crystalline cohomology, (Algebraic Geometry, Arcata 1974, Proc. Sympos. Pure Math., vol. 29, (1975), AMS), 459-478 · Zbl 0326.14005
[11] Ito, H., On automorphisms of supersingular K3 surfaces, Osaka J. Math., 34, 717-724, (1997) · Zbl 0909.14022
[12] Ito, H.; Liedtke, Ch., Elliptic K3 surfaces with \(p^n\)-torsion sections · Zbl 1258.14042
[13] Keum, J., Automorphisms of Jacobian Kummer surfaces, Compos. Math., 107, 269-288, (1997) · Zbl 0891.14013
[14] Keum, J., K3 surfaces with an order 60 automorphism and a characterization of supersingular K3 surfaces with Artin invariant 1, Math. Res. Lett., 21, 3, 509-520, (2014) · Zbl 1304.14046
[15] Keum, J., K3 surfaces with an automorphism of order 66, the maximum possible, J. Algebra, 426, 273-287, (2015) · Zbl 1314.14074
[16] Keum, J., Order 40 automorphisms of K3 surfaces, Development of Moduli Theory-Kyoto 2013, Adv. Stud. Pure Math., 69, 407-419, (2016) · Zbl 1434.14015
[17] Kondō, S., Automorphisms of algebraic K3 surfaces which act trivially on Picard groups, J. Math. Soc. Japan, 44, 75-98, (1992) · Zbl 0763.14021
[18] Kondō, S., The maximum order of finite groups of automorphisms of K3 surfaces, Amer. J. Math., 121, 6, 1245-1252, (1999) · Zbl 0978.14043
[19] Kondō, S., Maximal subgroups of the Mathieu group \(M_{23}\) and symplectic automorphisms of supersingular K3 surfaces, Int. Math. Res. Not. IMRN, 2006, (2006) · Zbl 1099.14033
[20] Machida, N.; Oguiso, K., On K3 surfaces admitting finite non-symplectic group actions, J. Math. Sci. Univ. Tokyo, 5, 273-297, (1998) · Zbl 0949.14022
[21] Mukai, S., Finite groups of automorphisms of K3 surfaces and the Mathieu group, Invent. Math., 94, 183-221, (1988) · Zbl 0705.14045
[22] Nikulin, V. V.; Nikulin, V. V., Finite groups of automorphisms of Kählerian surfaces of type K3, Uspekhi Mat. Nauk, Trans. Moscow Math. Soc., 38, 2, 71-135, (1980) · Zbl 0454.14017
[23] Nygaard, N., Higher deram-Witt complexes on supersingular K3 surfaces, Compos. Math., 42, 245-271, (1980/81) · Zbl 0482.14009
[24] Oguiso, K., A remark on global indices of \(\mathbb{Q}\)-Calabi-Yau 3-folds, Math. Proc. Cambridge Philos. Soc., 114, 427-429, (1993) · Zbl 0808.14031
[25] Ogus, A., Supersingular K3 crystals, (Journées de Géometrie Algébrique de Rennes, Astérisque, vol. 64, (1979)), 3-86 · Zbl 0435.14003
[26] Rudakov, A.; Shafarevich, I.; Shafarevich, I. R., Collected mathematical papers, Current Problems in Mathematics, vol. 18, 657-714, (1989), Springer-Verlag, Reprinted in
[27] Serre, J.-P., Le groupe de Cremona et ses sous-groupe finis, (Séminaire Bourbaki, Vol. 2008/2009, Exposés 997-1011, Astérisque, vol. 332, (2010)), 75-100, Exp. No. 1000 · Zbl 1257.14012
[28] Shioda, T., An explicit algorithm for computing the Picard number of certain algebraic surfaces, Amer. J. Math., 108, 415-432, (1986) · Zbl 0602.14033
[29] Shioda, T., On the Mordell-Weil lattices, Comment. Math. Univ. St. Pauli, 39, 211-240, (1990) · Zbl 0725.14017
[30] Silverman, J., The arithmetic of elliptic curves, Grad. Texts in Math., vol. 106, (1986), Springer-Verlag · Zbl 0585.14026
[31] Ueno, K., Classification theory of algebraic varieties and compact complex spaces, Lecture Notes in Math., vol. 439, (1975), Springer
[32] Ueno, K., A remark on automorphisms of Kummer surfaces in characteristic p, J. Math. Kyoto Univ., 26, 483-491, (1986) · Zbl 0612.14040
[33] G. Xiao, Non-symplectic involutions of a K3 surface, unpublished.
[34] Xiao, G., Galois covers between K3 surfaces, Ann. Inst. Fourier, 46, 1, 73-88, (1996) · Zbl 0845.14026
[35] Zhang, D.-Q., Automorphisms of K3 surfaces, AMS/IP Stud. Adv. Math., 39, 379-392, (2007) · Zbl 1126.14048
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.