Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-13T05:23:00.412Z Has data issue: false hasContentIssue false
  • English
  • Français

Classification of extremal elliptic K3 surfaces and fundamental groups of open K3 surfaces

Published online by Cambridge University Press:  22 January 2016

Ichiro Shimada  and
De-Qi Zhang
Ichiro Shimada
Affiliation:
Department of Mathematics, Faculty of Science, Hokkaido University, Sapporo, 060-0810, Japan, shimada@math.sci.hokudai.ac.jp
De-Qi Zhang
Affiliation:
Department of Mathematics, National University of Singapore, Lower KentRidge Road, 119260, Singapore, matzdq@math.nus.edu.sg
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We present a complete list of extremal elliptic K3 surfaces (Theorem 1.1). As an application, we give a sufficient condition for the topological fundamental group of complement to an ADE-configuration of smooth rational curves on a K3 surface to be trivial (Proposition 4.1 and Theorem 4.3).

Keywords

14J28
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 161 , March 2001 , pp. 23 - 54
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2001

References

[1] Artal-Bartolo, E., Tokunaga, H. and Zhang, D. Q., Miranda-Persson’s problem on extremal elliptic K3 surfaces, preprint. http://xxx.lanl.gov/list/math.AG, 9809065.Google Scholar
[2] Bourbaki, N., Eléments de mathématique. Groupes et algebres de Lie. Chapitre IV-VI, Hermann, Paris, 1968.Google Scholar
[3] Conway, J. H. and Sloane, N. J. A., Sphere packings, lattices and groups, Second edition, Grundlehren der Mathematischen Wissenschaften, 290, Springer, New York, 1993.Google Scholar
[4] Fujiki, A., Finite automorphism groups of complex tori of dimension two, Publ. Res. Inst. Math. Sci., 24 (1988), no. 1, 1–97.Google Scholar
[5] Kondō, S., Automorphisms of algebraic K3 surfaces which act trivially on Picard groups, J. Math. Soc. Japan, 44 (1992), no. 1, 75–98.Google Scholar
[6] Kondō, S., Niemeier lattices, Mathieu groups, and finite groups of symplectic automorphisms of K3 surfaces, With an appendix by Shigeru Mukai, Duke Math. J., 92 (1998), no. 3, 593–603.Google Scholar
[7] Miranda, R. and Persson, U., Mordell-Weil groups of extremal elliptic K3 surfaces, Problems in the theory of surfaces and their classification (Cortona, 1988), Sympos. Math., XXXII, Academic Press, London (1991), pp. 167–192.Google Scholar
[8] Morrison, D. R., On K3 surfaces with large Picard number, Invent. Math., 75 (1984), no. 1, 105–121.Google Scholar
[9] Mukai, S., Finite groups of automorphisms of K3 surfaces and the Mathieu group, Invent. Math., 94 (1988), no. 1, 183–221.Google Scholar
[10] Nikulin, V. V., Finite automorphism groups of Kähler K3 surfaces, Trans. Moscow Math. Soc, Issue 2 (1980), 71–135.Google Scholar
[11] Nikulin, V. V., Integer symmetric bilinear forms and some of their applications, Math. USSR Izvestija, 14 (1980), no. 1, 103–167.Google Scholar
[12] Nishiyama, K., The Jacobian fibrations on some K3 surfaces and their Mordell-Weil groups, Japan. J. Math. (N.S.), 22 (1996), no. 2, 293–347.Google Scholar
[13] Nori, M. V., Zariski’s conjecture and related problems, Ann. Sci. Ecole Norm. Sup. (4), 16 (1983), no. 2, 305–344.Google Scholar
[14] Piateskii-Shapiro, I. and Shafarevich, I. R., A Torelli theorem for algebraic surfaces of type K3, Math. USSR Izv., 35 (1971), 530–572.Google Scholar
[15] Serre, J.-P., A course in arithmetic, Graduate Texts in Mathematics, 7, Springer, New York, 1973.Google Scholar
[16] Shioda, T. and Inose, H., On singular K3 surfaces. Complex analysis and algebraic geometry, Iwanami Shoten, Tokyo, 1977, pp. 119–136.Google Scholar
[17] Todorov, A. N., Applications of the Kähler-Einstein-Calabi-Yau metric to moduli of K3 surfaces, Invent. Math., 61 (1980), no. 3, 251–265.Google Scholar
[18] Xiao, G., Galois covers between K3 surfaces, Ann. Inst. Fourier (Grenoble), 46 (1996), no. 1, 73–88.CrossRefGoogle Scholar
[19] Ye, Q., On extremal elliptic K3 surfaces, preprint. http://xxx.lanl.gov/abs/math.AG, 9901081.Google Scholar