Nagoya Mathematical Journal

Classification of extremal elliptic {$K3$} surfaces and fundamental groups of open {$K3$} surfaces

Ichiro Shimada and De-Qi Zhang

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Abstract

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).

Article information

Source
Nagoya Math. J., Volume 161 (2001), 23-54.

Dates
First available in Project Euclid: 27 April 2005

Permanent link to this document
https://projecteuclid.org/euclid.nmj/1114631551

Mathematical Reviews number (MathSciNet)
MR1820211

Zentralblatt MATH identifier
1064.14503

Subjects
Primary: 14J27: Elliptic surfaces
Secondary: 14J28: $K3$ surfaces and Enriques surfaces

Citation

Shimada, Ichiro; Zhang, De-Qi. Classification of extremal elliptic {$K3$} surfaces and fundamental groups of open {$K3$} surfaces. Nagoya Math. J. 161 (2001), 23--54. https://projecteuclid.org/euclid.nmj/1114631551


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References

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