Advertisement

Journal of Mathematical Sciences

, Volume 95, Issue 1, pp 2028–2048 | Cite as

K3 surfaces with interesting groups of automorphisms

  • V. V. Nikulin
Article

Abstract

By the fundamental result of I. I. Piatetsky-Shapiro and I. R. Shafarevich (1971), the automorphism group Aut(X) of aK3 surfaceX over ℂ and its action on the Picard latticeS X are prescribed by the Picard latticeS X . We use this result and our method (1980) to show the finiteness of the set of Picard latticesS X of rank ≥ 3 such that the automorphism group Aut(X) of theK3 surfaceX has a nontrivial invariant sublatticeS0 inS X where the group Aut(X) acts as a finite group. For hyperbolic and parabolic latticesS0, this has been proved by the author before (1980, 1995). Thus we extend these results to negative sublatticesS0.

We give several examples of Picard latticesS X with parabolic and negativeS0.

We also formulate the corresponding finiteness result for reflective hyperbolic lattices of hyperbolic type over purely real algebraic number fields. We give many examples of reflective hyperbolic lattices of the hyperbolic type.

These results are important for the theory of Lorentzian Kac-Moody algebras and mirror symmetry.

Keywords

Mirror Symmetry Automorphism Group Finite Group Number Field Algebraic Number 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    R. Borcherds, “Generalized Kac-Moody algebras,”J. Algebra,115, 501–512 (1988).Google Scholar
  2. 2.
    R. Borcherds, “The monster Lie algebra,”Adv. Math.,83, 30–47 (1990).Google Scholar
  3. 3.
    R. Borcherds, “The monstrous moonshine and monstrous Lie superalgebras,”Invent. Math.,109, 405–444 (1992).Google Scholar
  4. 4.
    R. Borcherds, “Sporadic groups and string theory,” In:Proc. European Congress of Mathematics (1992), pp. 411–421.Google Scholar
  5. 5.
    R. Borcherds, “Automorphic forms onO s+2,2 and infinite products,”Invent. Math.,120, 161–213 (1995).Google Scholar
  6. 6.
    R. Borcherds, “The moduli space of Enriques surfaces and the fake monster Lie superalgebra,”Topology,35, No. 3, 699–710 (1996).Google Scholar
  7. 7.
    V. A. Gritsenko, “Modular forms and moduli spaces of Abelian andK3 surfaces,”Algebra Analiz,6, No. 6, 65–102 (1994).Google Scholar
  8. 8.
    V. Gritsenko and K. Hulek, “Commutator coverings of Siegel threefolds,” Preprint RIMS Kyoto University RIMS-1128 (1997); alg-geom/9702007.Google Scholar
  9. 9.
    V. A. Gritsenko and V. V. Nikulin, “Siegel automorphic form correction of some Lorentzian Kac-Moody Lie algebras,”Amer. J. Math., (to appear); alg-geom/9504006.Google Scholar
  10. 10.
    V. A. Gritsenko and V. V. Nikulin, “Siegel automorphic form correction of a Lorentzian Kac-Moody algebra,”C. R. Acad. Sci. Paris. Sér. A-B,321, 1151–1156 (1995).Google Scholar
  11. 11.
    V. A. Gritsenko and V. V. Nikulin, “K3 surfaces, Lorentzian Kac-Moody algebras, and mirror symmetry,”Math. Res. Lett.,3, No. 2, 211–229 (1996); alg-geom/9510008.Google Scholar
  12. 12.
    V. A. Gritsenko and V. V. Nikulin, “The Igusa modular forms and ‘the simplest’ Lorentzian Kac-Moody algebras,”Mat. Sb.,187, No. 11 (1996); alg-geom/9603010.Google Scholar
  13. 13.
    V. A. Gritsenko and V. V. Nikulin, “Automorphic forms and Lorentzian Kac-Moody algebras, I,” Preprint RIMS Kyoto Univ. RIMS-1116 (1996); alg-geom/9610022.Google Scholar
  14. 14.
    V. A. Gritsenko and V. V. Nikulin, “Automorphic forms and Lorentzian Kac-Moody algebras, II,” Preprint RIMS Kyoto Univ. RIMS-1122 (1996); alg-geom/9611028.Google Scholar
  15. 15.
    V. A. Gritsenko and V. V. Nikulin, “The arithmetic mirror symmetry and Calabi-Yau manifolds,” Preprint RIMS Kyoto Univ. RIMS-1129 (1997); alg-geom/9612002.Google Scholar
  16. 16.
    V. Kac,Infinite Dimensional Lie Algebras, Cambridge Univ. Press (1990).Google Scholar
  17. 17.
    Vic. S. Kulikov, “Degenerations ofK3 surfaces and Enriques surfaces,”Izv. Akad. Nauk SSSR, Ser. Mat.,41, 1008–1042 (1977).Google Scholar
  18. 18.
    V. V. Nikulin, “Finite automorphism groups of KählerK3 surfaces,”Tr. Mosk. Mat. Obshch.,37, 73–137 (1979).Google Scholar
  19. 19.
    V. V. Nikulin, “Integral symmetric bilinear forms and some of their geometric applications,”Izv. Akad. Nauk SSSR, Ser. Mat.,43, 111–177 (1979).Google Scholar
  20. 20.
    V. V. Nikulin, “On factor groups of the automorphism groups of hyperbolic forms modulo subgroups generated by 2-reflections,”Dokl. Akad. Nauk SSSR,248, 1307–1309 (1979).Google Scholar
  21. 21.
    V. V. Nikulin, “On the quotient groups of the automorphism groups of hyperbolic forms by the subgroups generated by 2-reflections. Algebraic-geometric applications,” In:Sovremennye Problemy Matematiki. Itogi Nauki i Tekhn., Vol. 18, All-Union Institute for Scientific and Technical Information (VINITI), Moscow (1981), pp. 3–114.Google Scholar
  22. 22.
    V. V. Nikulin, “On arithmetic groups generated by reflections in Lobachevsky spaces,”Izv. Akad. Nauk SSSR, Ser. Mat.,44, 637–669 (1980).Google Scholar
  23. 23.
    V. V. Nikulin, “On the classification of arithmetic groups generated by reflections in Lobachevsky spaces,”Izv. Akad. Nauk SSSR, Ser. Mat.,45, No. 1, 113–142 (1981).Google Scholar
  24. 24.
    V. V. Nikulin, “Involutions of integral quadratic forms and their applications to real algebraic geometry,”Izv. Akad. Nauk SSSR, Ser. Mat.,47, No. 1 (1983).Google Scholar
  25. 25.
    V. V. Nikulin, “Surfaces of typeK3 with finite automorphism group and Picard group of rank three,”Tr. Mat. Inst. Sov. Akad. Nauk,165, 113–142 (1984).Google Scholar
  26. 26.
    V. V. Nikulin, “Discrete reflection groups in Lobachevsky spaces and algebraic surfaces,” In:Proc. Int. Congr. Math. Berkeley 1986, Vol. 1, pp. 654–669.Google Scholar
  27. 27.
    V. V. Nikulin, “A lecture on Kac-Moody Lie algebras of the arithmetic type,” Preprint Queen's University, Canada #1994–16 (1994); alg-geom/9412003.Google Scholar
  28. 28.
    V. V. Nikulin, “Reflection groups in Lobachevsky spaces and the denominator identity for Lorentzian Kac-Moody algebras,”Izv. Ross. Akad. Nauk, Ser. Mat.,60, No. 2, 73–106 (1996); alg-geom/9503003.Google Scholar
  29. 29.
    V. V. Nikulin, “The remark on discriminants ofK3 surfaces moduli as sets of zeros of automorphic forms,” Preprint (1995); alg-geom/9512018.Google Scholar
  30. 30.
    V. V. Nikulin, “Basis of the diagram method for generalized reflection groups in Lobachevsky spaces and algebraic surfaces with nef anticanonical class,”Int. J. Math.,7, No. 1, 71–108 (1996); alg-geom/9405011.Google Scholar
  31. 31.
    V. V. Nikulin, “Diagram method for 3-folds and its application to the Kähler cone and Picard number of Calabi-Yau 3-folds. I,” In:Higher Dimensional Complex Varieties: Proc. of Intern. Confer. held in Trento, Italy, June 15–24, 1994 (M. Andreatta and Th. Peternell, eds.), de Gruyter (1996), pp. 261–328; alg-geom/9401010.Google Scholar
  32. 32.
    V. V. Nikulin, “K3 surfaces with interesting groups of automorphisms,” Preprint (1997); alg-geom/ 9701011.Google Scholar
  33. 33.
    M. S. Raghunatan,Discrete Subgroups of Lie Groups, Springer (1968).Google Scholar
  34. 34.
    I. I. Pjatetckii-Sapiro and I. R. Šafarevich, “A Torelli theorem for algebraic surfaces of typeK3,”Izv. Akad. Nauk SSSR, Ser. Mat.,35, 530–572 (1971).Google Scholar
  35. 35.
    E. B. Vinberg, “On groups of unit elements of certain quadratic forms,”Mat Sb.,87, 18–36 (1972).Google Scholar
  36. 36.
    E. B. Vinberg, “The absence of crystallographic reflection groups in Lobachevsky spaces of large dimension,”Tr. Mosk. Mat. Obshch.,47, 68–102 (1984).Google Scholar
  37. 37.
    E. B. Vinberg, “Hyperbolic reflection groups,”Usp. Mat. Nauk,40, 29–66 (1985).Google Scholar

Copyright information

© Kluwer Academic/Plenum Publishers 1999

Authors and Affiliations

  • V. V. Nikulin

There are no affiliations available

Personalised recommendations