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K3 surfaces with interesting groups of automorphisms

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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 sublatticeS 0 inS X where the group Aut(X) acts as a finite group. For hyperbolic and parabolic latticesS 0, this has been proved by the author before (1980, 1995). Thus we extend these results to negative sublatticesS 0.

We give several examples of Picard latticesS X with parabolic and negativeS 0.

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.

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References

  1. R. Borcherds, “Generalized Kac-Moody algebras,”J. Algebra,115, 501–512 (1988).

    Google Scholar 

  2. R. Borcherds, “The monster Lie algebra,”Adv. Math.,83, 30–47 (1990).

    Google Scholar 

  3. R. Borcherds, “The monstrous moonshine and monstrous Lie superalgebras,”Invent. Math.,109, 405–444 (1992).

    Google Scholar 

  4. R. Borcherds, “Sporadic groups and string theory,” In:Proc. European Congress of Mathematics (1992), pp. 411–421.

  5. R. Borcherds, “Automorphic forms onO s+2,2 and infinite products,”Invent. Math.,120, 161–213 (1995).

    Google Scholar 

  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. V. A. Gritsenko, “Modular forms and moduli spaces of Abelian andK3 surfaces,”Algebra Analiz,6, No. 6, 65–102 (1994).

    Google Scholar 

  8. V. Gritsenko and K. Hulek, “Commutator coverings of Siegel threefolds,” Preprint RIMS Kyoto University RIMS-1128 (1997); alg-geom/9702007.

  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.

  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. 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. 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. 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.

  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.

  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.

  16. V. Kac,Infinite Dimensional Lie Algebras, Cambridge Univ. Press (1990).

  17. Vic. S. Kulikov, “Degenerations ofK3 surfaces and Enriques surfaces,”Izv. Akad. Nauk SSSR, Ser. Mat.,41, 1008–1042 (1977).

    Google Scholar 

  18. V. V. Nikulin, “Finite automorphism groups of KählerK3 surfaces,”Tr. Mosk. Mat. Obshch.,37, 73–137 (1979).

    Google Scholar 

  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. 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. 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. 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. 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. 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).

  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. 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. 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.

  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. V. V. Nikulin, “The remark on discriminants ofK3 surfaces moduli as sets of zeros of automorphic forms,” Preprint (1995); alg-geom/9512018.

  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. 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.

  32. V. V. Nikulin, “K3 surfaces with interesting groups of automorphisms,” Preprint (1997); alg-geom/ 9701011.

  33. M. S. Raghunatan,Discrete Subgroups of Lie Groups, Springer (1968).

  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. E. B. Vinberg, “On groups of unit elements of certain quadratic forms,”Mat Sb.,87, 18–36 (1972).

    Google Scholar 

  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. E. B. Vinberg, “Hyperbolic reflection groups,”Usp. Mat. Nauk,40, 29–66 (1985).

    Google Scholar 

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Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 45, Algebraic Geometry-8, 1997.

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Nikulin, V.V. K3 surfaces with interesting groups of automorphisms. J Math Sci 95, 2028–2048 (1999). https://doi.org/10.1007/BF02169159

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