Advertisement

Mathematische Annalen

, 342:903 | Cite as

Non-symplectic automorphisms of order 3 on K3 surfaces

  • Michela Artebani
  • Alessandra SartiEmail author
Article

Abstract

In this paper we study K3 surfaces with a non-symplectic automorphism of order 3. In particular, we classify the topological structure of the fixed locus of such automorphisms and we show that it determines the action on cohomology. This allows us to describe the structure of the moduli space and to show that it has three irreducible components.

Mathematics Subject Classification (2000)

Primary 14J28; Secondary 14J50 14J10 

References

  1. 1.
    Alexeev, V., Nikulin, V.V.: Del Pezzo and K3 surfaces. MSJ Memoirs, Vol. 15 (2006)Google Scholar
  2. 2.
    Atiyah M.F., Singer I.M.: The Index of Elliptic Operators: III. Ann. Math. 2nd Ser. 87(3), 546–604 (1968)MathSciNetGoogle Scholar
  3. 3.
    Buell D.A.: Binary Quadratic Forms: Classical Theory and Modern Computations. Springer, Heidelberg (1989)zbMATHGoogle Scholar
  4. 4.
    Bredon G.E.: Introduction to Compact Transformation Groups. Academic Press, New York (1972)zbMATHGoogle Scholar
  5. 5.
    Conway J.H., Sloane N.J.A.: Sphere Packing, Lattices and Groups. Springer, New York (1998)Google Scholar
  6. 6.
    Conway J.H., Sloane N.J.A.: The Coxeter-Todd lattice, the Mitchell group, and related sphere packings. Math. Proc. Camb. Phil. Soc. 93, 421–440 (1983)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Dillies, J.: Automorphisms and Calabi-Yau Threefolds. PhD thesis, University of Pennsylvania, Pennsylvania (2006)Google Scholar
  8. 8.
    Dolgachev, I.V., Kondō, S.: Moduli spaces of K3 surfaces and complex ball quotients. Arithmetic and Geometry Around Hypergeometric Functions Lecture Notes of a CIMPA Summer School held at Galatasaray University, Istanbul, 2005, Series: Progress in Mathematics, vol. 260, pp. 43–100. Birkhauser Verlag, Basel (2007)Google Scholar
  9. 9.
    Deligne P., Mostow G.D.: Monodromy of hypergeometric functions and non-lattice integral monodromy. Publ. Math. IHES 63, 5–89 (1986)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Dolgachev I., Kondō S., van Geemen B.: A complex ball uniformization of the moduli space of cubic surfaces via periods of K3 surfaces. J. Reine Angew. Math. 588, 99–148 (2005)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Feit W.: Some lattices over Q(√  − 3). J. Algebra 52(1), 248–263 (1978)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Garbagnati A., Sarti A.: Symplectic automorphisms of prime order on K3 surfaces. J. Algebra 318, 323–350 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Kharlamov V.M.: The topological type of nonsingular surfaces in \({\mathbb {P}^3\mathbb {R}}\) of degree four. Funct. Anal. Appl. 10(4), 295–304 (1976)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Kondō S.: The moduli space of curves of genus 4 and Deligne–Mostow’s complex reflection groups. Algebraic Geometry. Azumino. Adv. Stud. Pure Math. 36, 383–400 (2000)Google Scholar
  15. 15.
    Kondō S.: Automorphisms of algebraic K3 surfaces which act trivially on Picard groups. J. Math. Soc. Japan 44(1), 75–98 (1992)zbMATHMathSciNetGoogle Scholar
  16. 16.
    Kulikov V.S.: Degenerations of K3 surfaces and Enriques surfaces. Izv. Akad. Nauk SSSR Ser. Mat. 41, 1008–1042 (1977)zbMATHMathSciNetGoogle Scholar
  17. 17.
    Miranda, R.: The basic theory of elliptic surfaces. ETS Editrice Pisa (1989)Google Scholar
  18. 18.
    Machida N., Oguiso K.: On K3 surfaces admitting finite non-symplectic group actions. J. Math. Sci. Univ. Tokyo 5(2), 273–297 (1998)zbMATHMathSciNetGoogle Scholar
  19. 19.
    Namikawa Y.: Periods of Enriques surfaces. Math. Ann. 270, 201–222 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Nikulin V.V.: Finite groups of automorphisms of Kählerian surfaces of type K3. Moscow Math. Soc. 38, 71–137 (1980)Google Scholar
  21. 21.
    Nikulin V.V.: Integral symmetric bilinear forms and some of their applications. Math. USSR Izv. 14, 103–167 (1980)zbMATHCrossRefGoogle Scholar
  22. 22.
    Nikulin V.V.: Factor groups of groups of the automorphisms of hyperbolic forms with respect to subgroups generated by 2-reflections. J. Soviet Math. 22, 1401–1475 (1983)zbMATHCrossRefGoogle Scholar
  23. 23.
    Oguiso K., Zhang D.-Q.: On Vorontsov’s theorem on K3 surfaces with non-symplectic group actions. Proc. Am. Math. Soc. 128(6), 1571–1580 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Oguiso, K., Zhang, D.-Q.: K3 surfaces with order 11 automorphisms. math.AG/9907020Google Scholar
  25. 25.
    Persson U., Pinkham H.: Degenerations of surfaces with trivial canonical bundle. Ann. Math. 113, 45–66 (1981)CrossRefMathSciNetGoogle Scholar
  26. 26.
    Rudakov, A.N., Shafarevich, I.: Surfaces of type K3 over fields of finite characteristic. In: Shafarevich I. Collected mathematical papers, pp. 657–714. Springer, Berlin (1989)Google Scholar
  27. 27.
    Saint-Donat B.: Projective models of K3 surfaces. Am. J. Math. 96(4), 602–639 (1974)zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Sterk H.: Finiteness results for algebraic K3 surfaces. Math. Zeitschrift 189, 507–513 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Zhang, D.-Q.: Quotients of K3 surfaces modulo involutions. Jpn. J. Math. (1999) (to appear)Google Scholar
  30. 30.
    Zhang D.-Q.: Normal algebraic surfaces with trivial tricanonical divisors. Publ. RIMS Kyoto Univ. 33, 427–442 (1997)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Departamento de MatemáticaUniversidad de ConcepciónConcepciónChile
  2. 2.Fachbereich für MathematikJohannes Gutenberg-UniversitätMainzGermany

Personalised recommendations