Journal of Mathematical Sciences

, Volume 81, Issue 3, pp 2599–2630 | Cite as

Mirror symmetry for lattice polarizedK3 surfaces

  • I. V. Dolgachev
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References

  1. 1.
    V. Arnold, “Critical points of smooth functions”, In:Proc. I.C.M., Vancouver (1974), pp. 18–39.Google Scholar
  2. 2.
    P. Aspinwall and D. Morrison,String theory on K3 surfaces, preprint IASSNS-hep-94/23 (1994).Google Scholar
  3. 3.
    P. Aspinwall and D. Morrison,Mirror symmetry and the moduli space of K3 surfaces, to appear.Google Scholar
  4. 4.
    W. Barth, C. Peters, and A. Van de Ven,Compact Complex Surfaces, Ergenbnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Band 4, Springer-Verlag (1984).Google Scholar
  5. 5.
    V. Batyrev, “Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties”,J. Alg. Geom.,3, 493–535 (1994).MATHMathSciNetGoogle Scholar
  6. 6.
    C. Borcea,K3 surfaces with involution and mirror pairs of Calabi-Yau manifolds, preprint Rider College.Google Scholar
  7. 7.
    F. Cossec and I. Dolgachev,Enriques surfaces I, Birkhäuser (1989).Google Scholar
  8. 8.
    I. Dolgachev and V. Nikulin, “Exceptional singularities of V. I. Arnold andK3 surfaces”, In:Proc. USSR Topological Conference, Minsk (1977).Google Scholar
  9. 9.
    I. Dolgachev, “Integral quadratic forms: applications to algebraic geometry”, In:Sem. Bourbaki, 1982/83, No. 611;Asterisque,105/106, Soc. Math. France, 251–275.Google Scholar
  10. 10.
    I. Dolgachev, “On algebraic properties of algebras of automorphic forms”, In:Modular Functions in Analysis and Number Theory, Lect. Notes Math. Stat., Vol. 5, Univ. Pittsburgh (1983), pp. 21–29.Google Scholar
  11. 11.
    Essays on Mirror Symmetry (ed. S.-T. Yau) Int. Press Co., Hong Kong (1992).Google Scholar
  12. 12.
    R. Fricke,Lehrbuch der Algebra, B. 3, Braunschweig (1928).Google Scholar
  13. 13.
    Géometrie des surfaces K3: modules et périodes”, Astérisque,126, Soc. Math. France (1985).Google Scholar
  14. 14.
    A. Giveon and D.-J. Smit, “Symmetries of the moduli space of (2,2) superstring vacua”,Nucl. Phys. B,349, 168–206 (1991).CrossRefMathSciNetGoogle Scholar
  15. 15.
    Ph. Griffiths, “Periods of integrals on algebraic manifolds, I, II”Amer. J. Math.,90, 568–626, 805–865 (1968).MATHMathSciNetGoogle Scholar
  16. 16.
    Ph. Griffiths and L. Tu, “Infinitesimal variation of Hodge structure”, In:Topics in Transcendental Algebraic Geometry, Ann. Math. Studies, Vol. 106, Princeton University Press (1984).Google Scholar
  17. 17.
    D. James, “On Witt's theorem for unimodular quadratic forms”,Pac. J. Math.,26:2, 303–316 (1968).MATHGoogle Scholar
  18. 18.
    P. G. Kluit, “On the normalizer of Γ0(n)”, In:Modular Functions of One Variable, V, Lect. Notes Math., Vol. 601, Springer (1977), pp. 239–246.Google Scholar
  19. 19.
    M. Kobayashi,Duality of weights, mirror symmetry and Arnold's strange duality, preprint (1994).Google Scholar
  20. 20.
    J. Lehner and W. Newman, “Weierstrass, points of Γ0(n)”,Ann. Math.,79, 360–368 (1964).MathSciNetGoogle Scholar
  21. 21.
    B. Lian and S.-T. Yau,Arithmetic properties of mirror map and quantum coupling, preprint hep-th (1994).Google Scholar
  22. 22.
    B. Lian and S.-T. Yau,Mirror maps, modular relations and hypergeometric series II, preprint hep-th (1994).Google Scholar
  23. 23.
    E. Martinec, “Criticality, catastrophes, and compactifications”, In:Physics and Mathematics of Strings, World Scientific (1990), pp. 389–433.Google Scholar
  24. 24.
    J. Milnor, “On the 3-dimensional Brieskorn manifolds”, In:Knots, Groups and 3-Manifolds, Ann. Math. Stud., Vol. 84, Princeton Univ. Press (1975), pp. 175–224.Google Scholar
  25. 25.
    D. Morrison, “OnK3 surfaces with large Picard number”,Invent. Math.,75, 105–121 (1984).CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    D. Morrison, “Mirror symmetry and rational curves on quintic 3-folds: A guide for mathematicians”,J. Amer. Math. Soc.,6, 223–247 (1993).MATHMathSciNetGoogle Scholar
  27. 27.
    M. Nagura and K. Sugiyama, “Mirror symmetry of theK3 surface”,Int. J. Mod. Phys. A,10, No. 2, 233–252 (1995).MathSciNetGoogle Scholar
  28. 28.
    Y. Namikawa, “Periods of Enriques surfaces”,Math. Ann.,270, 201–222 (1985).CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    W. Neumann, “Abelian covers of quasihomogeneous singularities”, In:Singularities, Proc. Symp. Pure Math., Vol. 40, Part 2, A.M.S., Providence (1983), pp. 233–243.Google Scholar
  30. 30.
    V. Nikulin, “Finite groups of automorphisms of KählerK3 surfaces”, In:Tr. Mosk. Mat. Obshch., Vol. 38 (1980), pp. 71–135.Google Scholar
  31. 31.
    V. Nikulin, “Integral quadratic forms and some of its geometric applications”,Izv. Akad. Nauk SSSR. Ser. Mat.,43, 103–167 (1979).MathSciNetGoogle Scholar
  32. 32.
    V. Nikulin, “On rational maps between K3 surfaces”, In:Constantin Caratheodory: An International Tribute, Vol. I, II, World Sci. Publ. (1991), pp. 964–995.Google Scholar
  33. 33.
    C. Peters and J. Stienstra, “A pencil ofK3 surfaces related to Apéry's recurrence for ζ(3) and Fermi surfaces for potential zero”, In:Arithmetics of Complex Manifolds, Lect. Notes Math., Vol. 1399, Springer-Verlag (1989).Google Scholar
  34. 34.
    H. Pinkham, “Singularités exceptionnelles, la dualité étrange d'Arnold et les surfacesK3”,C. R. Acad. Sci. Paris, Ser. A-B,284, 615–618 (1977).MATHMathSciNetGoogle Scholar
  35. 35.
    S.-S. Roan,Mirror symmetry and Arnold's duality, preprint MPI (1993).Google Scholar
  36. 36.
    Y. Ruan and G. Tian, “A mathematical theory of quantum cohomology”,Math. Res. Lett.,1, No. 2, 269–278 (1994).MathSciNetGoogle Scholar
  37. 37.
    F. Scattone, “On the compactification of moduli spaces for algebraicK3 surfaces”,Mem. A.M.S.,70, No. 374 (1987).Google Scholar
  38. 38.
    G. Shimura,Introduction to the Arithmetic Theory of Automorphic Functions, Vol. 11, Publ. Math. Soc. Japan (1971).Google Scholar
  39. 39.
    H. Sterk, “Lattices andK3 surfaces of degree 6”,Lin. Alg. Appl., 226–228, 297–309 (1995).Google Scholar
  40. 40.
    A. Todorov,Some ideas from mirror geometry applied to the moduli space of K3, preprint.Google Scholar
  41. 41.
    C. Voisin, “Miroirs et involutions sur les surfacesK3”, In:Journées de Géométrie Algébrique d'Orsay, Vol. 218;Astérisque, Soc. Math. France, 273–323 (1993).Google Scholar
  42. 42.
    T. Yonemura, “Hypersurface simpleK3 singularities”,Tôhoku Math. J.,42, 351–380 (1990).MATHMathSciNetGoogle Scholar

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© Plenum Publishing Corporation 1996

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  • I. V. Dolgachev

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