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GKZ Systems, Gröbner Fans, and Moduli Spaces of Calabi-Yau Hypersurfaces

  • Shinobu Hosono
Chapter
Part of the Progress in Mathematics book series (PM, volume 160)

Abstract

We present a detailed analysis of the GKZ (Gel’fand, Kapranov and Zelevinski) hypergeometric systems in the context of mirror symmetry of Calabi-Yau hypersurfaces in toric varieties. As an application, we will derive a concise formula for the prepotential about large complex structure limits.

Keywords

Modulus Space Toric Variety Polyhedral Cone Period Integral Regular Triangulation 
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References

  1. [Bat1]
    V.V. Batyrev, Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, J. Alg. Geom. 3 (1994), 493–535.MathSciNetMATHGoogle Scholar
  2. [Bat2]
    V.V. Batyrev, Variations of the mixed Hodge structure of affine hypersurfaces in algebraic tori, Duke Math. J. 69 (1993), 349–409.MathSciNetMATHCrossRefGoogle Scholar
  3. [BC]
    V.V. Batyrev and D.A. Cox, On the Hodge structure of projective hypersurfaces in toric varieties, Duke Math. J. 75 (1994), 293–338.MathSciNetMATHCrossRefGoogle Scholar
  4. [BFS]
    L.J. Billera, P. Filliman, and B. Sturmfels, Constructions and complexity of secondary polytopes, Adv. in Math. 83 (1990), 155–179.MathSciNetMATHCrossRefGoogle Scholar
  5. [BG]
    R.L. Bryant and P.A. Griffiths, Some observations on the infinitesimal period relations for regular threefolds with trivial canonical bundle, Arithmetic and Geometry, vol II, (M. Artin and J. Tate, eds.), Birkhäuser, Boston, 1983, pp. 77–102.Google Scholar
  6. [CdGP]
    P. Candelas, X.C. de la Ossa, P.S. Green, and L. Parkes, A pair of Calabi-Yau manifolds as an exactly soluble superconformai theory, Nucl. Phys. B 356 (1991), 21–74.CrossRefGoogle Scholar
  7. [CdFKM1]
    P. Candelas, X. de la Ossa, A. Font, S. Katz and D.R. Morrison, Mirror symmetry for two parameter models I, Nucl. Phys. B 416 (1994), 481–562.MATHCrossRefGoogle Scholar
  8. [CdFKM2]
    P. Candelas, X. de la Ossa, A. Font, S. Katz and D.R. Morrison, Mirror symmetry for two parameter models II, Nucl. Phys. B 429 (1994), 626–674.CrossRefGoogle Scholar
  9. [CLO]
    D. Cox, J. Little and D. O’Shea, Ideals, Varieties, and Algorithms, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1992.Google Scholar
  10. [Del]
    P. Deligne, Equations différentielles à points singuliers réguliers, Lecture Notes in Math., vol. 163, Springer-Verlag, New York, 1970.Google Scholar
  11. [Ful]
    W. Fulton, Introduction to Toric Varieties, Ann. of Math. Studies 131, Princeton University Press, Princeton, New Jersey, 1993.Google Scholar
  12. [GKZ1]
    I.M. Gel’fand, A.V. Zelevinski, and M.M. Kapranov, Equations of hypergeometric type and toric varieties, Funktsional Anal. i. Prilozhen. 23 (1989), 12–26; English transi. Functional Anal. Appl. 23 (1989), 94-106.CrossRefGoogle Scholar
  13. [GKZ2]
    I.M. Gel’fand, A.V. Zelevinski, and M.M. Kapranov, Discriminants, Resultants and Multidimensional Determinants, Birkhäuser, Boston, 1994.MATHCrossRefGoogle Scholar
  14. [GY]
    B. Greene and S.-T. Yau, eds, Mirror Symmetry II, Studies in Advanced Mathematics, Amer. Math. Soc./International Press, 1996.Google Scholar
  15. [HKTY1]
    S. Hosono, A. Klemm, S. Theisen, and S.-T. Yau, Mirror symmetry, mirror map and applications to Calabi-Yau hypersurfaces, Commun. Math. Phys. 167 (1995), 301–350.MathSciNetMATHCrossRefGoogle Scholar
  16. [HKTY2]
    S. Hosono, A. Klemm, S. Theisen, and S.-T. Yau, Mirror symmetry, mirror map and applications to complete intersection Calabi-Yau spaces, Nucl. Phys. B 433 (1995), 501–554.MathSciNetCrossRefGoogle Scholar
  17. [HLY1]
    S. Hosono, B.H. Lian, and S.-T. Yau, GKZ-Generalized Hypergeometric Systems in Mirror Symmetry of Calabi-Yau Hypersurfaces, Commun. Math. Phys. 182 (1996),535–577.MathSciNetMATHCrossRefGoogle Scholar
  18. [HLY2]
    S. Hosono, B.H. Lian, and S.-T. Yau, Maximal Degeneracy Points of GKZ Systems, J. Amer. Math. Soc. 10 (1997), 427–443.MathSciNetMATHCrossRefGoogle Scholar
  19. [HLY3]
    S. Hosono, B.H. Lian, and S.-T. Yau, Type HA monodromy of Calabi-Yau manifolds, (1997) to appear.Google Scholar
  20. [HSS]
    S. Hosono, M.-H. Saito and J. Stienstra, On Mirror Symmetry Conjecture for Schoen’s Calabi-Yau 3-folds, Submitted to The Proceedings of Taniguchi Symposium, “Integrable Systems and Algebraic Geometry”, Kobe/Kyoto(1997), alg-geom/9709027.Google Scholar
  21. [Mor]
    D.R. Morrison, Picard-Fuchs equations and mirror maps for hypersur-faces, Essays on Mirror Manifolds (S.-T. Yau, ed.), Internal Press, Hong Kong, (1992), 241–264.Google Scholar
  22. [Oda]
    T. Oda, Convex bodies and Algebraic Geometry, An Introduction to the Theory of Toric Varieties, A Series of Modern Surveys in Mathematics, Springer-Verlag New York, 1985.Google Scholar
  23. [OP]
    T. Oda and H.S. Park, Linear Gale transform and Gelfand-Kapranov-Zelevinski decompositions, Tôhoku Math. J. 43 (1991), 375–399.MathSciNetMATHCrossRefGoogle Scholar
  24. [Saito]
    K. Saito, Primitive forms for a universal unfolding of a function with an isolated critical point, J. Fac. Sci. Univ. Tokyo 28 (1981), 775–792.MATHGoogle Scholar
  25. [Sti]
    J. Stienstra, Resonance in Hypergeometric Systems related to Mirror Symmetry, preprint, to be published in the proceedings of Kinosaki conference on Algebraic Geometry, Nov. 11–15, 1996.Google Scholar
  26. [Str]
    A. Strominger, Special Geometry, Commun. Math. Phys. 133 (1990), 163–180.MathSciNetMATHCrossRefGoogle Scholar
  27. [Stu1]
    B. Sturmfels, Göbner bases of toric varieties, Tôhoku Math. J. 43 (1991), 249–261.MathSciNetMATHCrossRefGoogle Scholar
  28. [Stu2]
    B. Sturmfels, Gröbner bases and convex polytopes, University Lecture Series vol. 8, Amer. Math. Soc, 1996.Google Scholar
  29. [Tian]
    G. Tian, Smoothness of the universal deformation space of compact Calabi-Yau manifolds and its Peterson-Weil metric, in Mathematical Aspects of String Theory, S.-T. Yau (ed.), Singapore, World Scientific 1988.Google Scholar
  30. [Zie]
    G.M. Ziegler, Lectures on Polytopes, Graduate Texts in Mathematics 152, Springer-Verlag, New York, 1994.Google Scholar

Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • Shinobu Hosono
    • 1
  1. 1.Department of MathematicsToyama UniversityToyamaJapan

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