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.
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References
V.V. Batyrev, Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, J. Alg. Geom. 3 (1994), 493–535.
V.V. Batyrev, Variations of the mixed Hodge structure of affine hypersurfaces in algebraic tori, Duke Math. J. 69 (1993), 349–409.
V.V. Batyrev and D.A. Cox, On the Hodge structure of projective hypersurfaces in toric varieties, Duke Math. J. 75 (1994), 293–338.
L.J. Billera, P. Filliman, and B. Sturmfels, Constructions and complexity of secondary polytopes, Adv. in Math. 83 (1990), 155–179.
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.
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.
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.
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.
D. Cox, J. Little and D. O’Shea, Ideals, Varieties, and Algorithms, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1992.
P. Deligne, Equations différentielles à points singuliers réguliers, Lecture Notes in Math., vol. 163, Springer-Verlag, New York, 1970.
W. Fulton, Introduction to Toric Varieties, Ann. of Math. Studies 131, Princeton University Press, Princeton, New Jersey, 1993.
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.
I.M. Gel’fand, A.V. Zelevinski, and M.M. Kapranov, Discriminants, Resultants and Multidimensional Determinants, Birkhäuser, Boston, 1994.
B. Greene and S.-T. Yau, eds, Mirror Symmetry II, Studies in Advanced Mathematics, Amer. Math. Soc./International Press, 1996.
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.
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.
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.
S. Hosono, B.H. Lian, and S.-T. Yau, Maximal Degeneracy Points of GKZ Systems, J. Amer. Math. Soc. 10 (1997), 427–443.
S. Hosono, B.H. Lian, and S.-T. Yau, Type HA monodromy of Calabi-Yau manifolds, (1997) to appear.
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.
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.
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.
T. Oda and H.S. Park, Linear Gale transform and Gelfand-Kapranov-Zelevinski decompositions, Tôhoku Math. J. 43 (1991), 375–399.
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.
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.
A. Strominger, Special Geometry, Commun. Math. Phys. 133 (1990), 163–180.
B. Sturmfels, Göbner bases of toric varieties, Tôhoku Math. J. 43 (1991), 249–261.
B. Sturmfels, Gröbner bases and convex polytopes, University Lecture Series vol. 8, Amer. Math. Soc, 1996.
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.
G.M. Ziegler, Lectures on Polytopes, Graduate Texts in Mathematics 152, Springer-Verlag, New York, 1994.
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Hosono, S. (1998). GKZ Systems, Gröbner Fans, and Moduli Spaces of Calabi-Yau Hypersurfaces. In: Kashiwara, M., Matsuo, A., Saito, K., Satake, I. (eds) Topological Field Theory, Primitive Forms and Related Topics. Progress in Mathematics, vol 160. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0705-4_8
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DOI: https://doi.org/10.1007/978-1-4612-0705-4_8
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