Maximal degeneracy points of GKZ systems

Hosono, S.; Lian, B.; Yau, S.-T.

doi:10.1090/S0894-0347-97-00230-0

Maximal degeneracy points of GKZ systems
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by S. Hosono, B. H. Lian and S.-T. Yau

J. Amer. Math. Soc. 10 (1997), 427-443

DOI: https://doi.org/10.1090/S0894-0347-97-00230-0

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Abstract:

Motivated by mirror symmetry, we study certain integral representations of solutions to the Gel$^\prime$fand-Kapranov-Zelevinsky (GKZ) hypergeometric system. Some of these solutions arise as period integrals for Calabi-Yau manifolds in mirror symmetry. We prove that for a suitable compactification of the parameter space, there exist certain special boundary points, which we called maximal degeneracy points, at which all solutions but one become singular.

References

Victor V. Batyrev, Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, J. Algebraic Geom. 3 (1994), no. 3, 493–535. MR 1269718
Victor V. Batyrev, Variations of the mixed Hodge structure of affine hypersurfaces in algebraic tori, Duke Math. J. 69 (1993), no. 2, 349–409. MR 1203231, DOI 10.1215/S0012-7094-93-06917-7
—, Quantum cohomology rings of toric manifolds, preprint 1993.
P. Berglund, S. Katz and A. Klemm, Nucl. Phys. B456 (1995), 153–204.
Louis J. Billera, Paul Filliman, and Bernd Sturmfels, Constructions and complexity of secondary polytopes, Adv. Math. 83 (1990), no. 2, 155–179. MR 1074022, DOI 10.1016/0001-8708(90)90077-Z
Philip Candelas, Xenia C. de la Ossa, Paul S. Green, and Linda Parkes, A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory, Nuclear Phys. B 359 (1991), no. 1, 21–74. MR 1115626, DOI 10.1016/0550-3213(91)90292-6
Philip Candelas, Xenia de la Ossa, Anamaría Font, Sheldon Katz, and David R. Morrison, Mirror symmetry for two-parameter models. I, Nuclear Phys. B 416 (1994), no. 2, 481–538. MR 1274436, DOI 10.1016/0550-3213(94)90322-0
Philip Candelas, Anamaría Font, Sheldon Katz, and David R. Morrison, Mirror symmetry for two-parameter models. II, Nuclear Phys. B 429 (1994), no. 3, 626–674. MR 1302876, DOI 10.1016/0550-3213(94)90155-4
David Cox, John Little, and Donal O’Shea, Ideals, varieties, and algorithms, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1992. An introduction to computational algebraic geometry and commutative algebra. MR 1189133, DOI 10.1007/978-1-4757-2181-2
Anamaría Font, Periods and duality symmetries in Calabi-Yau compactifications, Nuclear Phys. B 391 (1993), no. 1-2, 358–388. MR 1208020, DOI 10.1016/0550-3213(93)90152-F
William Fulton, Introduction to toric varieties, Annals of Mathematics Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993. The William H. Roever Lectures in Geometry. MR 1234037, DOI 10.1515/9781400882526
I. M. Gel′fand, A. V. Zelevinskiĭ, and M. M. Kapranov, Hypergeometric functions and toric varieties, Funktsional. Anal. i Prilozhen. 23 (1989), no. 2, 12–26 (Russian); English transl., Funct. Anal. Appl. 23 (1989), no. 2, 94–106. MR 1011353, DOI 10.1007/BF01078777
I. M. Gel′fand, M. M. Kapranov, and A. V. Zelevinsky, Newton polytopes of the classical resultant and discriminant, Adv. Math. 84 (1990), no. 2, 237–254. MR 1080979, DOI 10.1016/0001-8708(90)90047-Q
B. R. Greene and M. R. Plesser, Duality in Calabi-Yau moduli space, Nuclear Phys. B 338 (1990), no. 1, 15–37. MR 1059831, DOI 10.1016/0550-3213(90)90622-K
S. Hosono, A. Klemm, S. Theisen, and S.-T. Yau, Mirror symmetry, mirror map and applications to Calabi-Yau hypersurfaces, Comm. Math. Phys. 167 (1995), no. 2, 301–350. MR 1316509
S. Hosono, A. Klemm, S. Theisen, and S.-T. Yau, Mirror symmetry, mirror map and applications to complete intersection Calabi-Yau spaces, Nuclear Phys. B 433 (1995), no. 3, 501–552. MR 1319280, DOI 10.1016/0550-3213(94)00440-P
S. Hosono, B. Lian and S. T. Yau, GKZ-Generalized Hypergeometric Systems in Mirror Symmetry of Calabi-Yau Hypersurfaces, Harvard Univ. preprint, alg-geom/9511001, to appear in CMP 1996.
Albrecht Klemm and Stefan Theisen, Considerations of one-modulus Calabi-Yau compactifications: Picard-Fuchs equations, Kähler potentials and mirror maps, Nuclear Phys. B 389 (1993), no. 1, 153–180. MR 1202211, DOI 10.1016/0550-3213(93)90289-2
David R. Morrison, Picard-Fuchs equations and mirror maps for hypersurfaces, Essays on mirror manifolds, Int. Press, Hong Kong, 1992, pp. 241–264. MR 1191426
D. Mumford, J. Fogarty, and F. Kirwan, Geometric invariant theory, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34, Springer-Verlag, Berlin, 1994. MR 1304906
Tadao Oda, Convex bodies and algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 15, Springer-Verlag, Berlin, 1988. An introduction to the theory of toric varieties; Translated from the Japanese. MR 922894
Tadao Oda and Hye Sook Park, Linear Gale transforms and Gel′fand-Kapranov-Zelevinskij decompositions, Tohoku Math. J. (2) 43 (1991), no. 3, 375–399. MR 1117211, DOI 10.2748/tmj/1178227461
Bernd Sturmfels, Gröbner bases of toric varieties, Tohoku Math. J. (2) 43 (1991), no. 2, 249–261. MR 1104431, DOI 10.2748/tmj/1178227496
—, Gröbner Bases and Convex Polytopes, AMS University Lecture Series, Vol. 8, Providence, RI, 1995.
Shing-Tung Yau (ed.), Essays on mirror manifolds, International Press, Hong Kong, 1992. MR 1191418