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
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
Bibliographic Information
- S. Hosono
- Affiliation: Department of Mathematics, Toyama University, Toyama 930, Japan
- Email: hosono@sci.toyama-u.ac.jp
- B. H. Lian
- Affiliation: Department of Mathematics, Brandeis University, Waltham, Massachusetts 02154
- Email: lian@max.math.brandeis.edu
- S.-T. Yau
- Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
- MR Author ID: 185480
- ORCID: 0000-0003-3394-2187
- Received by editor(s): May 28, 1996
- Received by editor(s) in revised form: November 13, 1996
- © Copyright 1997 by the authors
- Journal: J. Amer. Math. Soc. 10 (1997), 427-443
- MSC (1991): Primary 14C30, 32G20
- DOI: https://doi.org/10.1090/S0894-0347-97-00230-0
- MathSciNet review: 1423031