## Maximal degeneracy points of GKZ systems.(English)Zbl 0874.32007

Summary: Motivated by mirror symmetry, we study certain integral representations of solutions to the Gel’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.

### MSC:

 32G20 Period matrices, variation of Hodge structure; degenerations 14C30 Transcendental methods, Hodge theory (algebro-geometric aspects) 14M25 Toric varieties, Newton polyhedra, Okounkov bodies
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### References:

 [1] Victor V. Batyrev, Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, J. Algebraic Geom. 3 (1994), no. 3, 493 – 535. · Zbl 0829.14023 [2] Victor V. Batyrev, Variations of the mixed Hodge structure of affine hypersurfaces in algebraic tori, Duke Math. J. 69 (1993), no. 2, 349 – 409. · Zbl 0812.14035 [3] -, Quantum cohomology rings of toric manifolds, preprint 1993. · Zbl 0806.14041 [4] P. Berglund, S. Katz and A. Klemm, Nucl. Phys. B456 (1995), 153-204. CMP 96:07 [5] Louis J. Billera, Paul Filliman, and Bernd Sturmfels, Constructions and complexity of secondary polytopes, Adv. Math. 83 (1990), no. 2, 155 – 179. · Zbl 0714.52004 [6] 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. · Zbl 1098.32506 [7] 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. · Zbl 0899.14017 [8] 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. · Zbl 1020.32506 [9] 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. · Zbl 0756.13017 [10] Anamaría Font, Periods and duality symmetries in Calabi-Yau compactifications, Nuclear Phys. B 391 (1993), no. 1-2, 358 – 388. · Zbl 1360.32009 [11] 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. · Zbl 0813.14039 [12] I. M. Gel$$^{\prime}$$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. · Zbl 0721.33006 [13] I. M. Gel$$^{\prime}$$fand, M. M. Kapranov, and A. V. Zelevinsky, Newton polytopes of the classical resultant and discriminant, Adv. Math. 84 (1990), no. 2, 237 – 254. , https://doi.org/10.1016/0001-8708(90)90047-Q I. M. Gel$$^{\prime}$$fand, M. M. Kapranov, and A. V. Zelevinsky, Generalized Euler integrals and \?-hypergeometric functions, Adv. Math. 84 (1990), no. 2, 255 – 271. · Zbl 0741.33011 [14] B. R. Greene and M. R. Plesser, Duality in Calabi-Yau moduli space, Nuclear Phys. B 338 (1990), no. 1, 15 – 37. [15] 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. · Zbl 0814.53056 [16] 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. · Zbl 1020.32508 [17] 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. · Zbl 0870.14028 [18] 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. [19] David R. Morrison, Picard-Fuchs equations and mirror maps for hypersurfaces, Essays on mirror manifolds, Int. Press, Hong Kong, 1992, pp. 241 – 264. · Zbl 0841.32013 [20] 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. · Zbl 0797.14004 [21] 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. · Zbl 0628.52002 [22] Tadao Oda and Hye Sook Park, Linear Gale transforms and Gel$$^{\prime}$$fand-Kapranov-Zelevinskij decompositions, Tohoku Math. J. (2) 43 (1991), no. 3, 375 – 399. · Zbl 0782.52006 [23] Bernd Sturmfels, Gröbner bases of toric varieties, Tohoku Math. J. (2) 43 (1991), no. 2, 249 – 261. · Zbl 0714.14034 [24] -, Gröbner Bases and Convex Polytopes, AMS University Lecture Series, Vol. 8, Providence, RI, 1995. CMP 96:05 [25] Shing-Tung Yau , Essays on mirror manifolds, International Press, Hong Kong, 1992. · Zbl 0816.00010
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