Abstract
This text is based on lectures by the author in the Summer School Algebraic Geometry and Hypergeometric Functions in Istanbul in June 2005. It gives a review of some of the basic aspects of the theory of hypergeometric structures of Gelfand, Kapranov and Zelevinsky, including Differential Equations, Integrals and Series, with emphasis on the latter. The Secondary Fan is constructed and subsequently used to describe the ‘geography’ of the domains of convergence of the Γ-series. A solution to certain Resonance Problems is presented and applied in the context of Mirror Symmetry. Many examples and some exercises are given throughout the paper.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
V. Batyrev, Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, J. Alg. Geom. 3 (1994), 493–535.
V. Batyrev, Variations of mixed Hodge structure of affine hypersurfaces in algebraic tori, Duke Math. J. 69 (1993), 349–409.
V. Batyrev, L. Borisov, Dual Cones and Mirror Symmetry for Generalized Calabi-Yau Manifolds, pp. 71–86 in Mirror Symmetry II, B. Greene, S.-T. Yau (eds.), Studies in Advanced Math., vol. 1, American Math. Soc. / International Press, 1997.
A. de Boo, Solving GKZ-hypergeometric systems using relative Stanley-Reisner ideals, PhD thesis, Utrecht University, November 1999 (unpublished).
L. Borisov, P. Horja, Mellin-Barnes Integrals as Fourier-Mukai Transforms, math.AG/0510486.
R. Bryant, P. Griffiths, Some observations on the infinitesimal period relations for regular threefolds with trivial canonical bundle, in: Arithmetic and Geometry, vol. II, M. Artin and J. Tate (eds.), Progress in Math. vol. 36, Birkhäuser, Boston, 1983, pp. 77–102.
Ph. Candelas, X. de la Ossa, P. Green, L. Parkes, A Pair of Calabi-Yau Manifolds as an Exactly Soluble Superconformal Theory, in: Essays on Mirror Manifolds, S-T. Yau (ed.), International Press, Hong Kong, 1992.
D. Cox, S. Katz, Mirror Symmetry and Algebraic Geometry, Mathematical Surveys and Monographs, vol. 68, American Math. Soc., Providence, Rhode Island, 1999.
W. Fulton, Introduction to Toric Varieties, Annals of Mathematics Studies 131, Princeton University Press, Princeton, New Jersey, 1993.
I.M. Gelfand, A.V. Zelevinskii, M.M. Kapranov, Hypergeometric functions and toral manifolds, Functional Analysis and its Applications 23 (1989), 94–106.
correction to [10], Funct. Analysis and its Appl. 27 (1993), 295.
I.M. Gelfand, M.M. Kapranov, A.V. Zelevinsky, Generalized Euler Integrals and AHypergeometric Functions, Advances in Math. 84 (1990), 255–271.
I.M. Gelfand, M.M. Kapranov, A.V. Zelevinsky, Discriminants, Resultants and Multidimensional Determinants, Birkhäuser Boston, 1994.
I.M. Gelfand, M.M. Kapranov, A.V. Zelevinsky, Hypergeometric Functions, Toric Varieties and Newton Polyhedra in: Special Functions, M. Kashiwara, T. Miwa (eds.), ICM-90 Satellite Conference Proceedings, Springer-Verlag Tokyo, 1991.
V. Guillemin, Moment Maps and Combinatorial Invariants of Hamiltonian T n-spaces, Progress in Math. vol. 22, Birkhäuser Boston, 1994.
S. Hosono, A. Klemm, S. Theisen, S.-T. Yau, Mirror Symmetry, Mirror Map and Applications to Calabi-Yau Hypersurfaces, Commun. Math. Phys. 167 (1995), 301–350; also hep-th/9308122.
M. Kontsevich, Homological Algebra of Mirror Symmetry, in: Proceedings of the International Congress of Mathematicians Zürich 1994, Birkhäuser Basel, 1995.
A. Libgober, J. Teitelbaum, Lines on Calabi-Yau Complete Intersections, Mirror Symmetry, and Picard-Fuchs Equations, Int. Math. Res. Notices 1 (1993), 29–39.
E. Looijenga, Uniformization by Lauricella Functions — An Overview of the Theory of Deligne-Mostow, this volume; also math.CV/0507534.
K. Mayr, Über die Lösung algebraischer Gleichungssysteme durch hypergeometrische Funktionen, Monatshefte für Math. 45 (1937), 280–313.
M. Passare, A. Tsikh, Algebraic Equations and Hypergeometric Series, in: The Legacy of Niels Henrik Abel, O.A. Laudal and R. Piene (eds.), Springer Verlag, Berlin, 2004.
J. Richter-Gebert, B. Sturmfels, T. Theobald, First steps in tropical geometry, math.AG/0306366.
R.P. Stanley, Combinatorics and Commutative Algebra (second edition), Progress in Math. 41, Birkhäuser Boston, 1996.
J. Stienstra, Resonant Hypergeometric Systems and Mirror Symmetry, in: Integrable Systems and Algebraic Geometry, Proceedings of the Taniguchi Symposium 1997, M.-H. Saito, Y. Shimizu, K. Ueno (eds.); World Scientific, Singapore, 1998; also alg-geom/9711002.
B. Sturmfels, Gröbner Bases and Convex Polytopes, University Lecture Series vol. 8, American Math. Soc., 1996.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2007 Birkhäuser Verlag Basel/Switzerland
About this chapter
Cite this chapter
Stienstra, J. (2007). GKZ Hypergeometric Structures. In: Holzapfel, RP., Uludağ, A.M., Yoshida, M. (eds) Arithmetic and Geometry Around Hypergeometric Functions. Progress in Mathematics, vol 260. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-8284-1_12
Download citation
DOI: https://doi.org/10.1007/978-3-7643-8284-1_12
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-7643-8283-4
Online ISBN: 978-3-7643-8284-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)