Skip to main content
SpringerLink
Log in

Picard–Fuchs Equations of Special One-Parameter Families of Invertible Polynomials

  • Chapter
  • First Online:
Book cover Arithmetic and Geometry of K3 Surfaces and Calabi–Yau Threefolds

Part of the book series: Fields Institute Communications ((FIC,volume 67))

Abstract

In this article we calculate the Picard–Fuchs equation of hypersurfaces defined by certain one-parameter families associated to invertible polynomials. For this we deduce the Picard–Fuchs equation from the GKZ system. As consequences of our work and facts from the literature, we show a relation between the Picard–Fuchs equation, the Poincaré series and the monodromy in the space of period integrals.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. V.I. Arnold, Critical points of smooth functions, and their normal forms. Uspehi Mat. Nauk 30(5(185)), 3–65 (1975)

    Google Scholar 

  2. V.I. Arnold, S.M. Guseĭn-Zade, A.N. Varchenko, in Singularities of Differentiable Maps. Vol. I. Monographs in Mathematics, vol. 82 (Birkhäuser, Boston, 1985)

    Google Scholar 

  3. V.V. Batyrev, Dual polyhedra and mirror symmetry for Calabi–Yau hypersurfaces in toric varieties. J. Algebr. Geom. 3(3), 493–535 (1994)

    MathSciNet  MATH  Google Scholar 

  4. V.V. Batyrev, L.A. Borisov, Dual cones and mirror symmetry for generalized Calabi–Yau manifolds, in Mirror Symmetry, II. AMS/IP Studies in Advanced Mathematics (American Mathematical Society, Providence, 1997), pp. 71–86

    Google Scholar 

  5. P. Berglund, T. Hübsch, A generalized construction of mirror manifolds, in Essays on Mirror Manifolds (International Press, Hong Kong, 1992), pp. 388–407

    Google Scholar 

  6. L.A. Borisov, Berglund–Hübsch mirror symmetry via vertex algebras. Preprint (2010), arXiv:1007.2633v3

    Google Scholar 

  7. Y.-H. Chen, Y. Yang, N. Yui, Monodromy of Picard–Fuchs differential equations for Calabi–Yau threefolds. J. Reine Angew. Math. 616, 167–203 (2008)

    MathSciNet  MATH  Google Scholar 

  8. A. Chiodo, Y. Ruan, LG/CY correspondence: the state space isomorphism. Adv. Math. 7(2), 57–218 (2011)

    MathSciNet  Google Scholar 

  9. A. Corti, V. Golyshev, Hypergeometric equations and weighted projective spaces. Sci. China Math. 54(8), 1577–1590 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. D.A. Cox, S. Katz, in Mirror Symmetry and Algebraic Geometry. Mathematical Surveys and Monographs, vol. 68 (American Mathematical Society, Providence, 1999)

    Google Scholar 

  11. V.I. Danilov, A.G. Khovanskiĭ, Newton polyhedra and an algorithm for calculating Hodge–Deligne numbers. Izv. Akad. Nauk SSSR Ser. Mat. 50(5), 925–945 (1986)

    MathSciNet  Google Scholar 

  12. P. Deligne, in Equations différentielles à points singuliers réguliers. Lecture Notes in Mathematics, vol. 163 (Springer, Berlin, 1970)

    Google Scholar 

  13. I. Dolgachev, Weighted projective varieties, in Group Actions and Vector Fields, Vancouver, BC, 1981. Lecture Notes in Mathematics, vol. 956 (Springer, Berlin, 1982), pp. 34–71

    Google Scholar 

  14. I.V. Dolgachev, Mirror symmetry for lattice polarized K3 surfaces. J. Math. Sci. 81(3), 2599–2630 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  15. W. Ebeling, Strange duality, mirror symmetry, and the Leech lattice, in Singularity Theory, Liverpool, 1996. London Mathematical Society Lecture Note Series, vol. 263 (Cambridge University Press, Cambridge, 1999), pp. 55–77

    Google Scholar 

  16. W. Ebeling, A. Takahashi, Strange duality of weighted homogeneous polynomials. Compos. Math., 147, 1413–1433 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. H. Fan, T. Jarvis, Y. Ruan, The Witten equation, mirror symmetry and quantum singularity theory. Preprint (2009), arXiv:0712.4021v3

    Google Scholar 

  18. S. Gährs, Picard–Fuchs equations of special one-parameter families of invertible polynomials. Preprint (2011), arXiv:1109.3462

    Google Scholar 

  19. I.M. Gel’fand, A.V. Zelevinskiĭ, M.M. Kapranov, Hypergeometric functions and toric varieties. Funktsional. Anal. i Prilozhen. 23(2), 12–26 (1989)

    Google Scholar 

  20. I.M. Gel’fand, A.V. Zelevinskiĭ, M.M. Kapranov, Generalized Euler integrals and A-hypergeometric functions. Adv. Math. 84(2), 255–271 (1990)

    Google Scholar 

  21. I.M. Gel’fand, A.V. Zelevinskiĭ, M.M. Kapranov, Hypergeometric functions, toric varieties and Newton polyhedra, in Special Functions, Okayama, 1990. ICM-90 Satellite Conference Proceedings (Springer, Tokyo, 1991), pp. 104–121

    Google Scholar 

  22. I.M. Gel’fand, A.V. Zelevinskiĭ, M.M. Kapranov, Correction to “Hypergeometric functions and toric varieties”. Funktsional. Anal. i Prilozhen. 27(4), 91 (1993)

    Google Scholar 

  23. S. Hosono, GKZ systems, Gröbner fans, and moduli spaces of Calabi–Yau hypersurfaces, in Topological Field Theory, Primitive Forms and Related Topics, Kyoto, 1996. Progress in Mathematics, vol. 160 (Birkhäuser, Boston, 1998), pp. 239–265

    Google Scholar 

  24. M. Krawitz, FJRW rings and Landau–Ginzburg Mirror Symmetry. Preprint (2009), arXiv:0906.0796

    Google Scholar 

  25. M. Krawitz, N. Priddis, P. Acosta, N. Bergin, H. Rathnakumara, FJRW-rings and mirror symmetry. Comm. Math. Phys. 296, 145–174 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  26. M. Kreuzer, H. Skarke, On the classification of quasihomogeneous functions. Comm. Math. Phys. 150(1), 137–147 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  27. D.R. Morrison, Picard–Fuchs equations and mirror maps for hypersurfaces, in Essays on Mirror Manifolds (International Press, Hong Kong, 1992), pp. 241–264

    Google Scholar 

  28. D.R. Morrison, Geometric aspects of mirror symmetry, in Mathematics Unlimited—2001 and Beyond (Springer, Berlin, 2001), pp. 899–918

    Google Scholar 

  29. J. Stienstra, GKZ hypergeometric structures, in Arithmetic and Geometry Around Hypergeometric Functions. Progress in Mathematics, vol. 260 (Birkhäuser, Basel, 2007), pp. 313–371

    Google Scholar 

  30. N. Yui, Arithmetic of certain Calabi–Yau varieties and mirror symmetry, in Arithmetic Algebraic Geometry, Park City, UT, 1999. IAS/Park City Mathematics Series, vol. 9 (American Mathematical Society, Providence, 2001), pp. 507–569

    Google Scholar 

Download references

Acknowledgements

This work was supported in part by DFG-RTG 1463. This article was part of my Ph.D. thesis and I would like to thank my supervisor Prof. Wolfgang Ebeling for his support. During my research I was supported by DFG RTG 1463. I would also like to thank Prof. Noriko Yui and Prof. Ragnar-Olaf Buchweitz, who were gave me input during my stay at the Fields Institute in Toronto. In addition I would like to thank the referee for helpful comments.

Author information

Authors and Affiliations

  1. Institut für algebraische Geometrie, Leibniz Universität Hannover, Welfengarten 1, 30167, Hannover, Germany

    Swantje Gährs

Authors
  1. Swantje Gährs
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Swantje Gährs .

Editor information

Editors and Affiliations

  1. , Mathematics Department, Stony Brook University, N/A, Stony Brook, 11794-3651, New York, USA

    Radu Laza

  2. , Institut für Algebraische Geometrie, Leibniz Universität Hannover, Welfengarten 1, Hannover, 30167, Germany

    Matthias Schütt

  3. , Department of Math & Stats, Queen's University, University Ave., Kingston, K7L 3N6, Ontario, Canada

    Noriko Yui

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer Science+Business Media New York

About this chapter

Cite this chapter

Gährs, S. (2013). Picard–Fuchs Equations of Special One-Parameter Families of Invertible Polynomials. In: Laza, R., Schütt, M., Yui, N. (eds) Arithmetic and Geometry of K3 Surfaces and Calabi–Yau Threefolds. Fields Institute Communications, vol 67. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6403-7_9

Download citation

Publish with us

Policies and ethics

Search

Navigation