Mathematische Zeitschrift

, Volume 288, Issue 3–4, pp 889–913 | Cite as

Global spectra, polytopes and stacky invariants

  • Antoine Douai


Given a convex polytope, we define its geometric spectrum, a stacky version of Batyrev’s stringy E-functions, and we prove a stacky version of a formula of Libgober and Wood about the E-polynomial of a smooth projective variety. As an application, we get a closed formula for the variance of the geometric spectrum and a Noether’s formula for two dimensional Fano polytopes (polytopes whose vertices are primitive lattice points; a Fano polytope is not necessarily smooth). We also show that this geometric spectrum is equal to the algebraic spectrum (the spectrum at infinity of a tame Laurent polynomial whose Newton polytope is the polytope alluded to). This gives an explanation and some positive answers to Hertling’s conjecture about the variance of the spectrum of tame regular functions.


Mirror symmetry Toric varieties Polytopes Orbifold cohomology Spectrum of regular tame functions 

Mathematics Subject Classification

32S40 14J33 34M35 14C40 


  1. 1.
    Batyrev, V.: Stringy Hodge numbers of varieties with Gorenstein canonical singularities. In: Saito, M.-H., et al. (eds.) Integrable Systems and Algebraic Geometry. Proceedings of the 41st Taniguchi Symposium, Japan 1997. World Scientific, Singapore, pp. 1–32 (1998)Google Scholar
  2. 2.
    Batyrev, V.: Stringy Hodge numbers and Virasoro algebra. Math. Res. Lett. 7, 155–164 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Batyrev, V., Schaller, K.: Stringy Chern classes of singular toric varieties and their applications. arXiv:1607.04135
  4. 4.
    Beck, M., Robbins, S.: Computing the Continuous Discretely. Springer, New York (2007)Google Scholar
  5. 5.
    Borisov, L., Chen, L., Smith, G.: The orbifold Chow ring of toric Deligne–Mumford stacks. J. Am. Soc. 18(1), 193–215 (2005)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Cox, D., Little, J., Schenck, A.: Toric Varieties, vol. 124. American Mathematical Society, Providence (2010)zbMATHGoogle Scholar
  7. 7.
    Dimca, A.: Monodromy and Hodge theory of regular functions. In: Siersma, D., et al. (eds.) New Developments in Singularity Theory, pp. 257–278. Kluwer Academic Publishers, Dordrecht (2001)CrossRefGoogle Scholar
  8. 8.
    Douai, A.: Global spectra, polytopes and stacky invariants. arXiv:1603.08693
  9. 9.
    Douai, A.: Quantum differential systems and some applications to mirror symmetry. arXiv:1203.5920
  10. 10.
    Douai, A.: Quantum differential systems and rational structures. Manuscr. Math. 145(3), 285–317 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Douai, A., Mann, E.: The small quantum cohomology of a weighted projective space, a mirror D-module and their classical limits. Geometriae Dedicata 164, 187–226 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Douai, A., Sabbah, C.: Gauss–Manin systems, Brieskorn lattices and Frobenius structures I. Ann. Inst. Fourier 53(4), 1055–1116 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Douai, A., Sabbah, C.: Gauss–Manin systems, Brieskorn lattices and Frobenius structures II. In: Hertling, C., Marcolli, M. (eds.) Frobenius Manifolds. Aspects of Mathematics E, vol. 36 (2004)Google Scholar
  14. 14.
    Fulton, W.: Introduction to Toric Varieties. Annals of Mathematics Studies, vol. 131. Princeton University Press, Princeton (1993)Google Scholar
  15. 15.
    Givental, A.: Homological geometry and mirror symmetry. Talk at ICM-94. In: Proceedings of ICM-94 Zurich, pp. 472–480. Birkhäuser, Basel (1995)Google Scholar
  16. 16.
    Hertling, C.: Frobenius Manifolds and Moduli Spaces for Singularities. Cambridge Tracts in Mathematics, vol. 151. Cambridge University Press, Cambridge (2002)Google Scholar
  17. 17.
    Hertling, C.: Frobenius manifolds and variance of the spectral numbers. In: Siersma, D., et al. (eds.) New Developments in Singularity Theory, pp. 235–255. Kluwer Academic Publishers, Dordrecht (2001)CrossRefGoogle Scholar
  18. 18.
    Hori, K., Vafa, C.: Mirror symmetry. arXiv:hep-th/0002222
  19. 19.
    Kouchnirenko, A.G.: Polyèdres de Newton et nombres de Milnor. Invent. Math. 32, 1–31 (1976)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Libgober, A.S., Wood, J.W.: Uniqueness of the complex structure on Kähler manifolds of certain homotopy types. J. Differ. Geom. 32(1), 139–154 (1990)CrossRefzbMATHGoogle Scholar
  21. 21.
    Mustaţă, M., Payne, S.: Ehrhart polynomials and stringy Betti numbers. Mathematische Annalen 333(4), 787–795 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Nemethi, A., Sabbah, C.: Semicontinuity of the spectrum at infinity. Abh. Math. Sem. Univ. Hambg. 69, 25–35 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Reichelt, T., Sevenheck, C.: Logarithmic Frobenius manifolds, hypergeometric systems and quantum \({\cal{D}} \)-modules. J. Algebraic Geom. 24, 201–281 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Sabbah, C.: Hypergeometric periods for a tame polynomial. Portugaliae Mathematica, Nova Serie 63(2), 173–226 (2006)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Stapledon, A.: Weighted Ehrhart theory and orbifold cohomology. Adv. Math. 219, 63–88 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Veys, W.: Arc Spaces, Motivic Integration and Stringy Invariants. Advanced Studies in Pure Mathematics, vol. 43, pp. 529–572. The Mathematical Society of Japan, Tokyo (2006)Google Scholar

Copyright information

© Springer-Verlag GmbH Deutschland 2017

Authors and Affiliations

  1. 1.Université Côte d’Azur, CNRS, LJAD, Parc ValroseNice Cedex 2France

Personalised recommendations