Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 045, 23 pages      arXiv:1612.09439
Contribution to the Special Issue on Modular Forms and String Theory in honor of Noriko Yui

Hodge Numbers from Picard-Fuchs Equations

Charles F. Doran a, Andrew Harder b and Alan Thompson cd
a) Department of Mathematical and Statistical Sciences, 632 CAB, University of Alberta, Edmonton, AB, T6G 2G1, Canada
b) Department of Mathematics, University of Miami, 1365 Memorial Drive, Ungar 515, Coral Gables, FL, 33146, USA
c) Mathematics Institute, Zeeman Building, University of Warwick, Coventry, CV4 7AL, UK
d) DPMMS, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WB, UK

Received January 20, 2017, in final form June 12, 2017; Published online June 18, 2017

Given a variation of Hodge structure over $\mathbb{P}^1$ with Hodge numbers $(1,1,\dots,1)$, we show how to compute the degrees of the Deligne extension of its Hodge bundles, following Eskin-Kontsevich-Möller-Zorich, by using the local exponents of the corresponding Picard-Fuchs equation. This allows us to compute the Hodge numbers of Zucker's Hodge structure on the corresponding parabolic cohomology groups. We also apply this to families of elliptic curves, K3 surfaces and Calabi-Yau threefolds.

Key words: variation of Hodge structures; Calabi-Yau manifolds.

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  1. Brav C., Thomas H., Thin monodromy in Sp(4), Compos. Math. 150 (2014), 333-343, arXiv:1210.0523.
  2. Cox D.A., Zucker S., Intersection numbers of sections of elliptic surfaces, Invent. Math. 53 (1979), 1-44.
  3. Del Angel P.L., Müller-Stach S., Van Straten D., Zuo K., Hodge classes associated to 1-parameter families of Calabi-Yau 3-folds, Acta Math. Vietnam. 35 (2010), 7-22, arXiv:0911.0277.
  4. Deligne P., Équations différentielles à points singuliers réguliers, Lecture Notes in Math., Vol. 163, Springer-Verlag, Berlin - New York, 1970.
  5. Doran B., Doran C.F., Harder A., Picard-Fuchs uniformization of modular subvarieties, in Uniformization, Riemann-Hilbert Correspondence, Calabi-Yau Manifolds, and Picard-Fuchs Equations (Institut Mittag-Leffler, 2017), Editors L. Ji, S.-T. Yau, to appear.
  6. Doran C.F., Picard-Fuchs uniformization: modularity of the mirror map and mirror-moonshine, in The Arithmetic and Geometry of Algebraic Cycles (Banff, AB, 1998), CRM Proc. Lecture Notes, Vol. 24, Amer. Math. Soc., Providence, RI, 2000, 257-281, math.AG/9812162.
  7. Doran C.F., Harder A., Novoseltsev A.Y., Thompson A., Families of lattice polarized K3 surfaces with monodromy, Int. Math. Res. Not. 2015 (2015), 12265-12318, arXiv:1312.6434.
  8. Doran C.F., Harder A., Novoseltsev A.Y., Thompson A., Calabi-Yau threefolds fibred by Kummer surfaces associated to products of elliptic curves, in String-Math 2014, Proc. Sympos. Pure Math., Vol. 93, Amer. Math. Soc., Providence, RI, 2016, 263-287, arXiv:1501.04024.
  9. Doran C.F., Harder A., Novoseltsev A.Y., Thompson A., Calabi-Yau threefolds fibred by mirror quartic K3 surfaces, Adv. Math. 298 (2016), 369-392, arXiv:1501.04019.
  10. Doran C.F., Harder A., Novoseltsev A.Y., Thompson A., Calabi-Yau threefolds fibred by high rank lattice Polarized K3 surfaces, arXiv:1701.03279.
  11. Doran C.F., Malmendier A., Calabi-Yau manifolds realizing symplectically rigid monodromy tuples, arXiv:1503.07500.
  12. Doran C.F., Morgan J.W., Mirror symmetry and integral variations of Hodge structure underlying one-parameter families of Calabi-Yau threefolds, in Mirror Symmetry. V, AMS/IP Stud. Adv. Math., Vol. 38, Editors J.D. Lewis, S.-T. Yau, N. Yui, Amer. Math. Soc., Providence, RI, 2006, 517-537, math.AG/0505272.
  13. Eskin A., Kontsevich M., Moeller M., Zorich A., Lower bounds for Lyapunov exponents of flat bundles on curves, arXiv:1609.01170.
  14. Filip S., Families of K3 surfaces and Lyapunov exponents, arXiv:1412.1779.
  15. Fougeron C., Parabolic degrees and Lyapunov exponents for hypergeometric local systems, arXiv:1701.08387.
  16. Fujino O., A canonical bundle formula for certain algebraic fiber spaces and its applications, Nagoya Math. J. 172 (2003), 129-171.
  17. Green M., Griffiths P., Kerr M., Some enumerative global properties of variations of Hodge structures, Mosc. Math. J. 9 (2009), 469-530.
  18. Griffiths P.A., Periods of integrals on algebraic manifolds. I. Construction and properties of the modular varieties, Amer. J. Math. 90 (1968), 568-626.
  19. Griffiths P.A., Periods of integrals on algebraic manifolds. II. Local study of the period mapping, Amer. J. Math. 90 (1968), 805-865.
  20. Griffiths P.A., On the periods of certain rational integrals. I, Ann. of Math. 90 (1969), 460-495.
  21. Griffiths P.A., On the periods of certain rational integrals. II, Ann. of Math. 90 (1969), 496-541.
  22. Griffiths P.A., Periods of integrals on algebraic manifolds. III. Some global differential-geometric properties of the period mapping, Inst. Hautes Études Sci. Publ. Math. 38 (1970), 125-180.
  23. Hollborn H., Müller-Stach S., Hodge numbers for the cohomology of Calabi-Yau type local systems, in Algebraic and Complex Geometry, Springer Proc. Math. Stat., Vol. 71, Springer, Cham, 2014, 225-240, arXiv:1302.3047.
  24. Kappes A., Lyapunov exponents of rank 2-variations of Hodge structures and modular embeddings, Ann. Inst. Fourier (Grenoble) 64 (2014), 2037-2066, arXiv:1303.1088.
  25. Miranda R., The basic theory of elliptic surfaces, Dottorato di Ricerca in Matematica, ETS Editrice, Pisa, 1989.
  26. Morrison D.R., Walcher J., D-branes and normal functions, Adv. Theor. Math. Phys. 13 (2009), 553-598, arXiv:0709.4028.
  27. Narumiya N., Shiga H., The mirror map for a family of $K3$ surfaces induced from the simplest 3-dimensional reflexive polytope, in Proceedings on Moonshine and Related Topics (Montréal, QC, 1999), CRM Proc. Lecture Notes, Vol. 30, Amer. Math. Soc., Providence, RI, 2001, 139-161.
  28. Singh S., Arithmeticity of four hypergeometric monodromy groups associated to Calabi-Yau threefolds, Int. Math. Res. Not. 2015 (2015), 8874-8889, arXiv:1308.4039.
  29. Singh S., Venkataramana T.N., Arithmeticity of certain symplectic hypergeometric groups, Duke Math. J. 163 (2014), 591-617, arXiv:1208.6460.
  30. Walcher J., Extended holomorphic anomaly and loop amplitudes in open topological string, Nuclear Phys. B 817 (2009), 167-207, arXiv:0705.4098.
  31. Zucker S., Hodge theory with degenerating coefficients. $L_{2}$ cohomology in the Poincaré metric, Ann. of Math. 109 (1979), 415-476.

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