Skip to main content
SpringerLink
Log in

Geometric variations of local systems and elliptic surfaces

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

Geometric variations of local systems are families of variations of Hodge structure; they typically correspond to fibrations of Kähler manifolds for which each fibre itself is fibred by codimension one Kähler manifolds. In this article, we introduce the formalism of geometric variations of local systems and then specialize the theory to study families of elliptic surfaces. We interpret a construction of twisted elliptic surface families used by Besser—Livné in terms of the middle convolution functor, and use explicit methods to calculate the variations of Hodge structure underlying the universal families of MN-polarized K3 surfaces. Finally, we explain the connection between geometric variations of local systems and geometric isomonodromic deformations, which were originally considered by the first author in 1999.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. Besser and R. Livné, Universal Kummer families over Shimura curves, in Arithmetic and Geometry of K3 Surfaces and Calabi—Yau Threefolds Fields Institute Communications, Vol. 67, Springer, New York, 2013, pp. 201–265.

    Chapter  Google Scholar 

  2. A. Besserand R. Livné, Picard—Fuchs equations of families of QM abelian surfaces, Communications in Number Theory and Physics 12 (2018), 829–856.

    Article  MathSciNet  Google Scholar 

  3. J. S. Birman, Braids, Links, and Mapping Class Groups, Annals of Mathematics Studies, Vol. 82, Princeton University Press, Pronceton, NJ, 1975.

    Book  Google Scholar 

  4. M. Bogner and S. Reiter, On symplectically rigid local systems of rank four and Calabi—Yau operators, Journal of Symbolic Computation 48 (2013), 64–100.

    Article  MathSciNet  Google Scholar 

  5. J. Carlson, S. Müller-Stach and C. Peters, Period Mappings and Period Domains, Cambridge Studies in Advanced Mathematics, Vol. 168, Cambridge University Press, Cambridge, 2017.

    Book  Google Scholar 

  6. A. Clingher, C. F. Doran, J. Lewis and U. Whitcher, Normal Forms, K3 surface moduli, and modular parametrizations, in Groups and Symmetries, CRM Proceedings & Lecture Notes, Vol. 47, American Mathematical Society, Providence, RI, 2009, pp. 81–98.

    Chapter  Google Scholar 

  7. P. Deligne, Équations différentielles à points singuliers réguliers, Lecture Notes in Mathematics, Vol. 163, Springer, Berlin—Heidelberg, 1970.

    Book  Google Scholar 

  8. M. Dettweiler, On the middle convolution of local systems, https://arxiv.org/abs/0810.3334.

  9. M. Dettweiler and S. Wewers, Variation of local systems and parabolic cohomology, Israel Journal of Mathematics 156 (2006), 157–185.

    Article  MathSciNet  Google Scholar 

  10. M. Dettweiler and S. Wewers, Variation of parabolic cohomology and Poincare duality, in Groupes de Galois arithmétiques et différenti’els, Séminaires et Congrès, Vol. 13, Société Mathématique de France, Paris, 2006, pp. 145–164.

    Google Scholar 

  11. I. V. Dolgachev, Mirror symmetry for lattice polarized K3 surfaces, Journal of Mathematical Sciences 81 (1996), 2599–2630.

    Article  MathSciNet  Google Scholar 

  12. C. F. Doran, Picard—Fuchs uniformization and geometric isomonodromic deformations: Modularity and variation of the mirror map, Ph. D. Thesis, Harvard University, Cambridge, MA, 1999.

    Google Scholar 

  13. C. F. Doran, Algebraic and geometric isomonodromic deformations, Journal of Differential Geometry 59 (2001), 33–85.

    Article  MathSciNet  Google Scholar 

  14. C. F. Doran, A. Harder, A. Y. Novoseltsev and A. Thompson, Families of lattice polarized K3 surfaces with monodromy, International Mathematics Research Notices 2015 (2015), 12265–12318.

    MathSciNet  Google Scholar 

  15. C. F. Doran, A. Harder, A. Y. Novoseltsev and A. Thompson, Calabi—Yau threefolds fibred by high rank lattice polarized K3 surfaces, Mathematische Zeitschrift 294 (2020), 783–815.

    Article  MathSciNet  Google Scholar 

  16. C. F. Doran, J. Kostiuk and F. You, The Doran—Harder—Thompson conjecture for toric complete intersections, https://arxiv.org/abs/1910.11955.

  17. C. F. Doran and A. Malmendier, Calabi—Yau manifolds realizing symplectically rigid monodromy tuples, Advances in Theoretical and Mathematical Physics 23 (2019), 1271–1359.

    Article  MathSciNet  Google Scholar 

  18. S. Herfurtner, Elliptic surfaces with four singular fibres, Mathematische Annalen 291 (1991), 319–342.

    Article  MathSciNet  Google Scholar 

  19. N. O. Ilten, J. Lewis and V. Przyjalkowski, Toric degenerations of Fano threefolds giving weak Landau—Ginzburg models, Journal of Algebra 374 (2013), 104–121.

    Article  MathSciNet  Google Scholar 

  20. N. M. Katz, Rigid Local Systems, Annals of Mathematics Studies, Vol. 139, Princeton University Press, Princeton, NJ, 1996.

    Book  Google Scholar 

  21. K. Kodaira, On compact Complex Analytic Surfaces. I, Annals of Mathematics 71 (1960), 111–152.

    Article  MathSciNet  Google Scholar 

  22. J. A. Kostiuk, Geometric Variations of Local Systems, Ph D. thesis, University of Alberta, Edmonton, AB, 2018.

    Google Scholar 

  23. B. Malgrange, Sur les deformations isomonodromiques. I. Singularités regulières, in Mathematics and Physics (Paris, 1979/1982), Progress in Mathematics, Vol. 37, Birkhäuser, Boston, MA, 1983, pp. 401–426.

    Google Scholar 

  24. M. Schütt and T. Shioda, Elliptic surfaces, in Algebraic geometry in East Asia—Seoul 2008, Advanced Studies in Pure Mathematics, Vol. 60, Mathematical Society of Japan, Tokyo, 2010, pp. 51–160.

    Chapter  Google Scholar 

  25. C. T. Simpson, Katz’s middle convolution algorithm, Pure and Applied Mathematics Quarterly 5 (2009), 781–852.

    Article  MathSciNet  Google Scholar 

  26. P. F. Stiller, Elliptic curves over function fields and the Picard number, American Journal of Mathematics 102 (1980), 565–593.

    Article  MathSciNet  Google Scholar 

  27. P. F. Stiller, Differential equations associated with elliptic surfaces, Journal of the Mathematical Society of Japan 33 (1981), 203–233.

    Article  MathSciNet  Google Scholar 

  28. P. Stiller, Special values of Dirichlet series, monodromy, and the periods of automorphic forms, Memoirs of the American Mathematical Society 49 (1984).

  29. P. F. Stiller, The Picard numbers of elliptic surfaces with many symmetries, Pacific Journal of Mathematics 128 (1987), 157–189.

    Article  MathSciNet  Google Scholar 

  30. A. Thompson, S. A. Filippini and H. Ruddat, An introduction to Hodge structures, in Calabi—Yau Varieties: Arithmetic, Geometry and Physics, Fields Institute Monographs, Vol. 34, Springer, New York, 2015, pp. 83–130.

    Google Scholar 

  31. S. Zucker, Hodge theory with degenerating coefficients: L2cohomology in the Poincaré metric, Annals of Mathematics 109 (1979), 415–476.

    Article  MathSciNet  Google Scholar 

  32. S. Zucker and D. A. Cox, Intersection numbers of sections of elliptic surfaces, Inventiones mathematicae 53 (1979), 1–44.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, T6G 2G1, Canada

    Charles F. Doran

  2. Center of Mathematical Sciences and Applications, Harvard University, Cambridge, MA, 02138, USA

    Charles F. Doran

  3. Department of Mathematics, Brown University, Box 1917, 151 Thayer Street, Providence, RI, 02912, USA

    Jordan Kostiuk

Authors
  1. Charles F. Doran
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jordan Kostiuk
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jordan Kostiuk.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Doran, C.F., Kostiuk, J. Geometric variations of local systems and elliptic surfaces. Isr. J. Math. 258, 1–79 (2023). https://doi.org/10.1007/s11856-023-2466-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-023-2466-z

Search

Navigation