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

SIGMA 16 (2020), 125, 42 pages      arXiv:1910.13535      https://doi.org/10.3842/SIGMA.2020.125

A Map Between Moduli Spaces of Connections

Frank Loray a and Valente Ramírez b
a) Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France
b) University of Twente, Department of Applied Mathematics, 7500 AE Enschede, The Netherlands

Received December 17, 2019, in final form November 24, 2020; Published online December 02, 2020

Abstract
We are interested in studying moduli spaces of rank 2 logarithmic connections on elliptic curves having two poles. To do so, we investigate certain logarithmic rank 2 connections defined on the Riemann sphere and a transformation rule to lift such connections to an elliptic curve. The transformation is as follows: given an elliptic curve $C$ with elliptic quotient $\pi\colon C\to{\mathbb P}^1$, and the logarithmic connection $(E,\nabla)$ on ${\mathbb P}^1$, we may pullback the connection to the elliptic curve to obtain a new connection $(\pi^*E, \pi^*\nabla)$ on $C$. After suitable birational modifications we bring the connection to a particular normal form. The whole transformation is equivariant with respect to bundle automorphisms and therefore defines a map between the corresponding moduli spaces of connections. The aim of this paper is to describe the moduli spaces involved and compute explicit expressions for the above map in the case where the target space is the moduli space of rank 2 logarithmic connections on an elliptic curve $C$ with two simple poles and trivial determinant.

Key words: moduli spaces; parabolic connection; logarithmic connection; parabolic vector bundle; parabolic Higgs bundle; elliptic curve.

pdf (771 kb)   tex (120 kb)  

References

  1. Arinkin D., Lysenko S., Isomorphisms between moduli spaces of ${\rm SL}(2)$-bundles with connections on ${\bf P}^1\setminus \{x_1,\dots, x_4\}$, Math. Res. Lett. 4 (1997), 181-190.
  2. Arinkin D., Lysenko S., On the moduli of ${\rm SL}(2)$-bundles with connections on ${\bf P}^1\setminus\{x_1,\dots,x_4\}$, Int. Math. Res. Not. 1997 (1997), 983-999.
  3. Biswas I., Mukherjee A., Symplectic structures on moduli spaces of parabolic Higgs bundles and Hilbert scheme, Comm. Math. Phys. 240 (2003), 149-159, arXiv:math.AG/0302207.
  4. Boalch P., Symplectic manifolds and isomonodromic deformations, Adv. Math. 163 (2001), 137-205, arXiv:2002.00052.
  5. Diarra K., Loray F., Ramified covers and tame isomonodromic solutions on curves, Trans. Moscow Math. Soc. 2015 (2015), 219-236, arXiv:1412.8196.
  6. Fassarella T., Loray F., Flat parabolic vector bundles on elliptic curves, J. Reine Angew. Math. 761 (2020), 81-122, arXiv:1707.00820.
  7. Fernández Vargas N., Geometry of the moduli of parabolic bundles on elliptic curves, Trans. Amer. Math. Soc., to appear, arXiv:1611.05417.
  8. Goldman W.M., The symplectic nature of fundamental groups of surfaces, Adv. Math. 54 (1984), 200-225.
  9. Goldman W.M., Ergodic theory on moduli spaces, Ann. of Math. 146 (1997), 475-507.
  10. Heu V., Loray F., Flat rank two vector bundles on genus two curves, Mem. Amer. Math. Soc. 259 (2019), v+103 pages, arXiv:1401.2449.
  11. Hitchin N.J., Twistor spaces, Einstein metrics and isomonodromic deformations, J. Differential Geom. 42 (1995), 30-112.
  12. Inaba M.-A., Moduli of parabolic connections on curves and the Riemann-Hilbert correspondence, J. Algebraic Geom. 22 (2013), 407-480, arXiv:math.AG/0602004.
  13. Inaba M.-A., Iwasaki K., Saito M.-H., Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI. I, Publ. Res. Inst. Math. Sci. 42 (2006), 987-1089, arXiv:math.AG/0309342.
  14. Iwasaki K., Fuchsian moduli on a Riemann surface — its Poisson structure and Poincaré-Lefschetz duality, Pacific J. Math. 155 (1992), 319-340.
  15. Iwasaki K., A modular group action on cubic surfaces and the monodromy of the Painlevé VI equation, Proc. Japan Acad. Ser. A Math. Sci. 78 (2002), 131-135.
  16. Loray F., Isomonodromic deformations of Lamé connections, Painlevé VI equation and Okamoto symmetry, Izv. Math. 80 (2016), 113-166, arXiv:1410.4976.
  17. Loray F., Saito M.H., Lagrangian fibrations in duality on moduli spaces of rank 2 logarithmic connections over the projective line, Int. Math. Res. Not. 2015 (2015), 995-1043, arXiv:1302.4113.
  18. Loray F., van der Put M., Ulmer F., The Lamé family of connections on the projective line, Ann. Fac. Sci. Toulouse Math. 17 (2008), 371-409.
  19. Machu F.-X., Monodromy of a class of logarithmic connections on an elliptic curve, SIGMA 3 (2007), 082, 31 pages, arXiv:0708.2186.
  20. Mazzocco M., Vidunas R., Cubic and quartic transformations of the sixth Painlevé equation in terms of Riemann-Hilbert correspondence, Stud. Appl. Math. 130 (2013), 17-48, arXiv:1011.6036.
  21. Nitsure N., Moduli of semistable logarithmic connections, J. Amer. Math. Soc. 6 (1993), 597-609.
  22. Ramirez V., Connections on elliptic curves, GitHub repository, 2019, available at https://github.com/valentermz/Connections-on-elliptic-curves.
  23. The Sage Developers, The Sage Mathematics Software System, Version 8.7, 2019, available at https://www.sagemath.org.
  24. Tsuda T., Okamoto K., Sakai H., Folding transformations of the Painlevé equations, Math. Ann. 331 (2005), 713-738.
  25. Tu L.W., Semistable bundles over an elliptic curve, Adv. Math. 98 (1993), 1-26.

Previous article  Next article  Contents of Volume 16 (2020)