Skip to main content
SpringerLink
Log in

The twistor geometry of parabolic structures in rank two

  • Published:
Proceedings - Mathematical Sciences Aims and scope Submit manuscript

Abstract

Let X be a quasi-projective curve, compactified to (Y, D) with \(X=Y-D\). We construct a Deligne–Hitchin twistor space out of moduli spaces of framed \(\lambda \)-connections of rank 2 over Y with logarithmic singularities and quasi-parabolic structure along D. To do this, one should divide by a Hecke-gauge groupoid. Tame harmonic bundles on X give preferred sections, and the relative tangent bundle along a preferred section has a mixed twistor structure with weights 0, 1, 2. The weight 2 piece corresponds to the deformations of the KMS structure including parabolic weights and the residues of the \(\lambda \)-connection.

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. Balaji V and Seshadri C S, Moduli of parahoric ${\mathscr {G}}$-torsors on a compact Riemann surface, J. Algebraic Geom. 24(1) (2015) 1–49

    Article  MathSciNet  Google Scholar 

  2. Beck F, Biswas I, Heller S and Röser M, Geometry of the space of sections of twistor spaces with circle action, arXiv preprint arXiv:2102.10853 (2021)

  3. Beck F, Heller S and Röser M, Energy of sections of the DeligneHitchin twistor space, Math. Ann. 380(3) (2021) 1169–1214

    Article  MathSciNet  Google Scholar 

  4. Bhosle U, Parabolic vector bundles on curves, Arkiv för matematik 27(1) (1989) 15–22

    Article  MathSciNet  Google Scholar 

  5. Bhosle U and Ramanathan A, Moduli of parabolic $G$-bundles on curves, Math. Zeitschrift 202(2) (1989) 161–180

    Article  MathSciNet  Google Scholar 

  6. Biswas I, Parabolic bundles as orbifold bundles, Duke Math. J. 88(2) (1997) 305–325

    Article  MathSciNet  Google Scholar 

  7. Biswas I, Heller S and Röser M, Real holomorphic sections of the Deligne–Hitchin twistor space, Comm. Math. Phys. 366(3) (2019) 1099–1133

    Article  MathSciNet  Google Scholar 

  8. Biswas I, Inaba M-A, Komyo A and Saito M-H, On the moduli spaces of framed logarithmic connections on a Riemann surface, arXiv preprint arXiv:2103.12121 (2021)

  9. Boalch P, Riemann–Hilbert for tame complex parahoric connections, Transformation Groups 16(1) (2011) 27–50

    Article  MathSciNet  Google Scholar 

  10. Budur N, Lerer L and Wang H, Absolute sets of rigid local systems, arXiv preprint arXiv:2104.00168 (2021)

  11. Deligne P, Equations différentielles à points singuliers réguliers, Lecture Notes in Math., vol. 163 (1970) (Springer)

  12. Esnault H and Viehweg E, Logarithmic de Rham complexes and vanishing theorems, Inventiones 86(1) (1986) 161–194

    Article  MathSciNet  Google Scholar 

  13. Fassarella T and Loray F, Flat parabolic vector bundles on elliptic curves, J. Reine Angew. Math. 761 (2020) 81–122

    Article  MathSciNet  Google Scholar 

  14. Fock V and Goncharov A, Moduli spaces of local systems and higher Teichmüller theory, Publ. Math. IHES 103 (2006) 1–211

    Article  Google Scholar 

  15. Fernandez Herrero A, On the quasicompactness of the moduli stack of logarithmic \(G\)-connections over a curve, arXiv preprint arXiv:2002.11832 (2020)

  16. Hai P H, Dos Santos J P and Tâm P T, Algebraic theory of formal regular-singular connections with parameters, arXiv preprint arXiv:2107.06474 (2021)

  17. Hitchin N, The self-duality equations on a Riemann surface, Proc. London Math. Soc. 3(1) (1987) 59–126

    Article  MathSciNet  Google Scholar 

  18. Hitchin N, Karlhede A, Lindström U and Roček M, Hyperkähler metrics and supersymmetry, Comm. Math. Phys. 108(4) (1987) 535–589

    Article  MathSciNet  Google Scholar 

  19. Holdt M, Hyperkähler metrics on the moduli space of weakly parabolic Higgs bundles, arXiv preprint arXiv:2106.16017 (2021)

  20. Hu Z, Huang P and Zong R, Moduli spaces of parabolic bundles over ${\mathbb{P}}^1$ with five marked points, arXiv preprint arXiv:2108.08994 (2021)

  21. Inaba M-A, Iwasaki K and 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

    Article  MathSciNet  Google Scholar 

  22. Jeffrey L, Extended moduli spaces of flat connections on Riemann surfaces, Math. Ann. 298(1) (1994) 667–692

    Article  MathSciNet  Google Scholar 

  23. Kollár J, Cone theorems and bug-eyed covers, J. Algebraic Geom. 1(2) (1992) 293–323

    MathSciNet  MATH  Google Scholar 

  24. Konno H, Construction of the moduli space of stable parabolic Higgs bundles on a Riemann surface, J. Math. Soc. Japan 45(2) (1993) 253–276

    Article  MathSciNet  Google Scholar 

  25. Maruyama M and Yokogawa K, Moduli of parabolic stable sheaves, Math. Ann. 293(1) (1992) 77–99

    Article  MathSciNet  Google Scholar 

  26. Matsumoto T, Birational geometry of moduli spaces of rank $2$ logarithmic connections, arXiv preprint arXiv:2105.06892 (2021)

  27. Mehta V and Seshadri C S, Moduli of vector bundles on curves with parabolic structures, Math. Ann. 248(3) (1980) 205–239

    Article  MathSciNet  Google Scholar 

  28. Misra S and Singh A, On the relative logarithmic connections and relative residue formula, arXiv preprint arXiv:2108.06321 (2021)

  29. Mochizuki T, Asymptotic behaviour of bame harmonic bundles and an application to pure twistor ${\mathscr {D}}$-modules, Parts 1 and 2, Memoirs AMS, vol. 185 (2007) (Providence: AMS) pp. 869–870

  30. Mochizuki T, Mixed Twistor ${\mathscr {D}}$-Modules, Lecture Notes in Math., vol. 2125 (2015) (Springer)

  31. Nitsure N, Cohomology of the moduli of parabolic vector bundles, Proc. Indian Acad. Sci. – Math. Sci. 95(1) (1986) 61–77

    Article  MathSciNet  Google Scholar 

  32. Nitsure N, Moduli of semistable logarithmic connections, J. Amer. Math. Soc. 6 (1993) 597–609

    Article  MathSciNet  Google Scholar 

  33. Nitsure N, Moduli of regular holonomic ${\mathscr {D}}$-modules with normal crossing singularities, Duke Math. J. 99(1) (1999) 1–39

    Article  MathSciNet  Google Scholar 

  34. Nitsure N and Sabbah C, Moduli of pre-${\mathscr {D}}$-modules, perverse sheaves and the Riemann–Hilbert morphism, I, Math. Ann. 306 (1996) 47–73

  35. Pridham J P, Deforming $\ell $-adic representations of the fundamental group of a smooth variety, J. Algebraic Geom. 15(3) (2006) 415–442

    Article  MathSciNet  Google Scholar 

  36. Seshadri C S, Moduli of vector bundles on curves with parabolic structures, Bull. Amer. Math. Soc. 83(1) (1977) 124–126

    Article  MathSciNet  Google Scholar 

  37. Sabbah C, Polarizable twistor ${\mathscr {D}}$-modules, Astérisque 300 (2005) (Paris: SMF)

  38. Sabbah C, Aspects algébriques de la division des distributions, http://www.math.polytechnique.fr/xups/xups03-03.pdf

  39. Simpson C, Harmonic bundles on noncompact curves, J. Amer. Math. Soc. 3(3) (1990) 713–770

    Article  MathSciNet  Google Scholar 

  40. Simpson C, The Hodge filtration on nonabelian cohomology, in: Algebraic geometry (Santa Cruz, 1995), Proc. Sympos. Pure Math. (1997) (Providence: AMS) pp. 217–281

  41. Simpson C, Mixed twistor structures, preprint arXiv:alg-geom/9705006 (1997)

  42. Simpson C, A weight two phenomenon for the moduli of rank one local systems on open varieties, in: From Hodge theory to integrability and TQFT tt*-geometry, Proc. Sympos. Pure Math., vol. 78 (2008) (Providence: AMS) pp. 175–214

  43. Singh A, Moduli space of logarithmic connections singular over a finite subset of a compact Riemann surface, arXiv preprint arXiv:1912.01288 (2019)

Download references

Acknowledgements

This article is based upon the work supported by a grant from the Institute for Advanced Study. This is supported by the Agence Nationale de la Recherche program 3ia Côte d’Azur ANR-19-P3IA-0002 and Hodgefun ANR-16-CE40-0011, European Research Council Horizons 2020 Grant 670624 (Mai Gehrke’s DuaLL project) and the International Centre for Theoretical Sciences program ICTS/mbrs2020/02. The author would like to thank the many colleagues who have contributed, through numerous discussions and inspiring articles, to this work. He would also like to thank Takuro Mochizuki, for several improvements to a first draft of this work. Many thanks go to the referee for important suggestions and corrections.

Author information

Authors and Affiliations

  1. Université Côte d’Azur, CNRS, LJAD, 06108, Nice, France

    Carlos Simpson

Authors
  1. Carlos Simpson
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Carlos Simpson.

Additional information

Communicated by V. Balaji.

Note from the Chief Editor: This article is part of the “Special Issue in Memory of Professor C S Seshadri”.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Simpson, C. The twistor geometry of parabolic structures in rank two. Proc Math Sci 132, 54 (2022). https://doi.org/10.1007/s12044-022-00696-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12044-022-00696-1

Keywords

2010 Mathematics Subject Classification

Search

Navigation