Introduction to Nonabelian Hodge Theory

Flat connections, Higgs bundles and complex variations of Hodge structure
  • Alberto García-RabosoEmail author
  • Steven RayanEmail author
Part of the Fields Institute Monographs book series (FIM, volume 34)


Hodge theory bridges the topological, smooth and holomorphic worlds. In the abelian case of the preceding chapter, these are embodied by the Betti, de Rham and Dolbeault cohomology groups, respectively, of a smooth compact Kähler manifold, X.



We thank Chuck Doran, David Morrison, Radu Laza, and Johannes Walcher for organizing the Workshop on Hodge Theory in String Theory, as well as Alan Thompson for arranging the concentrated graduate course in which we gave these lectures. We are indebted to Noriko Yui for encouraging us to contribute these notes, and for organizing a vibrant and productive thematic program at the Fields Institute. These notes benefited from useful comments and suggestions by Philip Boalch, Tamás Hausel, Tony Pantev, and an anonymous referee. We also thank Marco Gualtieri and Lisa Jeffrey for their support and for providing a stimulating working environment in Toronto.


  1. 1.
    Adams, M.R., Harnad, J., Hurtubise, J.: Isospectral Hamiltonian flows in finite and infinite dimensions. II. Integration of flows. Commun. Math. Phys. 134(3), 555–585 (1990).
  2. 2.
    Atiyah, M.F., Bott, R.: The Yang-Mills equations over Riemann surfaces. Philos. Trans. R. Soc. Lond. Ser. A 308(1505), 523–615 (1983). doi:10.1098/rsta.1983.0017.
  3. 3.
    Beauville, A.: Jacobiennes des courbes spectrales et systèmes hamiltoniens complètement intégrables. Acta Math. 164(3–4), 211–235 (1990). doi:10.1007/BF02392754.
  4. 4.
    Beauville, A., Narasimhan, M.S., Ramanan, S.: Spectral curves and the generalised theta divisor. J. Reine Angew. Math. 398, 169–179 (1989). doi:10.1515/crll.1989.398.169.
  5. 5.
    Biquard, O.: Fibrés de Higgs et connexions intégrables: le cas logarithmique (diviseur lisse). Ann. Sci. École Norm. Sup. (4) 30(1), 41–96 (1997). doi:10.1016/S0012-9593(97)89915-6.
  6. 6.
    Biquard, O., Boalch, P.: Wild non-abelian Hodge theory on curves. Compos. Math. 140(1), 179–204 (2004). doi:10.1112/S0010437X03000010.
  7. 7.
    Boalch, P.: Hyperkahler manifolds and nonabelian Hodge theory of (irregular) curves. arXiv:1203.6607 [math.AG] (2012).
  8. 8.
    Boden, H.U., Yokogawa, K.: Moduli spaces of parabolic Higgs bundles and parabolic K(D) pairs over smooth curves. I. Int. J. Math. 7(5), 573–598 (1996). doi:10.1142/S0129167X96000311.
  9. 9.
    Bonsdorff, J.: Autodual connection in the Fourier transform of a Higgs bundle. Asian J. Math. 14(2), 153–173 (2010). doi:10.4310/AJM.2010.v14.n2.a1.
  10. 10.
    Bottacin, F.: Symplectic geometry on moduli spaces of stable pairs. Ann. Sci. École Norm. Sup. (4) 28(4), 391–433 (1995).
  11. 11.
    Bradlow, S.B., García-Prada, O., Gothen, P.B.: Moduli spaces of holomorphic triples over compact Riemann surfaces. Math. Ann. 328(1–2), 299–351 (2004). doi:10.1007/s00208-003-0484-z.
  12. 12.
    Chuang, W.Y., Diaconescu, D.E., Pan, G.: Wallcrossing and cohomology of the moduli space of Hitchin pairs. Commun. Number Theory Phys. 5(1), 1–56 (2011). doi:10.4310/CNTP.2011.v5.n1.a1.
  13. 13.
    Corlette, K.: Flat G-bundles with canonical metrics. J. Differ. Geom. 28(3), 361–382 (1988).
  14. 14.
    Diederich, K., Ohsawa, T.: Harmonic mappings and disc bundles over compact Kähler manifolds. Publ. Res. Inst. Math. Sci. 21(4), 819–833 (1985). doi:10.2977/prims/1195178932.
  15. 15.
    Donagi, R.: Spectral covers. In: Current Topics in Complex Algebraic Geometry (Berkeley, CA, 1992/93). Mathematical Sciences Research Institute publications, vol. 28, pp. 65–86. Cambridge University Press, Cambridge (1995)Google Scholar
  16. 16.
    Donagi, R., Markman, E.: Spectral covers, algebraically completely integrable, Hamiltonian systems, and moduli of bundles. In: Integrable Systems and Quantum Groups (Montecatini Terme, 1993). Lecture Notes in Mathematics, vol. 1620, pp. 1–119. Springer, Berlin (1996). doi:10.1007/BFb0094792.
  17. 17.
    Donagi, R., Pantev, T.: Geometric Langlands and non-abelian Hodge theory. In: Surveys in Differential Geometry. Vol. XIII. Geometry, Analysis, and Algebraic Geometry: Forty years of the Journal of Differential Geometry. Surveys in Differential Geometry, vol. 13, pp. 85–116. International Press, Somerville (2009). doi:10.4310/SDG.2008.v13.n1.a3.
  18. 18.
    Donagi, R., Pantev, T.: Langlands duality for Hitchin systems. Invent. Math. 189(3), 653–735 (2012). doi:10.1007/s00222-012-0373-8.
  19. 19.
    Donaldson, S.K.: Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles. Proc. Lond. Math. Soc. (3) 50(1), 1–26 (1985). doi:10.1112/plms/s3-50.1.1.
  20. 20.
    Donaldson, S.K.: Infinite determinants, stable bundles and curvature. Duke Math. J. 54(1), 231–247 (1987). doi:10.1215/S0012-7094-87-05414-7.
  21. 21.
    Donaldson, S.K.: Twisted harmonic maps and the self-duality equations. Proc. Lond. Math. Soc. (3) 55(1), 127–131 (1987). doi:10.1112/plms/s3-55.1.127.
  22. 22.
    Frenkel, E.: Gauge theory and Langlands duality. Astérisque (332), Exp. No. 1010, ix–x, 369–403 (2010)Google Scholar
  23. 23.
    García-Prada, O., Gothen, P.B., Muñoz, V.: Betti numbers of the moduli space of rank 3 parabolic Higgs bundles. Mem. Am. Math. Soc. 187(879), viii+80 (2007). doi:10.1090/memo/0879.
  24. 24.
    García-Prada, O., Heinloth J., Schmitt, A.: On the motives of moduli of chains and Higgs bundles. J. Eur. Math. Soc. (JEMS) 16(12), 2617–2668 (2014). doi:10.4171/JEMS/494. Scholar
  25. 25.
    Goldman, W.M., Xia, E.Z.: Rank one Higgs bundles and representations of fundamental groups of Riemann surfaces. Mem. Am. Math. Soc. 193(904), viii+69 (2008). doi:10.1090/memo/0904.
  26. 26.
    Gothen, P.B.: The Betti numbers of the moduli space of stable rank 3 Higgs bundles on a Riemann surface. Int. J. Math. 5(6), 861–875 (1994). doi:10.1142/S0129167X94000449.
  27. 27.
    Gothen, P.B.: The topology of Higgs bundle moduli spaces. Ph.D. thesis, University of Warwick (1995)Google Scholar
  28. 28.
    Harder, G., Narasimhan, M.S.: On the cohomology groups of moduli spaces of vector bundles on curves. Math. Ann. 212, 215–248 (1974/75)Google Scholar
  29. 29.
    Hausel, T.: Geometry of the moduli space of Higgs bundles. Ph.D. thesis, Cambridge University (1998)Google Scholar
  30. 30.
    Hausel, T.: Global topology of the Hitchin system. In: Farkas, G., Morrison, I. (eds.) Handbook of Moduli II. International Press, Somerville (2013)Google Scholar
  31. 31.
    Hausel, T., Rodriguez-Villegas, F.: Mixed Hodge polynomials of character varieties. Invent. Math. 174(3), 555–624 (2008). doi:10.1007/s00222-008-0142-x. With an appendix by Nicholas M. Katz
  32. 32.
    Hausel, T., Thaddeus, M.: Mirror symmetry, Langlands duality, and the Hitchin system. Invent. Math. 153(1), 197–229 (2003). doi:10.1007/s00222-003-0286-7.
  33. 33.
    Hausel, T., Thaddeus, M.: Relations in the cohomology ring of the moduli space of rank 2 Higgs bundles. J. Am. Math. Soc. 16(2), 303–327 (electronic) (2003). doi:10.1090/S0894-0347-02-00417-4.
  34. 34.
    Hausel, T., Thaddeus, M.: Generators for the cohomology ring of the moduli space of rank 2 Higgs bundles. Proc. Lond. Math. Soc. (3) 88(3), 632–658 (2004). doi:10.1112/S0024611503014618.
  35. 35.
    Hitchin, N.J.: The self-duality equations on a Riemann surface. Proc. Lond. Math. Soc. (3) 55(1), 59–126 (1987). doi:10.1112/plms/s3-55.1.59. 3-55.1.59
  36. 36.
    Hitchin, N.J.: Stable bundles and integrable systems. Duke Math. J. 54(1), 91–114 (1987). doi:10.1215/S0012-7094-87-05408-1.
  37. 37.
    Hitchin, N.J.: Langlands duality and \(G_{2}\) spectral curves. Q. J. Math. 58(3), 319–344 (2007). doi:10.1093/qmath/ham016.
  38. 38.
    Hitchin, N.J.: Generalized holomorphic bundles and the B-field action. J. Geom. Phys. 61(1), 352–362 (2011). doi:10.1016/j.geomphys.2010.10.014.
  39. 39.
    Hitchin, N.J., Segal, G.B., Ward, R.S.: Integrable systems. Twistors, loop groups, and Riemann surfaces, Lectures from the Instructional Conference held at the University of Oxford, Oxford, September 1997. Graduate Texts in Mathematics, 4, x+136 (1999)Google Scholar
  40. 40.
    Huybrechts, D.: Complex Geometry. An Introduction. Universitext. Springer, Berlin (2005)Google Scholar
  41. 41.
    Jost, J., Yau, S.T.: Harmonic mappings and Kähler manifolds. Math. Ann. 262(2), 145–166 (1983). doi:10.1007/BF01455308.
  42. 42.
    Jost, J., Yau, S.T.: Harmonic maps and group representations. In: Differential Geometry. Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 52, pp. 241–259. Longman Science and Technology, Harlow (1991)Google Scholar
  43. 43.
    Jost, J., Zuo, K.: Harmonic maps of infinite energy and rigidity results for representations of fundamental groups of quasiprojective varieties. J. Differ. Geom. 47(3), 469–503 (1997).
  44. 44.
    Kapustin, A., Witten, E.: Electric-magnetic duality and the geometric Langlands program. Commun. Number Theory Phys. 1(1), 1–236 (2007). doi:10.4310/CNTP.2007.v1.n1.a1.
  45. 45.
    Kobayashi, S.: Differential Geometry of Complex Vector Bundles. Publications of the Mathematical Society of Japan, vol. 15. Princeton University Press, Princeton; Iwanami Shoten, Tokyo (1987)Google Scholar
  46. 46.
    Konno, H.: Construction of the moduli space of stable parabolic Higgs bundles on a Riemann surface. J. Math. Soc. Jpn. 45(2), 253–276 (1993). doi:10.2969/jmsj/04520253.
  47. 47.
    Koszul, J.L., Malgrange, B.: Sur certaines structures fibrées complexes. Arch. Math. (Basel) 9, 102–109 (1958)Google Scholar
  48. 48.
    Logares, M., Martens, J.: Moduli of parabolic Higgs bundles and Atiyah algebroids. J. Reine Angew. Math. 649, 89–116 (2010). doi:10.1515/CRELLE.2010.090.
  49. 49.
    Markman, E.: Spectral curves and integrable systems. Compos. Math. 93(3), 255–290 (1994).
  50. 50.
    Markman, E.: Generators of the cohomology ring of moduli spaces of sheaves on symplectic surfaces. J. Reine Angew. Math. 544, 61–82 (2002). doi:10.1515/crll.2002.028.
  51. 51.
    Mehta, V.B., Ramanathan, A.: Restriction of stable sheaves and representations of the fundamental group. Invent. Math. 77(1), 163–172 (1984). doi:10.1007/BF01389140.
  52. 52.
    Mehta, V.B., Seshadri, C.S.: Moduli of vector bundles on curves with parabolic structures. Math. Ann. 248(3), 205–239 (1980). doi:10.1007/BF01420526.
  53. 53.
    Mochizuki, T.: Kobayashi-Hitchin correspondence for tame harmonic bundles and an application. Astérisque (309), viii+117 (2006)Google Scholar
  54. 54.
    Mochizuki, T.: Kobayashi-Hitchin correspondence for tame harmonic bundles. II. Geom. Topol. 13(1), 359–455 (2009). doi:10.2140/gt.2009.13.359.
  55. 55.
    Mochizuki, T.: Wild harmonic bundles and wild pure twistor D-modules. Astérisque (340), x+607 (2011)Google Scholar
  56. 56.
    Nakajima, H.: Hyper-Kähler structures on moduli spaces of parabolic Higgs bundles on Riemann surfaces. In: Moduli of Vector Bundles (Sanda, 1994; Kyoto, 1994). Lecture Notes in Pure and Applied Mathematics, vol. 179, pp. 199–208. Dekker, New York (1996)Google Scholar
  57. 57.
    Narasimhan, M.S., Seshadri, C.S.: Stable and unitary vector bundles on a compact Riemann surface. Ann. Math. (2) 82, 540–567 (1965)Google Scholar
  58. 58.
    Nasatyr, B., Steer, B.: Orbifold Riemann surfaces and the Yang-Mills-Higgs equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 22(4), 595–643 (1995).
  59. 59.
    Newlander, A., Nirenberg, L.: Complex analytic coordinates in almost complex manifolds. Ann. Math. (2) 65, 391–404 (1957)Google Scholar
  60. 60.
    Ngô, B.C.: Le lemme fondamental pour les algèbres de Lie. Publ. Math. Inst. Hautes Études Sci. (111), 1–169 (2010). doi:10.1007/s10240-010-0026-7.
  61. 61.
    Nitsure, N.: Moduli space of semistable pairs on a curve. Proc. Lond. Math. Soc. (3) 62(2), 275–300 (1991). doi:10.1112/plms/s3-62.2.275.
  62. 62.
    Rayan, S.: Co-Higgs bundles on \(\mathbb{P}^{1}\). N.Y. J. Math. 19, 925–945 (2013).
  63. 63.
    Seshadri, C.S.: Space of unitary vector bundles on a compact Riemann surface. Ann. Math. (2) 85, 303–336 (1967)Google Scholar
  64. 64.
    Simpson, C.T.: Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization. J. Am. Math. Soc. 1(4), 867–918 (1988). doi:10.2307/1990994.
  65. 65.
    Simpson, C.T.: Harmonic bundles on noncompact curves. J. Am. Math. Soc. 3(3), 713–770 (1990). doi:10.2307/1990935.
  66. 66.
    Simpson, C.T.: Nonabelian Hodge theory. In: Proceedings of the International Congress of Mathematicians, Kyoto, 1990, Vol. I, II, pp. 747–756. Mathematical Society of Japan, Tokyo (1991)Google Scholar
  67. 67.
    Simpson, C.T.: Higgs bundles and local systems. Inst. Hautes Études Sci. Publ. Math. (75), 5–95 (1992).
  68. 68.
    Simpson, C.T.: Moduli of representations of the fundamental group of a smooth projective variety. I. Inst. Hautes Études Sci. Publ. Math. (79), 47–129 (1994).
  69. 69.
    Simpson, C.T.: Moduli of representations of the fundamental group of a smooth projective variety. II. Inst. Hautes Études Sci. Publ. Math. (80), 5–79 (1995/1994).
  70. 70.
    Strominger, A., Yau, S.T., Zaslow, E.: Mirror symmetry is T-duality. Nucl. Phys. B 479(1-2), 243–259 (1996). doi:10.1016/0550-3213(96)00434-8.
  71. 71.
    Thaddeus, M.: Stable pairs, linear systems and the Verlinde formula. Invent. Math. 117(2), 317–353 (1994). doi:10.1007/BF01232244.
  72. 72.
    Uhlenbeck, K., Yau, S.T.: On the existence of Hermitian-Yang-Mills connections in stable vector bundles. Commun. Pure Appl. Math. 39(S, suppl.), S257–S293 (1986). doi:10.1002/cpa.3160390714. Frontiers of the mathematical sciences: 1985 (New York, 1985)
  73. 73.
    Wells Jr., R.O.: Differential Analysis on Complex Manifolds. Graduate Texts in Mathematics, vol. 65, 3rd edn. Springer, New York (2008). doi:10.1007/978-0-387-73892-5. With a new appendix by Oscar García-Prada
  74. 74.
    Witten, E.: Gauge theory and wild ramification. Anal. Appl. (Singap.) 6(4), 429–501 (2008). doi:10.1142/S0219530508001195.

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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TorontoTorontoCanada

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