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Hodge polynomials of SL\((2,\mathbb{C })\)-character varieties for curves of small genus

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Abstract

We compute the E-polynomials of the moduli spaces of representations of the fundamental group of a complex curve into \(\mathrm{\text{ SL} }(2,{\mathbb{C }}),\) for the case of small genus \(g,\) and allowing the holonomy around a fixed point to be any matrix of \(\mathrm{\text{ SL} }(2,{\mathbb{C }}),\) that is \(\,\text{ Id}\,, -\,\text{ Id}\,,\) diagonalisable, or of either of the two Jordan types. For this, we introduce a new geometric technique, based on stratifying the space of representations, and on the analysis of the behaviour of the E-polynomial under fibrations.

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References

  1. Arapura, D.: The Leray spectral sequence is motivic. Invent. Math. 160, 567–589 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. Boden, H., Yokogawa, K.: Moduli spaces of parabolic Higgs bundles and parabolic K(D) pairs over smooth curves: I. Int. J. Math. 7, 573–598 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  3. de Cataldo, A., Hausel, T., Migliorini, L.: Topology of Hitchin systems and Hodge theory of character varieties: the case \(A_{1}\). Ann. Math. (2) 175(3), 1329–1407 (2012)

    Google Scholar 

  4. Corlette, K.: Flat \(G\)-bundles with cannonical metrics. J. Diff. Geom. 28, 361–382 (1988)

    MathSciNet  MATH  Google Scholar 

  5. Deligne, P.: Équations différentielles á points singuliers réguliers. In: Lecture Notes in Mathematics, vol. 163. Springer, Berlin (1970)

  6. Deligne, P.: Théorie de Hodge II. Publ. Math. I.H.E.S. 40, 5–5 (1971)

  7. Deligne, P.: Théorie de Hodge III. Publ. Math. I.H.E.S. 44, 5–77 (1974)

  8. Donaldson, S.: Twisted harmonic maps and the self duality equations. Proc. Lond. Math. Soc. 55, 127–131 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  9. 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, viii+80 (2007)

  10. García-Prada, O., Heinloth, J., Schmitt, A.: On the motives of moduli of chains and Higgs bundles, preprint, arXiv:1104.5558

  11. Gothen, P.B.: The Betti numbers of the moduli space of rank 3 Higgs bundles. Int. J. Math. 5, 861–875 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hausel, T.: Mirror symmetry and Langlands duality in the non-Abelian Hodge theory of a curve, Geometric methods in algebra and number theory, pp. 193–217. In: Progr. Math., vol. 235. Birkhauser, Basel (2005)

  13. Hausel, T., Letellier, E., Rodriguez-Villegas, F.: Arithmetic harmonic analysis on character and quiver varieties. Duke Math. J. 160(2), 323–400 (2011)

    Google Scholar 

  14. Hausel, T.: Rodriguez Villegas, F.: Mixed Hodge polynomials of character varieties. Invent. Math. 174, 555–624 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hausel, T., Thaddeus, M.: Mirror symmetry, Langlands duality and Hitchin systems. Invent. Math. 153, 197–229 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  16. Hitchin, N.J.: The self-duality equations on a Riemann surface. Proc. Lond. Math. Soc. 55(3), 59–126 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hitchin, N.J.: Stable bundles and integrable systems. Duke Math. J. 54, 91–114 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  18. Mereb, M.: On the E-polynomials of a family of Character Varieties, Ph. D. dissertation, arXiv:1006.1286

  19. Mehta, M.: Hodge structure on the cohomology of the moduli space of Higgs bundles, preprint, arXiv:math.AG/0112111

  20. Muñoz, V., Ortega, D., Vázquez-Gallo, M.-J.: Hodge polynomials of the moduli space of pairs. Int. J. Math. 18, 695–721 (2007)

    Article  MATH  Google Scholar 

  21. Muñoz, V., Ortega, D., Vázquez-Gallo, M.-J.: Hodge polynomials of the moduli space of triples of rank (2,2). Q. J. Math. 60, 235–272 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Muñoz, V.: The \(SL(2,{\mathbb{C}})\)-character varieties of torus knots. Rev. Mat. Complut. 22, 489–497 (2009)

    MathSciNet  MATH  Google Scholar 

  23. Peters, C., Steenbrink, J.: Mixed Hodge Structures, vol. 52. Springer, Berlin (2007)

    Google Scholar 

  24. Simpson, C.: Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization. J. Am. Math. Soc. 1, 867–918 (1988)

    Article  MATH  Google Scholar 

  25. Simpson, C.: Higgs bundles and local systems. Inst. Hautes Études Sci. Publ. Math. 75, 5–95 (1992)

    Article  MATH  Google Scholar 

  26. Simpson, C.: Moduli space of representations of the fundamental group of a smooth projective variety II. Inst. Hautes Études Sci. Publ. Math. 80, 5–79 (1995)

    MATH  Google Scholar 

  27. Simpson, C.: Harmonic bundles on noncompact curves. J. Am. Math. Soc. 3, 713–770 (1990)

    Article  MATH  Google Scholar 

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Acknowledgments

We thank Tamás Hausel and Richard Thomas for helpful comments. In particular, several conversations with T. Hausel have been invaluable to confirm the correctness of our polynomials. We also thank the referees for their careful reading of the paper.

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Authors and Affiliations

  1. Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), C/Nicolas Cabrera 15, 28049 , Madrid, Spain

    Marina Logares

  2. Facultad de Matemáticas, Universidad Complutense de Madrid, Plaza Ciencias 3, 28040 , Madrid, Spain

    Vicente Muñoz

  3. Department of Mathematical Sciences, University of Liverpool, Peach Street, Liverpool, L69 7ZL, UK

    P. E. Newstead

Authors
  1. Marina Logares
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  2. Vicente Muñoz
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  3. P. E. Newstead
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Corresponding author

Correspondence to Vicente Muñoz.

Additional information

M. Logares is supported by an i-Math Future contract and partially supported by FCT (Portugal) through project PTDC/MAT/099275/2008 and (Spain) project MTM2010-17717. V. Muñoz is supported by (Spanish MICINN) research project MTM2010-17389. M. Logares and P. E. Newstead authors would like to thank the Isaac Newton Institute, where this work was completed during the Moduli Spaces programme.

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Logares, M., Muñoz, V. & Newstead, P.E. Hodge polynomials of SL\((2,\mathbb{C })\)-character varieties for curves of small genus. Rev Mat Complut 26, 635–703 (2013). https://doi.org/10.1007/s13163-013-0115-5

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