Mixed Hodge polynomials of character varieties: With an appendix by Nicholas M. Katz

T. Hausel, F. Rodríguez Villegas, Inventiones Mathematicae 174 (2008) 555–624.


Journal Article | Published
Author
;
Abstract
We calculate the E-polynomials of certain twisted GL(n,ℂ)-character varieties Mn of Riemann surfaces by counting points over finite fields using the character table of the finite group of Lie-type GL(n, q) and a theorem proved in the appendix by N. Katz. We deduce from this calculation several geometric results, for example, the value of the topological Euler characteristic of the associated PGL(n,ℂ)-character variety. The calculation also leads to several conjectures about the cohomology of Mn: an explicit conjecture for its mixed Hodge polynomial; a conjectured curious hard Lefschetz theorem and a conjecture relating the pure part to absolutely indecomposable representations of a certain quiver. We prove these conjectures for n=2.
Publishing Year
Date Published
2008-12-01
Journal Title
Inventiones Mathematicae
Acknowledgement
The first author was supported by NSF grants DMS-0305505 and DMS- 0604775 an Alfred Sloan Fellowship and a Royal Society University Research Fellowship. The second author was supported by an NSF grant DMS-0200605.
Volume
174
Issue
3
Page
555 - 624
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Cite this

Hausel T, Rodríguez Villegas F. Mixed Hodge polynomials of character varieties: With an appendix by Nicholas M. Katz. Inventiones Mathematicae. 2008;174(3):555-624. doi:10.1007/s00222-008-0142-x
Hausel, T., & Rodríguez Villegas, F. (2008). Mixed Hodge polynomials of character varieties: With an appendix by Nicholas M. Katz. Inventiones Mathematicae, 174(3), 555–624. https://doi.org/10.1007/s00222-008-0142-x
Hausel, Tamas, and Fernando Rodríguez Villegas. “Mixed Hodge Polynomials of Character Varieties: With an Appendix by Nicholas M. Katz.” Inventiones Mathematicae 174, no. 3 (2008): 555–624. https://doi.org/10.1007/s00222-008-0142-x.
T. Hausel and F. Rodríguez Villegas, “Mixed Hodge polynomials of character varieties: With an appendix by Nicholas M. Katz,” Inventiones Mathematicae, vol. 174, no. 3, pp. 555–624, 2008.
Hausel T, Rodríguez Villegas F. 2008. Mixed Hodge polynomials of character varieties: With an appendix by Nicholas M. Katz. Inventiones Mathematicae. 174(3), 555–624.
Hausel, Tamas, and Fernando Rodríguez Villegas. “Mixed Hodge Polynomials of Character Varieties: With an Appendix by Nicholas M. Katz.” Inventiones Mathematicae, vol. 174, no. 3, Springer, 2008, pp. 555–624, doi:10.1007/s00222-008-0142-x.

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