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Relations in the cohomology ring of the moduli space of rank 2 Higgs bundles


Authors: Tamás Hausel and Michael Thaddeus
Journal: J. Amer. Math. Soc. 16 (2003), 303-329
MSC (2000): Primary 14H60; Secondary 14D20, 14H81, 32Q55, 58D27
DOI: https://doi.org/10.1090/S0894-0347-02-00417-4
Published electronically: December 3, 2002
MathSciNet review: 1949162
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Abstract: The moduli space of stable bundles of rank $2$ and degree $1$ on a Riemann surface has rational cohomology generated by the so-called universal classes. The work of Baranovsky, King-Newstead, Siebert-Tian and Zagier provided a complete set of relations between these classes, expressed in terms of a recursion in the genus. This paper accomplishes the same thing for the noncompact moduli spaces of Higgs bundles, in the sense of Hitchin and Simpson. There are many more independent relations than for stable bundles, but in a sense the answer is simpler, since the formulas are completely explicit, not recursive. The results of Kirwan on equivariant cohomology for holomorphic circle actions are of key importance.


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Additional Information

Tamás Hausel
Affiliation: Department of Mathematics, University of California, Berkeley, California 94720
Address at time of publication: Department of Mathematics, University of Texas, RLM 11.168, 26th and Speedway, Austin, Texas 78712
Email: hausel@math.utexas.edu

Michael Thaddeus
Affiliation: Department of Mathematics, Columbia University, 2990 Broadway, New York, New York 10027
Email: thaddeus@math.columbia.edu

DOI: https://doi.org/10.1090/S0894-0347-02-00417-4
Received by editor(s): June 10, 2002
Published electronically: December 3, 2002
Additional Notes: The first author was supported by NSF grant DMS–97–29992
The second author was supported by NSF grant DMS–98–08529
Article copyright: © Copyright 2002 American Mathematical Society

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