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

Hausel, Tamás; Thaddeus, Michael

doi:10.1090/S0894-0347-02-00417-4

Relations in the cohomology ring of the moduli space of rank 2 Higgs bundles
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by Tamás Hausel and Michael Thaddeus

J. Amer. Math. Soc. 16 (2003), 303-329

DOI: https://doi.org/10.1090/S0894-0347-02-00417-4

Published electronically: December 3, 2002

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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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