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

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

2000-03-16

arxiv.org/abs/math/0003094v2

Mathematics

Algebraic Geometry

31 pages, LaTeX. Changes in title and introduction only

Scientific paper

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 non-compact 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. Together, Parts I and II describe the cohomology rings of spaces of rank 2 Higgs bundles at essentially the same level of detail as is known for stable bundles.

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