Generators for the cohomology ring of the moduli space of rank 2 higgs bundles

article Generators for the cohomology ring of the moduli space of rank 2 higgs bundles published Tamas Hausel author 4A0666D8-F248-11E8-B48F-1D18A9856A87 Michael Thaddeus author The moduli space of stable vector bundles on a Riemann surface is smooth when the rank and degree are coprime, and is diffeomorphic to the space of unitary connections of central constant curvature. A classic result of Newstead and Atiyah and Bott asserts that its rational cohomology ring is generated by the universal classes, that is, by the Kunneth components of the Chern classes of the universal bundle. This paper studies the larger, non-compact moduli space of Higgs bundles, as introduced by Hitchin and Simpson, with values in the canonical bundle K. This is diffeomorphic to the space of all connections of central constant curvature, whether unitary or not. The main result of the paper is that, in the rank 2 case, the rational cohomology ring of this space is again generated by universal classes. The spaces of Higgs bundles with values in K(n) for n > 0 turn out to be essential to the story. Indeed, we show that their direct limit has the homotopy type of the classifying space of the gauge group, and hence has cohomology generated by universal classes. 2000 Mathematics Subject Classification 14H60 (primary), 14D20, 14H81, 32Q55, 58D27 (secondary). Oxford University Press2004 Proceedings of the London Mathematical Society10.1112/S0024611503014618 883632 - 658 yes T. Hausel, M. Thaddeus, Proceedings of the London Mathematical Society 88 (2004) 632–658. Hausel, Tamás, and Michael Thaddeus. “Generators for the Cohomology Ring of the Moduli Space of Rank 2 Higgs Bundles.” Proceedings of the London Mathematical Society. Oxford University Press, 2004. <a href="https://doi.org/10.1112/S0024611503014618">https://doi.org/10.1112/S0024611503014618</a>. Hausel T, Thaddeus M. Generators for the cohomology ring of the moduli space of rank 2 higgs bundles. Proceedings of the London Mathematical Society. 2004;88(3):632-658. doi:<a href="https://doi.org/10.1112/S0024611503014618">10.1112/S0024611503014618</a> Hausel, Tamás, and Michael Thaddeus. “Generators for the Cohomology Ring of the Moduli Space of Rank 2 Higgs Bundles.” Proceedings of the London Mathematical Society, vol. 88, no. 3, Oxford University Press, 2004, pp. 632–58, doi:<a href="https://doi.org/10.1112/S0024611503014618">10.1112/S0024611503014618</a>. Hausel T, Thaddeus M. 2004. Generators for the cohomology ring of the moduli space of rank 2 higgs bundles. Proceedings of the London Mathematical Society. 88(3), 632–658. T. Hausel and M. Thaddeus, “Generators for the cohomology ring of the moduli space of rank 2 higgs bundles,” Proceedings of the London Mathematical Society, vol. 88, no. 3. Oxford University Press, pp. 632–658, 2004. Hausel, T., & Thaddeus, M. (2004). Generators for the cohomology ring of the moduli space of rank 2 higgs bundles. Proceedings of the London Mathematical Society. Oxford University Press. <a href="https://doi.org/10.1112/S0024611503014618">https://doi.org/10.1112/S0024611503014618</a> 14642018-12-11T11:52:10Z2021-01-12T06:50:55Z