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The rank stable topology of instantons on $\overline{\mathbf{CP}}^2$


Authors: Jim Bryan and Marc Sanders
Journal: Proc. Amer. Math. Soc. 125 (1997), 3763-3768
MSC (1991): Primary 58D27, 53C07, 55R45, 14Dxx
DOI: https://doi.org/10.1090/S0002-9939-97-04156-7
MathSciNet review: 1425114
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Abstract: Let $% \mathcal{M} _{k}^{n}$ be the moduli space of based (anti-self-dual) instantons on $% \overline{\mathbf {CP}}^2$ of charge $k$ and rank $n$. There is a natural inclusion $% \mathcal{M} _{k}^{n}\hookrightarrow % \mathcal{M}_{k}^{n+1}$. We show that the direct limit space $% \mathcal{M}_k^\infty$ is homotopy equivalent to $BU(k)\times BU(k)$. Let $% \ell _{\infty}$ be a line in the complex projective plane and let $% \widetilde{% {\mathbf C} % {\mathbf{P}}}^{2}$ be the blow-up at a point away from $% \ell _{\infty}$. $% \mathcal{M} _{k}^{n}$ can be alternatively described as the moduli space of rank $n$ holomorphic bundles on $% \widetilde{% \mathbf{C}% \mathbf{P}}^{2}$ with $c_{1}=0$ and $c_{2}=k$ and with a fixed holomorphic trivialization on $% \ell _{\infty}$.


References [Enhancements On Off] (What's this?)

  • [Br] Bryan, J. Symplectic Geometry and the Relative Donaldson Invariants of $\overline{% \mathbb{CP}}^2$, to appear in Forum Math.
  • [Bu] Buchdahl, N. Instantons on $n% \mathbb{CP}^2$. J. Diff. Geo. 37(1993), 669-687. MR 94e:53025
  • [Fr-Mo] Robert Friedman and John Morgan. The Differential Topology of Complex Surfaces. Springer-Verlag, 1994.
  • [Ki] King, A. Instantons and Holomorphic Bundles on the Blown Up Plane, Ph.D. Thesis, Oxford, 1989.
  • [Kir] Kirwan, F. Geometric invariant theory and the Atiyah-Jones conjecture, Proceedings of the Sophus Lie Memorial Conference (Oslo, 1992), editors O. A. Laudal and B. Jahren, Scandinavian University Press, 1994, 161-188.
  • [Sa] Sanders, M., Classifying spaces and dirac operators coupled to instantons, Trans. of the A.M.S. Vol. 347, No. 10 1995. MR 96m:58030
  • [Ta] Taubes, C. The stable topology of self-dual moduli spaces J. Diff. Geo. 29 (1989), 163-230. MR 90f:58023
  • [Ti] Tian, Y. The Atiyah-Jones Conjecture for classical groups and Bott periodicity, to appear in J. Diff. Geo.

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Additional Information

Jim Bryan
Affiliation: Mathematical Sciences Research Institute, 1000 Centennial Drive, Berkeley, California 94720-5070
Email: jbryan@msri.org

Marc Sanders
Affiliation: Department of Mathematics and Computer Science, Dickinson College, Carlisle, Pennsylvania 17013
Email: sandersm@dickinson.edu

DOI: https://doi.org/10.1090/S0002-9939-97-04156-7
Received by editor(s): August 2, 1996
Communicated by: Ronald A. Fintushel
Article copyright: © Copyright 1997 American Mathematical Society

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