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On the converse to a theorem of Atiyah and Bott

Author(s): Robert Friedman; John W. Morgan
Journal: J. Algebraic Geom. 11 (2002), 257-292.
Posted: November 19, 2001
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Abstract | References | Additional information

Abstract: Let $G$ be a complex reductive group and let $C$be a smooth curve of genus at least one. We prove a converse to a theorem of Atiyah-Bott concerning the stratification of the space of holomorphic $G$-bundles on $C$. In case the genus of $C$ is one, we establish that there is a stratification in the strong sense. The paper concludes with a characterization of the minimally unstable strata in case $G$ is simple

References:

1.
J. F. Adams, Lectures on Lie Groups, Benjamin, New York, 1969.

2.
M. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Phil. Trans. Roy. Soc. London A 308 (1982), 523-615.

3.
N. Bourbaki, Groupes et Algèbres de Lie, Chap. 4, 5, et 6, Masson, Paris, 1981.

4.
R. Friedman and J.W. Morgan, Holomorphic principal bundles over elliptic curves, math.AG/9811130.

5.
R. Friedman and J.W. Morgan, Holomorphic principal bundles over elliptic curves II: The parabolic construction, math.AG/0006174.

6.
A. Ramanathan, Stable principal bundles on a compact Riemann surface, Math. Ann. 213 (1975), 129-152.

7.
A. Ramanathan, Deformation of principal bundles on the projective line, Invent. Math. 71 (1983), 165-191.

8.
S. Shatz, The decomposition and specialization of algebraic families of vector bundles, Comp. Math. 35 (1977), 163-187.

Additional Information:

Robert Friedman
Affiliation: Department of Mathematics, Columbia University, New York, New York 10027
Email: rf@math.columbia.edu

John W. Morgan
Affiliation: Department of Mathematics, Columbia University, New York, New York 10027
Email: jm@math.columbia.edu

PII: S 1056-3911(01)00304-6
Received by editor(s): June 19, 2000
Posted: November 19, 2001
Additional Notes: The first author was partially supported by NSF grant DMS-99-70437. The second author was partially supported by NSF grant DMS-97-04507.

Journal of Algebraic Geometry
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