On the converse to a theorem of Atiyah and Bott
Authors:
Robert Friedman and John W. Morgan
Journal:
J. Algebraic Geom. 11 (2002), 257-292
DOI:
https://doi.org/10.1090/S1056-3911-01-00304-6
Published electronically:
November 19, 2001
MathSciNet review:
1874115
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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
Adams J. F. Adams, Lectures on Lie Groups, Benjamin, New York, 1969.
AtBo M. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Phil. Trans. Roy. Soc. London A 308 (1982), 523–615.
Bour N. Bourbaki, Groupes et Algèbres de Lie, Chap. 4, 5, et 6, Masson, Paris, 1981.
FM R. Friedman and J.W. Morgan, Holomorphic principal bundles over elliptic curves, math.AG/9811130.
FMII R. Friedman and J.W. Morgan, Holomorphic principal bundles over elliptic curves II: The parabolic construction, math.AG/0006174.
Ra A. Ramanathan, Stable principal bundles on a compact Riemann surface, Math. Ann. 213 (1975), 129–152.
Ra2 A. Ramanathan, Deformation of principal bundles on the projective line, Invent. Math. 71 (1983), 165–191.
Shatz S. Shatz, The decomposition and specialization of algebraic families of vector bundles, Comp. Math. 35 (1977), 163–187.
Adams J. F. Adams, Lectures on Lie Groups, Benjamin, New York, 1969.
AtBo M. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Phil. Trans. Roy. Soc. London A 308 (1982), 523–615.
Bour N. Bourbaki, Groupes et Algèbres de Lie, Chap. 4, 5, et 6, Masson, Paris, 1981.
FM R. Friedman and J.W. Morgan, Holomorphic principal bundles over elliptic curves, math.AG/9811130.
FMII R. Friedman and J.W. Morgan, Holomorphic principal bundles over elliptic curves II: The parabolic construction, math.AG/0006174.
Ra A. Ramanathan, Stable principal bundles on a compact Riemann surface, Math. Ann. 213 (1975), 129–152.
Ra2 A. Ramanathan, Deformation of principal bundles on the projective line, Invent. Math. 71 (1983), 165–191.
Shatz 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
Received by editor(s):
June 19, 2000
Published electronically:
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.
