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 be a complex reductive group and let 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 -bundles on . In case the genus of 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 is simple

**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

DOI:
https://doi.org/10.1090/S1056-3911-01-00304-6

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.