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Local Systems on $\mathbb P$1 − S for S a Finite Set

Published online by Cambridge University Press:  04 December 2007

Prakash Belkale
Prakash Belkale
Affiliation:
Department of Mathematics, University of Utah, 155 S 1400 E, Salt Lake City, UT 84112-0090, U.S.A. E-mail: belkale@math.utah.edu
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Abstract

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I give the necessary and sufficient conditions for the existence of Unitary local systems with prescribed local monodromies on $\mathbb P$1 − S where S is a finite set. This is used to give an algorithm to decide if a rigid local system on $\mathbb P$1 − S has finite global monodromy, thereby answering a question of N. Katz. The methods of this article (use of Harder–Narasimhan filtrations) are used to strengthen Klyachko's theorem on sums of Hermitian matrices. In the Appendix, I give a reformulation of Mehta–Seshadri theorem in the SU(n) setting.

Keywords

vector bundleHarder–Narasimhan filtrationrepresentations of fundamental groupsKlyachko's theoremlocal systems
Type
Research Article
Information
Compositio Mathematica , Volume 129 , Issue 1 , October 2001 , pp. 67 - 86
Copyright
© 2001 Kluwer Academic Publishers