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Bulletin of the London Mathematical Society Advance Access originally published online on March 11, 2008
Bulletin of the London Mathematical Society 2008 40(1):163-171; doi:10.1112/blms/bdm116
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© 2008 London Mathematical Society

Simplicity of stable principal sheaves

Indranil Biswas

School of Mathematics
Tata Institute of Fundamental Research
Homi Bhabha Road
Mumbai 400005
India
indranil@math.tifr.res.in

Tomás L. Gómez

IMAFF
Consejo Superior de Investigaciones
Científicas
Serrano 113bis
28006 Madrid
Spain
Facultad de Matemáticas
Universidad Complutense de Madrid
28040 Madrid
Spain

Received 26 November 2005. Revision received 13 July 2007.

Let M be a compact connected Kähler manifold, and let G be a connected complex reductive linear algebraic group. We prove that a principal G-sheaf on M admits an admissible Einstein–Hermitian connection if and only if the principal G-sheaf is polystable. Using this it is shown that the holomorphic sections of the adjoint vector bundle of a stable principal G-sheaf on M are given by the center of the Lie algebra of G. The Bogomolov inequality is shown to be valid for polystable principal G-sheaves.

2000 Mathematics Subject Classification 32L05, 14F05.


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