Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 013, 7 pages      arXiv:1310.7472      https://doi.org/10.3842/SIGMA.2014.013

Semistability of Principal Bundles on a Kähler Manifold with a Non-Connected Structure Group

Indranil Biswas a and Tomás L. Gómez b
a) School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India
b) Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), Nicolás Cabrera 15, Campus Cantoblanco UAM, 28049 Madrid, Spain

Received October 29, 2013, in final form February 07, 2014; Published online February 12, 2014

Abstract
We investigate principal G-bundles on a compact Kähler manifold, where G is a complex algebraic group such that the connected component of it containing the identity element is reductive. Defining (semi)stability of such bundles, it is shown that a principal G-bundle EG admits an Einstein-Hermitian connection if and only if EG is polystable. We give an equivalent formulation of the (semi)stability condition. A question is to compare this definition with that of [Gómez T.L., Langer A., Schmitt A.H.W., Sols I., Ramanujan Math. Soc. Lect. Notes Ser., Vol. 10, Ramanujan Math. Soc., Mysore, 2010, 281-371].

Key words: Einstein-Hermitian connection; principal bundle; parabolic subgroup; (semi)stability.

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