Indranil Biswas, Tomás L. Gómez, “Semistability of Principal Bundles on a Kähler Manifold with a Non-Connected Structure Group”, SIGMA, 2014, Volume 10,013, 7 pp.

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

Indranil Biswas^a, 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

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 $E_G$ admits an Einstein–Hermitian connection if and only if $E_G$ 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].

Keywords: Einstein–Hermitian connection; principal bundle; parabolic subgroup; (semi)stability.

MSC: 53C07; 14F05

Received: October 29, 2013; in final form February 7, 2014; Published online February 12, 2014

Language: English

DOI: 10.3842/SIGMA.2014.013