Biswas, Indranil ; Bruzzo, Ugo
(2010)
*On semistable principal bundles over a complex projective manifold, II*
Geometriae Dedicata, 146
(1).
pp. 27-41.
ISSN 0046-5755

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Official URL: http://www.springerlink.com/content/x72323505mhg26...

Related URL: http://dx.doi.org/10.1007/s10711-009-9424-8

## Abstract

Let (X, ω) be a compact connected Kahler manifold of complex dimension d and E_{G} → X a holomorphic principal G-bundle, where G is a connected reductive linear algebraic group defined over C. Let Z(G) denote the center of G. We prove that the following three statements are equivalent: (1) There is a parabolic subgroup P ⊂ G and a holomorphic reduction of structure group E_{P} ⊂ E_{G} to P, such that the corresponding L(P)/Z(G)-bundle E_{L(P)/Z(G)}: = E_{P}(L(P)/Z(G) → X admits a unitary flat connection, where L(P) is the Levi quotient of P. (2) The adjoint vector bundle ad(E _{G}) is numerically flat. (3) The principal G-bundle E _{G }is pseudostable, and ∫_{X} C_{2}(ad(E_{G})ω^{d-2} = 0 If X is a complex projective manifold, and ω represents a rational cohomology class, then the third statement is equivalent to the statement that E _{G} is semistable with c_{2}(ad(E_{G} )) = 0.

Item Type: | Article |
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Source: | Copyright of this article belongs to Springer-Verlag. |

Keywords: | Principal Bundle; Pseudostability; Numerical Effectiveness |

ID Code: | 3638 |

Deposited On: | 18 Oct 2010 10:16 |

Last Modified: | 27 Jan 2011 06:20 |

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