On semistable principal bundles over a complex projective manifold,II

Let (X, ω) be a compact connected Kähler manifold of complex dimension d and \({E_G\,\longrightarrow\,X}\) a holomorphic principal G–bundle, where G is a connected reductive linear algebraic group defined over \({\mathbb{C}}\). Let Z(G) denote the center of G. We prove that the following three statements are equivalent:

1. (1)
   There is a parabolic subgroup \({P\,\subset\,G}\) and a holomorphic reduction of structure group \({E_P\,\subset\,E_G}\) to P, such that the corresponding L(P)/Z(G)–bundle

   $E_{L(P)/Z(G)}\,:=\,E_P(L(P)/Z(G))\,\longrightarrow\,X$

   admits a unitary flat connection, where L(P) is the Levi quotient of P.
    
2. (2)
   The adjoint vector bundle ad(E _G ) is numerically flat.
    
3. (3)
   The principal G–bundle E _G is pseudostable, and

   $\int\limits_X c_2({\rm ad}(E_G))\omega^{d-2}\,=\,0.$
    

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.