On Semistable Principal Bundles over a Complex Projective Manifold, Ii

Let (X, ω) be a compact connected Kähler manifold of complex dimension d and EG −→ 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 EP ⊂ EG to P , such that the corresponding L(P )/Z(G)–bundle EL(P )/Z(G) := EP (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(EG) is numerically flat. (3) The principal G–bundle EG is pseudostable, and ∫ X c2(ad(EG))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 EG is semistable with c2(ad(EG)) = 0.