On semistable principal bundles over a complex projective manifold, II

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 C₂(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 c₂(ad(E_G )) = 0.