On semistable principal bundles over a complex projective manifold, II

TL;DR

This paper establishes the equivalence of three conditions related to semistability, flat connections, and numerical flatness for principal G-bundles over compact Kaehler manifolds, deepening understanding of their geometric structure.

Contribution

It proves the equivalence of conditions involving reductions to parabolic subgroups, flat connections, and numerical flatness for principal G-bundles, extending prior results in complex geometry.

Findings

Equivalence of existence of flat connections and numerical flatness.

Characterization of pseudostability via characteristic classes.

Connection between reductions to parabolic subgroups and flatness.

Abstract

Let (X, \omega) be a compact connected Kaehler manifold of complex dimension d and E_G a holomorphic principal G-bundle on X, 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 of G and a holomorphic reduction of the structure group of E_G to P (say, E_P) such that the bundle obtained by extending the structure group of E_P to L(P)/Z(G) (where L(P) is the Levi quotient of P) admits a flat connection; (2) The adjoint vector bundle ad(E_G) is numerically flat; (3) The principal G-bundle E_G is pseudostable, and the degree of the charateristic class c_2(ad(E_G) is zero.