Numerical flatness and principal bundles on Fujiki manifolds

Abstract

Let $M$ be a compact connected Fujiki manifold, $G$ a semisimple affine algebraic group over $\mathbb C$ with one simple factor and $P$ a fixed proper parabolic subgroup of $G$. For a holomorphic principal $G$--bundle $E_G$ over $M$, let ${\mathcal E}_P$ be the holomorphic principal $P$-bundle $E_G\rightarrow E_G/P$ given by the quotient map. We prove that the following three statements are equivalent: (1) ${\rm ad}(E_G)$ is numerically flat, (2) the holomorphic line bundle $\bigwedge^{\rm top} {\rm ad}({\mathcal E}_P)^*$ is nef, and (3) for every reduced irreducible compact complex analytic space $Z$ with a K\"ahler form $\omega$, holomorphic map $\gamma : Z \rightarrow M$, and holomorphic reduction of structure group $E_P \subset \gamma^*E_G$ to $P$, the inequality ${\rm degree}({\rm ad}(E_P)) \leq 0$ holds.