Remarks on the geodesic-Einstein metrics of a relative ample line bundle (with an appendix by Xu Wang)

In this paper, we introduce the associated geodesic-Einstein flow for a relatively ample line bundle $L$ over the total space $\mathcal{X}$ of a holomorphic fibration and obtain a few properties of that flow. In particular, we prove that the pair $(\mathcal{X},L)$ is nonlinear semistable if the {associated} Donaldson type functional is bounded from below and the geodesic-Einstein flow has long-time {existence property}. We also define the associated $S$-classes and $C$-classes for $(\mathcal{X},L)$ and obtain two inequalities between them when $L$ admits a geodesic-Einstein metric. Finally, in the appendix of this paper, we prove that a relatively ample line bundle is geodesic-Einstein if and only if an associated infinite rank bundle is Hermitian-Einstein.