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

TL;DR

This paper introduces a flow for relatively ample line bundles over holomorphic fibrations, explores its properties, and establishes conditions for stability and existence of geodesic-Einstein metrics, linking them to Hermitian-Einstein bundles.

Contribution

It defines the geodesic-Einstein flow, studies its properties, and connects the existence of such metrics to stability and Hermitian-Einstein conditions for line bundles.

Findings

The geodesic-Einstein flow has long-time existence under certain conditions.

Boundedness of the Donaldson type functional implies nonlinear semistability.

Equations relating S-classes and C-classes are established for geodesic-Einstein line bundles.

Abstract

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.