Geodesic-Einstein metrics and nonlinear stabilities

TL;DR

This paper introduces geodesic-Einstein metrics on line bundles over holomorphic fibrations, linking their existence to nonlinear stability and generalizing classical Hermitian-Einstein metrics, with implications for vector bundle stability.

Contribution

It defines geodesic-Einstein metrics and nonlinear stability notions, and establishes their relationship, extending classical stability concepts to a broader geometric context.

Findings

Existence of geodesic-Einstein metrics characterized by a Donaldson type functional

Proved that a holomorphic vector bundle admits a Finsler-Einstein metric iff it admits a Hermitian-Einstein metric

Connected nonlinear stability of line bundles with the existence of special metrics

Abstract

In this paper, we introduce notions of nonlinear stabilities for a relative ample line bundle over a holomorphic fibration and define the notion of a geodesic-Einstein metric on this line bundle, which generalize the classical stabilities and Hermitian-Einstein metrics of holomorphic vector bundles. We introduce a Donaldson type functional and show that this functional attains its absolute minimum at geodesic-Einstein metrics, and we also discuss the relations between the existence of geodesic-Einstein metrics and the nonlinear stabilities of the line bundle. As an application, we will prove that a holomorphic vector bundle admits a Finsler-Einstein metric if and only if it admits a Hermitian-Einstein metric, which answers a problem posed by S. Kobayashi.