Uniqueness of balanced metrics on holomorphic vector bundles

TL;DR

This paper proves the uniqueness of omega-balanced metrics on holomorphic vector bundles over compact Kaehler manifolds, establishing a key property that supports existence and rigidity results in complex geometry.

Contribution

It demonstrates that omega-balanced metrics, if they exist on a holomorphic vector bundle, are unique, and applies this to prove existence, uniqueness, and rigidity in related geometric contexts.

Findings

Uniqueness of omega-balanced metrics on holomorphic vector bundles.

Existence and uniqueness results for certain direct sums of vector bundles.

Rigidity of omega-balanced Kaehler maps into Grassmannians.

Abstract

Let $E\to M$ be a holomorphic vector bundle over a compact Kaehler manifold $(M, \omega)$. We prove that if $E$ admits a $\omega$-balanced metric (in X. Wang's terminology) then it is unique. This result together with a result of L. Biliotti and A. Ghigi implies the existence and uniqueness of $\omega$-balanced metrics of certain direct sums of irreducible homogeneous vector bundles over rational homogeneous varieties. We finally apply our result to show the rigidity of $\omega$-balanced Kaehler maps into Grassmannians.