Uniqueness of balanced metrics on holomorphic vector bundles

Abstract

Let E → M be a holomorphic vector bundle over a compact Kähler manifold ( M , ω ) . We prove that if E admits a ω -balanced metric (in X. Wang’s terminology (Wang, 2005 [3])) then it is unique. This result together with Biliotti and Ghigi (2008) [14] implies the existence and uniqueness of ω -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 ω -balanced Kähler maps into Grassmannians.