
TL;DR
This paper investigates conditions under which a holomorphic vector bundle over a compact Kähler manifold admits a unique balanced metric, with applications to homogeneous bundles and embeddings into Grassmannians.
Contribution
It establishes a necessary and sufficient condition for the existence of a unique balanced metric on decomposed vector bundles and applies this to homogeneous bundles and Kähler embeddings.
Findings
Unique balanced metric exists if and only if rank-to-dimension ratios are equal across factors.
Proves existence and rigidity of balanced Kähler embeddings into Grassmannians.
Characterizes balanced metrics on decomposed bundles over homogeneous varieties.
Abstract
Let be a holomorphic vector bundle over a compact Kaehler manifold and let be its decomposition into irreducible factors. Suppose that each admits a -balanced metric in Donaldson-Wang terminology. In this paper we prove that admits a unique -balanced metric if and only if for all , where denotes the rank of and . We apply our result to the case of homogeneous vector bundles over a rational homogeneous variety and we show the existence and rigidity of balanced Kaehler embedding from into Grassmannians.
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