Balanced metrics on homogeneous vector bundles

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 $E\rightarrow M$ be a holomorphic vector bundle over a compact Kaehler manifold $(M, \omega)$ and let $E=E_1\oplus... \oplus E_m\rightarrow M$ be its decomposition into irreducible factors. Suppose that each $E_j$ admits a $\omega$-balanced metric in Donaldson-Wang terminology. In this paper we prove that $E$ admits a unique $\omega$-balanced metric if and only if $\frac{r_j}{N_j}=\frac{r_k}{N_k}$ for all $j, k=1, ..., m$, where $r_j$ denotes the rank of $E_j$ and $N_j=\dim H^0(M, E_j)$. We apply our result to the case of homogeneous vector bundles over a rational homogeneous variety $(M, \omega)$ and we show the existence and rigidity of balanced Kaehler embedding from $(M, \omega)$ into Grassmannians.