Let $E\to X$ be a holomorphic vector bundle. Denote by $\mathbb{P}(E)\to X$ its projectivisation and $\mathcal{O}\_E(1)\to \mathbb{P}(E)$ the associated tautological line bundle.


I would like to know whether we can characterize the fact that $E$ is Hermitian-Eintein using the bundle $\mathcal{O}\_E(1)$.


* On one hand we have a correspondance between Finsler metric on $E$ and Hermitian metrics on $\mathcal{O}\_E(1)$.
* On the other hand there exists a theorem which states "$E$ admits a Finsler-Einstein metric iff it admits a Hermitian-Einstein metric" (see Geodesic-Einstein metrics and nonlinear stabilities by Feng–Liu–Wan, Trans. AMS 2019, [link at AMS site](https://www.ams.org/journals/tran/2019-371-11/S0002-9947-2018-07658-9/home.html))


From this it seems that we should be able to characterize the existence of Hermitian-Einstein metrics on $E$ through the geometry of $\mathcal{O}\_E(1)$. But there is a priori no notion of Hermitian-Einstein metric on $\mathcal{O}\_E(1)$ as $\mathbb{P}(E)$ is no canonically Kahler.



Characterize Hermitian-Einstein metric on $E$ using the tautological bundle $\mathcal{O}_E(1)$
This is exactly the proposition 3.5 of the mentioned paper:


*If $\mathcal{O}\_{P(E^\*)}(1)$ admits a geodesic-Einstein metric, then the induced $L^2$ metric on $E$ is a Hermitian-Einstein metric.*


