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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)

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.

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  • $\begingroup$ There is probably something there, but maybe note that it won't be as obvious as "$\mathcal O_E(1)$ is Hermite-Einstein" because every holomorphic line bundle on a compact Kahler manifold can be equipped with a Hermite-Einstein metric. $\endgroup$
    – Gunnar Þór Magnússon
    Commented Jan 31, 2022 at 9:09
  • $\begingroup$ @GunnarÞórMagnússon Do you have a reference for that? $\endgroup$
    – BinAcker
    Commented Jan 31, 2022 at 10:32
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    $\begingroup$ Let $X$ be compact Kahler with Kahler metric $\omega$ and $L \to X$ a line bundle. Let $c$ be the $\omega$-harmonic representative of $c_1(L)$. By a lemma of Weil there exists a Hermitian metric $h$ on $L$ whose curvature form is $c$ (proved I think in Voisin or Huybrechts, but is done by taking a trivializing Cech cover, finding local representatives for the metric, and patching over overlaps). Because of the commutation identities, $\Lambda c$ will be a harmonic function, that is, a constant, so $(L,h)$ is Hermite-Einstein.. $\endgroup$
    – Gunnar Þór Magnússon
    Commented Jan 31, 2022 at 11:11

1 Answer 1

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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.

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