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Given a Hermitian metric on a holomorphic vector bundle we can easily define its Chern connection. But if we are given a connection $\mathcal{A}$,

$$[De=\mathcal{A}e,]$$

where $e$ is a holomorphic frame field, how do we know if we can find a Hermitian metric on this vector bundle, so that the connection $\mathcal{A}$ is the Chern connection of the metric, or other types of connections compatible with the metric?

In general vector bundles, I searched and find it's related to if the holonomy group is a subgroup of some unitary groups, and so related to curvatures by the theorem of Ambroise-Singer.

I'm think is there any simpler criterions in the case of holomorphic bundle?

Thanks for your attention!

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    $\begingroup$ I don't see why it should be simpler in the holomorphic case. Without having $U(n)$ holonomy, just knowing that the $(0,1)$-part of the connection is $\bar\partial$ won't tell you anything helpful, will it? Have you explicitly written down the line bundle case? $\endgroup$ – Ted Shifrin Apr 26 at 23:52
  • $\begingroup$ Hello Ted, thank you for the attention. In the case of line bundle, I think that the curvature form being a pure imaginary form is a necessary and sufficient condition. $\endgroup$ – William Shakespaghetti Apr 28 at 0:46

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