It is a simple fact that if $L \to B$ is a complex line bundle endowed with an Hermitian product and a compatible connection $\nabla$, then the curvature $F\_\nabla$ is imaginary (and so are the local connection $1$-forms). I am curious, though, if the following converse is true: if $\nabla$ is given such that $F\_\nabla$ is imaginary, is it possible to find an Hermitian product such that $\nabla$ be compatible with it?



Hermitic connections on complex line bundles with imaginary curvature form
Even flat connections don't have to arise from a Hermitian inner product; they can have holonomy not unitary.


