0
$\begingroup$

Let $E$ be a holomorphic line bundle over a compact, complex manifold $X$. Then $E$ is said to be a positive line bundle if and only if there exists a hermitian metric $h_X$ on $X$ and a hermitian metric $h_E$ on $E$ for which equation $\omega=\frac{i}{2\pi}\Theta$ holds, where $\omega$ is the fundamental form associated to $h_X$, and where $\Theta$ is the curvature form of the canonical connection on $E$ associated to $h_E$.

Some other definitions go like: $E$ is said to be a positive line bundle if $[\omega]=[\frac{i}{2\pi}\Theta]$.

I wonder if $[\omega]=[\frac{i}{2\pi}\Theta]$ can imply $\omega=\frac{i}{2\pi}\Theta$ by making a specific choice of the Hermitian metrics?

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.