# Different definitions of positive line bundle

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?