Positive line bundle

Remarks

If $e$ is a local holomorphic frame of $L$ and $\abs{e}_h^2 = e^{-\varphi}$, then $i \Theta_h = i \partial \bar\partial \varphi$. So $L$ is positive exactly when some metric has local weights $\varphi$ that are strictly plurisubharmonic. The form $\frac{i}{2\pi} \Theta_h$ represents the first Chern class $c_1(L)$, so positivity implies that $c_1(L)$ contains a Kähler form.

Example: $\mathcal{O}(1)$ on $\mathbb{P}^n$ with the metric dual to the standard metric on $\mathcal{O}(-1)$. On $\set{z_0 \ne 0}$ with affine coordinates $z$, the dual frame $e^*$ has $\abs{e^*}^2 = (1 + \abs{z}^2)^{-1}$, so $\varphi = \log(1 + \abs{z}^2)$. Then $i \partial \bar\partial \varphi$ is the Fubini–Study form, which is positive definite at each point.

Non-example: the trivial line bundle on a compact connected complex manifold $X$ of positive dimension. Any metric is $\abs{1}_h^2 = e^{-\varphi}$ for a global smooth function $\varphi$. At a maximum point of $\varphi$ the complex Hessian $(\partial^2 \varphi / \partial z_j \partial \bar z_k)$ is negative semidefinite, so it is not positive definite there.