= Solution
Use the integral normalization of the <Chern connection> curvature. For a <holomorphic local frame> $e$, put $h_e=h(e,e)>0$. The <fundamental form of a Hermitian holomorphic line bundle> is
$$
\boxed{\omega_{(\mathcal L,h)}=\frac{i}{2\pi}F_h
=-\frac{i}{2\pi}\partial\bar\partial\log h_e.}
$$
This follows from the <local formula for the Chern connection on a line bundle>, $F_h=\bar\partial\partial\log h_e$. It is a closed <real (1, 1)-form> representing the <First Chern class>. If $h_e=e^{-\varphi}$, the formula becomes $(i/(2\pi))\partial\bar\partial\varphi$.
The <Hermitian metric> is positive when this is a <positive real (1, 1)-form>: for every nonzero $v\in T^{1,0}M$,
$$
\boxed{-i\omega_{(\mathcal L,h)}(v,\bar v)>0.}
$$
Equivalently, the matrix $(\partial^2\varphi/\partial z_j\partial\bar z_k)$ is a <positive-definite matrix> in every frame. Such a metric makes $\mathcal L$ a <positive holomorphic line bundle>. Omitting the normalization factor $2\pi$ changes none of these positivity conditions.
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