{title2=$\omega_{(L,h)}=iF_h/(2\pi)$}

= Fundamental form of a Hermitian holomorphic line bundle
{synonym}

For a <holomorphic local frame> $e$ with $h_e=h(e,e)>0$, this form is
$$
\omega_{(L,h)}=-\frac{i}{2\pi}\partial\bar\partial\log h_e.
$$
It is a closed <real (1, 1)-form> representing the <First Chern class>. A positive <Hermitian metric> makes it a <positive real (1, 1)-form>. Replacing $h$ by $e^{-u}h$ adds $(i/(2\pi))\partial\bar\partial u$ to the form.


 Normalized curvature form of a Hermitian holomorphic line bundle

ID: normalized-curvature-form-of-a-hermitian-holomorphic-line-bundle