A Kähler–Hodge manifold $M$ can be defined as a Kähler manifold whose Kähler form $\omega$ is integral, namely $\omega\in H^{2}(M,\mathbb{Z})$. It is known then that there always exists a Hermitian line bundle $L\to M$ whose Chern class satisfies $c\_{1}(L) = [\omega]$. I have the following questions:


* Does anyone know a reference where the statements above are proven?
* How can this line bundle $L$, given a particular integral Kähler form $\omega$, be explicitly constructed?
* What happens if $\omega$ is not integral? Can one still construct some kind of $\mathbb{R}$-bundle whose curvature is $\omega$?


Thanks.



Line bundles over Kähler–Hodge manifolds
At least when $M$ is compact, which I assume here,
the standard reference for this subject is probably the book


* *Algebraic Geometry*, Griffiths & Harris


but the following books are excellent references as well :


* *Differential analysis on complex manifolds*, Wells
* *[Complex Analytic and Differential Geometry](https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf)*, Demailly
* *Hodge Theory and Complex Algebraic Geometry*, Voisin


A proof could run as follows:


* Hodge's theorem implies that there exists a complex line bundle $L$
on $M$ with first Chern class $c\_1(L)=\omega$;
* Endow $L$ with an hermitian metric $(L,\|\cdot\|\_0)$; its curvature form 
$c\_1(L,\|\cdot\|\_0)$ is a $(1,1)$-form cohomologous to $\omega$. By the $dd^c$-lemma, there exists a function $\phi$ such that $\omega= c\_1(L,\|\cdot\|\_0)+dd^c\phi$; then $\|\cdot\|=e^{-\phi}\|\cdot\|\_0$ is a hermitian metric on $L$ with curvature form equal to $\omega$.


For an explicit construction of $L$: the standard proof of Hodge's theorem is cohomological, and uses a long exact sequence in cohomology. Tracing down the arguments gives an actual construction if one is given an open covering of $M$ by contractible open sets.


The arguments really require that $\omega$ be integral. A generic complex torus of dimension $>1$ carries no complex line bundle, but has many Kähler forms.


