# Line bundles over Kähler–Hodge manifolds

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.

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 :

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.

• It seems that you are assuming that $M$ is compact, a restriction which was not in the question.
– abx
Feb 18 '15 at 10:52
• @abx: Indeed... I'll edit the answer.
– ACL
Feb 19 '15 at 10:02
• Thanks for the explanation. How crucial is that $M$ is a compact manifold? Does a similar result hold in for a non-compact base $M$? Mar 14 '15 at 23:00

If we consider a usual sympleetic manifold with $[\omega]\in H^2(M,\mathbb Z)$ integral then the Weil's theorem guarantees that there exists a Hermitian line bundle $L$ over $X$ with a conneetion $V$ such that $V$ is compatible with the Hermitan strueture on $L$ and induces the symplectic form by $\omega = curv(V)$.

For the second part of your question , The proof is in section "12.2 Integral closed 2-forms and line bundles" http://www.math.toronto.edu/~jeffrey/mat1312/lec14.gq.pdf

In third part of your question: Manifolds which are not quantizable then there is no such line bundle $L$ which first chern class be proportional to kahler class