I would like to find a proof for Remark 0.5 in the following article of Claire Voisin:


<https://webusers.imj-prg.fr/~claire.voisin/Articlesweb/fanosymp.pdf>


She writes in this remark the following:


**Remark 0.5** *A compact Kähler manifold $X$ which is rationally connected satisfies $H^2(X, {\cal O}\_X) = 0$, hence is projective.*


I understand that a Kähler manifold with $H^2(X, {\cal O}\_X) = 0$ is projective. However, I don't understand why a Kähler manifold that is rationally connected has $H^2(X, {\cal O}\_X) = 0$. Indeed, the definition for rational connectedness that Voisin is using is the following:


**Definition 0.3** *A compact Kähler manifold $X$ is rationally connected if for any two points $x, y\in X$, there exists a (maybe singular) rational curve $C\subset X$ with the property that $x\in C$, $y\in C$.*


So my question is the following: How to prove that $H^2(X, {\cal O}\_X)$ for a compact Kähler manifold $X$ that satisfies Definition 0.3? Is this easy/hard/well-known?


**PS.** As Donu Arapura correctly says below the vanishing of $H^2(X, {\cal O}\_X)$ for rationally connected projective manifolds is a classical fact. However I want a proof of such a vanishing for Kähler manifolds (to show that they are projective). So I want to know if this vanishing is a well known fact or a couple of pages are needed to prove it?



Rationally connected Kähler manifolds are projective
This result follows from Corollaire (p.212) of [this paper](http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN00209696X) by F. Campana, *Coréduction algébrique d'un espace analytique faiblement kählérien compact,* Invent. Math. 63 (1981), no. 2, 187–223.


I had the same problem of finiding a citable reference for this result some time ago, and this one did the job.


