Let $X$ be a compact complex manifold. Suppose that $X$ is rationally connected in the sense that any two points lie in the image of a rational curve $\mathbb{CP}^1 \to X$. Are there any non-Kähler examples of such $X$?


If $X$ is Kähler and rationally connected, then $X$ is projective. So I might suspect that being rationally connected may force some algebraic structure on $X$, but I cannot find any reference for this question.



Can a non-Kähler complex manifold be rationally connected?
I am writing my comment as a question. I have certainly explained these examples before on MathOverflow, since they show that the Kollár-Miyaoka-Mori conjecture cannot hold beyond Fujiki class $\mathcal{C}$ (roughly, in the setting of Kaehler manifolds).


Let $C$ be a copy of $\mathbb{CP}^1$. Let $E$ be a (geometric) holomorphic vector bundle over $C$ of rank $2$ that is ample. Denote by $E^\*$ the open complement in $E$ of the zero section. Consider the holomorphic, fiberwise, linear action of the discrete group $\mathbb{Z}$ on $E^\*$ by scaling by $2$ (or by any invertible complex number with modulus different from $1$). This action is free. Denote the quotient by $$\pi:E^\*\to X.$$ This morphism factors the projection from $E^\*$ to $C$, so there is an induced proper, holomorphic submersion, $$\rho:X\to C.$$


The compact complex manifold $X$ is not in Fujiki class $\mathcal{C}$ since the first Betti number equals $1$. Yet it is rationally connected in the sense that any two points are contained in the image of a holomorphic map from $\mathbb{CP}^1$.


Indeed, for any two points of $C$ (possibly the same point twice) and for any two elements of $E$ in the fibers over these points, there is a high-degree self-map $\mathbb{CP}^1\to C$ and distinct points of $\mathbb{CP}^1$ lying over the two points of $C$ (possibly the same point, which means the two distinct points are distinct preimages of this one point of $C$) such that the pullback of $E$ to $\mathbb{CP}^1$ is "very, very ample". Thus, there exists a global section of the pullback that is everywhere nonzero, and that has the specified values over the two distinct points of $\mathbb{CP}^1$. This global section defines a holomorphic map from $\mathbb{CP}^1$ to $E^\*$ that connects the two specified points of $E^\*$. The composition with $\pi$ is the desired holomorphic map to $X$.


