Let $M$ complex manifolds admitting a smooth, positive, proper plurisubharmonic exhaustion, $\rho:M\to[0,\infty)$, whose we have complex Monge-Ampere foliation $(\partial\bar\partial \rho)^n=0$. Patrizio, Giorgio;and Wong, Pit Mann in

Stability of the Monge-Ampère foliation,Mathematische Annalen,March 1983, Volume 263, Issue 1, pp 13–29

showed that if the volume function is continuous on the level sets of $\rho$, then the leaf space, $\mathcal L$, admits a Kähler form $\omega$, so that, if $\pi:M\to\mathcal L $ is the projection, we have $$\partial\bar \partial \log \rho=\pi^*\omega.$$

We have always the following theorem:

Theorem: If $\omega$ is non-negative, $\omega^{n-1}\neq 0$, $\omega^n=0$, and $d\omega=0$, then

$$\mathcal F=\text{ann}(\omega)=\{W\in TX|\omega(W,\bar W)=0, \forall W\in TX\}$$
define a foliation $\mathcal F$ on $X$ and each leaf of $\mathcal F$ being a Riemann surface(which is Kaehler always)

You can see my short not about fiberwise Calabi-Yau foliation and semi-Ricci flat metric introduced by Greene,Shapire,Vafa, and Yau

https://hal.archives-ouvertes.fr/hal-01551080

Take a holomorphic foliation map $\pi:X\to Y$ such that the leaves of the foliation coincide with the fiber of $\pi$, then the pull back of any Kahler metric on $Y$ to $X$ gives rise to a homogeneous holomorphic Monge-Ampère foliation and the degenerate Kahler form can be the pull back of a Kahler metric on $Y$. See Proposition 6.4 of the following paper of Ruan.

In fact by Theorem 1.3 in the following reference, when the homogeneous Monge-Ampère equation comes from a collapsing, the foliation is holomorphic.

In fact holomorphic foliation correspond to Cheeger-Fukaya-Gromov theory about collapsing Riemannian manifolds.

If you want to study the collapsing part of degeneration of K\"ahler-Einstein metrics, then you are in deal with holomorphic foliation(see Wei-Dong Ruan's paper in bellow) and also fiberwise Kahler-Einstein foliation (which is a foliation in fiber direction and may not be foliation in horizontal direction. See this preprint)

Wei-Dong Ruan, On the convergence and collapsing of Kähler metrics, J. Differential Geom.Volume 52, Number 1 (1999), 1-40.