References for holomorphic foliations
As far as I can remember right now, the great general introduction to the theory of holomorphic foliations is yet to be written. Anyway let me mention some of the books that I know and which you may find useful. Let me warn you that none of them address your specific question.
- Brunella - Birational geometry of foliations
- Suwa - Indices of vector fields and residues of holomorphic foliations
- Gomez-Mont, Bobadilla - Sistemas Dinamicos Holomorfos en Superficies ( in Spanish )
- Loray - Pseudo-groupe d'une singularité de feuilletage holomorphe en dimension deux (in French )
- Camacho, Sad - Pontos singulares de equações diferenciais analiticas ( in Portuguese )
- Lins Neto, Scárdua - Folheações algébricas complexas ( in Portuguese )
- Lins Neto - Componentes irredutíveis dos espaços de folheações ( in Portuguese )
Let me also mention that F. Touzet recently studied foliations admitting a transversal Kähler metric in this paper.
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.
A few years ago I was wondering about (kind of) the same question and could not really find a satisfactory introduction. I settled for introductions to the general theory of foliations. I suppose you know references for that.
There are a few papers on foliations in algebraic geometry, especially in characteristic $p$. I understand that that is not what you are asking, but perhaps algebraic ideas might give you something in the Kahler case.
In particular, Miyaoka's paper, MR927960 (89e:14011) 14E05 (14D99 14F10 14J40) Miyaoka, Yoichi Deformations of a morphism along a foliation and applications. Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), 245–268, Proc. Sympos. Pure Math., 46, Part 1, Amer. Math. Soc., Providence, RI, 1987. is probably basic.
In general, it seems to me, that having an algebraic foliation is a very strong property. For instance, Kebekus-Solá Conde-Toma show that with some additional positivity properties an algebraic foliation implies very strong restrictions on the underlying manifold.
Again, I understand that this is not what you are asking for, but perhaps the references in this latter paper give you something to start and you might find that elusive introduction. If you do, please let me know. :)