Question

I'm looking for an introduction to holomorphic foliations and foliations of complex manifolds.

Any little helps, but I'm particularily interested in problems of the type where we have a hermitian manifold $(X,h)$ (not necessarily compact) and a foliation $\mathcal F$ of $X$, such that the restriction of $h$ to any leaf of the foliation is Kahler. Anything that could help to describe existence of such foliations, or consequences of their existence, would be greatly appreciated. But, to begin with:

Is there any general introduction to the theory of foliations on complex manifolds?

Answer 1

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.

1. Brunella - Birational geometry of foliations
2. Suwa - Indices of vector fields and residues of holomorphic foliations
3. Gomez-Mont, Bobadilla - Sistemas Dinamicos Holomorfos en Superficies ( in Spanish )
4. Loray - Pseudo-groupe d'une singularité de feuilletage holomorphe en dimension deux (in French )
5. Camacho, Sad - Pontos singulares de equações diferenciais analiticas ( in Portuguese )
6. Lins Neto, Scárdua - Folheações algébricas complexas ( in Portuguese )
7. 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.

Answer 2

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. :)