# References for holomorphic foliations

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?

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.

• Thanks for the list. I think the last two might cause a little difficulty for many of us. :) – Sándor Kovács Jun 17 '11 at 23:22
• Indeed. I've never tried to read anything in Portuguese, this'll be fun. Thanks for the list! – Gunnar Þór Magnússon Jun 18 '11 at 7:49

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. :)

• Thanks Sándor, I'll take a look at those. This is a situation that seems to be important to my thesis; I tumble naturally upon a hermitian manifold which admits such a foliation, and heuristics suggest it should admit a second such foliation, the interplay between the two then being interesting. I'm having real trouble showing the existence of this second foliation in the most basic example, so I hoped there might be some general theory available. – Gunnar Þór Magnússon Jun 17 '11 at 17:31
• Indeed every projective manifold admits a lot of foliations, the most simple examples being the ones defined by the fibers of rational maps. What is indeed very rare is to have foliations with "positive" tangent sheaves. These are the ones that according to Miyaoka's and Bogomolov-McQuillan's Theorems impose strong restrictions on the underlying manifold. – Jorge Vitório Pereira Jun 17 '11 at 19:49
• Hi Jorge, yes, of course you are right. I was thinking foliations that are not coming from fibrations. – Sándor Kovács Jun 17 '11 at 23:21
• Hi Sándor. It is also relatively easy to produce other examples. Foliations of dimension one abound as we have just to take any rational section of the tangent bundle. If this section is generic enough then it leaves no proper algebraic subvariety of positive dimension invariant. For foliations of higher dimension it is harder to come up with examples but there are some ways to build them: wedge of rational closed $1$-forms, suspensions of representations of the fundamental group of complements, pull-backs from other varieties,... – Jorge Vitório Pereira Jun 18 '11 at 15:14
• Hi Jorge, obviously I know nothing about foliations. I'll have to ask you about these when I am in Brazil in October. Are you going to the ALGA meeting? – Sándor Kovács Jun 18 '11 at 15:27