Do transvers foliations induce complex structure? Hallo,
I have the following question: Let $M$ smooth analytic manifold of dimension 4n. Assume furthermore that $M$ admits two foliations $A$, $B$, both with leaves of dimension 2n such that the leaves of $A$ are transvers to the leaves of $B$ at each point. Also the leaves of $A$ are $n$-dim complex manifolds with complex structure $J_{\alpha}$ (for a leaf $\alpha \in A$) which vary smoothly if $\alpha$ vary in $A$. The leaves of $B$ are $n-$dim complex manifolds with complex structure $J_{\beta}$ (for a leaf $\beta \in B$) which vary smoothly if $\beta$ vary in $B$. My question is now: Is it possible to define a complex structure on te manifold $M$ by: for $x \in M$ there exists exactly one leaf $\alpha \in A$ that contains $x$ and one leaf $\beta \in B$ that contain $x$ and set for the complex structure $J := J_{\alpha, x} + J_{\beta, x}$. This is an almost complex structure. Is this structure integrable, if $J_{\alpha}$ and $J_{\beta}$ are integrable on each leaf? If not what assumption do I need in order to make it integrable? I hope for a lot of answers. Thanks in advance.
Marin  
 A: The answer is 'no, in general':  For example, if $(M^4,J)$ is any real-analytic almost complex $4$-manifold, one can easily construct (locally, in a neighborhood of any point of $M$) a pair $(A,B)$ of real-analytic, transverse foliations by pseudo-holomorphic curves, and these will satisfy your conditions.  However, when you do your construction, you'll get back the original $J$, which need not be integrable.
To get integrability of the $J$ you construct, you'll need to suppose the vanishing of its Nijnhuis tensor, part of which has to vanish already because of the integrability of the restriction of $J$ to the leaves of the two foliations, but that's not enough to force the entire Nijnhuis tensor to vanish, as the above example shows.
A: We start by giving an answer that rephrases the Nijenhuis integrability condition
using the foliations, $A,B$, but
the main point here  is
 to present a more geometric
criterion  for integrability, related to complex structures of moduli spaces. 
The main claim is that leafwise
holomorphicity of certain natural holonomy induced maps (ie transverse moduli)
give  necessary and sufficient conditions  for integrability. 
We finish with some more speculative questions/answers.
To pick up where Robert Bryant left off,
by Nijenhuis, Newlander--Nirenberg and the 
hypotheses 
on $A,B,J$, (leafwise complex   structures),
our problem reduces to the pair of systems of equations,
$$ (1) P_B^{0,1}(L_{X_i}Y_j)=0, \ (2) {\rm\    likewise\  with\    
A,B \  switched}$$
where $L_{X_i}$ is a Lie derivative, (tensored by $C$)
$ X_i,Y_j$ a  basis,  for $T^{1,0} M$, 
$ X_i\in T^{1,0} A$ is a J-holomorphic vector field along $A$, and
likewise for $Y_j$ along $B$, and
$P_A^{0,1}: T^C M \mapsto T^{0,1}  A$, 
($T^C M=T^{1,0} M \oplus T^{0,1} M $),
are  ($A,B, T^{i,j}$ adapted)  projections.
So integrability of $J$ is equivalent to formal
 vanishing of  these projected mixed Lie-bracket terms, and
the general idea here is that this
 strongly suggests putting things  in terms of
the holonomy of $A$ acting on  $B$
(or on $J_{|TB}$).
Likewise, switching 
$A,B$ provides the other needed equations.
We want to interpret this system of equations
 in terms of 
some kind of holomorphicity of {\it holonomy}, 
along a leaf.
I don't know any references dealing with this specific structure,
but what follows is related to things like holomorphic motions,
and HCMA (homogeneous complex Monge-Ampere equations). 
So what we will propose now may have some  new aspects, but it is not too far from things
already done in these related areas.
Now we consider more directly the relation of the holonomies  of $A$ and $B$ to integrability of $J$.
A (local) leaf $A_x \owns x$ parametrizes a family $B_y, y\in A_x$
of the (local) leaves it intersects, and the foliation $A$ near 
$A_x$ induces a map $H:A_x\to {\cal J}$ where ${\cal J}$ is the 
space of
complex structures on the (local) leaf $B_x$. 
(we just pullback restrictions of $J$ to $B_y$ by
the Holonomy,
ie a flow in $A$).
But  ${\cal J}$ itself
has a complex structure, 
as in the theory of moduli spaces; it comes from  the complex structure induced by $J$
on the Lie algebra of diffeos of $B_x$ (or vector-fields).
Basically, because ${\cal J}$ is a quotient
space of the space (or pseudogroup) of Diffeos of $B_x$.
{\bf Claim}: Given  $J$  integrable on $M$,


*

*If $H_A: A_x\to {\cal J}$ is holomorphic for all $x\in U$, an open set,
then
either $A$ or $B$ is transversely holomorphic on $U$.

*Conversely, if $B$ is transversely holomorphic on $U$
then $H_A: A_x\to {\cal J}$ is holomorphic on $U$. 
This shows that holomorphic $H_A$ is too strong for our purposes, but
a transversely  linearized version of the 2nd part of the claim will turn out to be just what we need.
{\bf  Proof } of claim: 
$H_A$ is a constant map iff $A$ is transversely holomorphic.
If $H_A$ is a non-constant map then since the holonomy of $B$
commutes (by the definition of $H_A$) with the holomorphic  maps 
$H_A: A_x\to {\cal J}$, the   holonomy of $B$ must be itself
holomorphic, proving the first part.  Notice that branch points of $H_A$  give
removable singularities    for holonomy ($L_Y J_{|A}$)  of $B$.
2nd part:
$B$ is a transversely holomorphic foliation by
smooth holomorphic varieties in a ball in $C^N$,
so up to a holomorphic coordinate change, $f$,
it is just a family of parallel n-planes.  Also $A_0$ can be taken to be a
k-plane for  the same  $f$,  ($x=0$ here).
Representing leaves of $A$ as graphs over  $A_0$  with values in  $B_0$ 
gives  the converse.
(Leafwise holomorphicity is the main point, but $H_A$ is even fully holomorphic.)  {\bf qed}.
We will weaken the holomorphicity property of  $H_A$ by 
just using the $J$ induced on the normal bundle of a leaf $NA_x$,
and we now consider holomorphicity of   the holonomy pullback
induced on  $(NA_x,J)$.
${{\cal J}_{N_xA}}={{\cal J}_N}$ is  the  homogeneous space, $SL(2n,R)/SL(n,C)$, with its  natural complex structure,
ie the 
space of
complex structures on the tangent space  $T_xB=T_xB_x$. 
Consider now
the  map $NH:A_x\to {{\cal J}_N}$, namely the restriction of
$H_A$ on the leaf $A_x$ to 1-jets  at $x$ of $B_x$.
{\bf Corollary}: 
Given  $J$  integrable on $M$,
$NH:A_x\to {{\cal J}_N}$  is A-leafwise holomorphic.
{\bf  Proof}:
Apply the 2nd part of the claim above, but using a transversely holomorphic foliation 
$B'$ transverse to $A_x$.  A  family of parallel n-planes suffices (working locally). {\bf qed}.
Now we note that there is no discrepancy between pulling-back $J$ from $NA$ instead of $TB$,
(even though  $TB$ is not holomorphic along a leaf),
and we will show that the Nijenhuis system above is equivalent to
holomorphicity of $NH$.
Given any $x\in M,A,B,J$ as in  the question, consider the
 holonomy $A'$  and complex structure $J'$,
induced on the normal bundle $NA_x$ of a leaf, and the associated transversal  linearization $B'$ of $B$
along $A_x$, obtained as  a limit of $A_x$--transverse rescalings of the original
$M,A,B,J$.  This uses a chart adapted to $A_x$ (as for $f$  above) but more to the point,
a holomorphic trivialization $\tau$ of $TM$ along $A_x$, adapted to $A_x$.  Observe that 
the limit $B'$ of $B$--transforms becomes holomorphic in the limit 
(the tangential components are scaled away, analogously to Poincare normal form constructions).
These special  cases of the original
problem, with  $x\in M'=NA_x,A',B',J'$, 
   always have transversely holomorphic $B'$, so the Corollary
provides leafwise holomorphic $H_{A'}$, when  $J$ is  integrable, and  conversely,
$J'$ is  integrable if   $H_{A'}$ is holomorphic.
Thus, by comparing, in the context of $M'$,  the 
Nijenhuis integrability condition in the ($A,B$ adapted) form above,
with the criterion suggested by the Corollary, it now seems clear  that these are
equivalent.
Note that the  transverse linearization, which gives  $M'$, stabilizes  the  Nijenhuis system  (equation  (1)) along a leaf
$A_x$, just as it stabilizes $H_{A'}$.
 (Furthermore, they have very similar formal structure and both provide the  integrability conditions.
It may be more direct   to  just do the calculation, but these geometric constructions may have
some further usefulness as we will see below.)
This leads us to the desired criterion:
{\bf  Main Claim}: 
$J$ is integrable on $U$,  an open set, 
iff
 the A-maps $NH:A_x\to {{\cal J}_N}$  are A-leafwise holomorphic, for all $x\in U$,
and likewise with
$A,B$ interchanged.
(ie holomorphicity of   the  A-maps and  B-maps  together is necessary and sufficient).
Our discussion of this criterion so far relies implicitly  on the Newlander--Nirenberg theorem.
We
propose to consider a direct geometric construction which might eliminate this dependence: given as  data 
 leafwise holomorphic maps, $NH_A,  NH_B$, the goal is to
 construct  transverse foliations ${\hat A},{\hat B}$  by holomorphic curves on a complex chart, ${\hat M}\subset C^N$
realizing this data. Then   to observe that it is diffeo to the given double foliation
with $J$. 
In the speculative finishing discussion below,  we sketch a  possible
new proof of the Newlander--Nirenberg theorem, but only for 4-manifolds,
and
a related possible application to moduli of these 
$ M,A,B,J$--structures.
To get started, we  use the
preceeding remarks of Robert Bryant giving transverse foliations by pseudo-holomorphic curves
in the almost complex case,  which  requires  4-dimensionality.  Then use the formal equivalence of the
Nijenhuis integrability condition to holomorphiciy of $NH_A,  NH_B$, to proceed and use the latter.
The idea for constructing ${\hat M}$ is quite simply to reverse the arguments that led
to  holomorphic maps, $NH_A,  NH_B$; this would realize transverse foliations ${\hat A},{\hat B}$
by \lq developing' graphs, reversing  the proof of the 1st claim (2nd part) above.
There is some ambiguity going backwards from $NH_A,  NH_B$ to ${\hat A},{\hat B}$;
one needs to choose a  lift  from almost complex structures to vector fields, recalling how
${\cal J}$ is a quotient
space of the space (or pseudogroup) of Diffeos of $B_x$, for example.  This  corresponds well
to the ambiguity, in the presence of ${\hat A},{\hat B}$, of choosing 
${\hat M}\subset C^N$ up to biholomorphisms.  Also to integrate the transverse vector fields (the lifts)
we  use trivializations $\tau$ as above to choose the lift from
$NA$ to $TM$.  One has to be careful that these can be chosen without any constraints beyond what
has been discussed here.
Note that  $\tau$  exists by a very simple case of  Newlander--Nirenberg;  the special case of
a transverse linearization,  $x\in M'=NA_x,A',B',J'$, as above, ie we  must use the integrability,  coming  from holomorphicity
of $H'= NH$.
Note that the integrability criterion of the corollary above  is nontrivial, since $NH$, along a single  leaf, can be arbitrarily
specified,  in the almost
complex, leafwise-integrable case.  The construction sketched here now seems to suggest
that the whole family of  $NH_A,  NH_B$
can  also  be specified quite freely.  This would mean that they give good moduli for 
these 
$ M,A,B,J$--structures.
To extend this reconstruction of an $ M,A,B,J$--structure to higher dimensions, 
 from data such as  holomorphic $NH_A$, one might use large families of (pseudo-)holomorphic
curves; we are guessing that  sprays of such curves that cover the tangent bundle of $M$ could have enough Jacobi-fields
(ie holonomy)  to formally express the Nijenhuis condition.
