TITLE: Foliations by holomorphic curves on complex surfaces
QUESTION [11 upvotes]: On a complex surface, does there exist a non-singular foliation by holomorphic curves that is NOT a holomorphic foliation, i.e. a transversally holomorphic foliation?
The surface should be compact and can admit holomorphic foliations.

REPLY [2 votes]: This is a question which is related to Lagrangian foliations on hyperkaehler manifolds, but much of the results are conjectures, or unpublished. Let $M$ be a K3, and $\omega$ a cohomology class on a boundary of a Kaehler cone, satisfying $\omega^2=0$ (such clases are called nef). There are quite a few of such classes on K3; for a general non-algebraic K3, for all classes with $\omega^2=0$, either $\omega$ or $-\omega$ is nef.
It follows from the general compactness arguments that $\omega$ is represented by a closed, positive (1,1)-current. If this current is continuous, it has exactly one positive and one vanishing eigenvalue, hence its kernels give a foliation by holomorphic curves. 
The tangent bundle of a general non-algebraic K3 has no rank 1 
coherent subsheaves, hence this K3 has no holomorphic foliations.
However, in Serge Cantat's thesis (unpublished) he showed an existence
of a Holder continuous positive closed (1,1)-form representing a class
with $\omega^2=0$, using McMullen's exotic holomorphic automorphisms on 
non-algebraic K3. This should give such a non-holomorphic foliation by holomorphic curves. 
I think that such foliations (without relation to K3)
were also discussed in Marco Brunella's paper
Brunella, Marco
On Kähler surfaces with semipositive Ricci curvature. 
Riv. Math. Univ. Parma (N.S.) 1 (2010), no. 2, 441–450. 
but it is not available anywhere, so I cannot check.