
5. Foliations

1. Problem 5.1.

[H. Jacobowitz] Let $(M^{2n+d},\mathcal{F}^d)$ be a transversely holomorphic foliation.

There is a "complexification" $\tilde M^{2n+2d}$ of $(M,\mathcal{F})$. Does $\tilde M$ have a complex structure which is consistent with the given foliation?
1. Remark. The CR analogue of this problem is as follows:

Let $V^{n+d}\subset T_{\mathbb{C}}M^{2n+d}$ be a distribution with $[V,V]\subseteq V$. Then the real foliation $\mathrm{Re}(V\cap\bar V)$ is complexifiable if and only if there exists a CR structure $\tilde V\subset V$, $\tilde V\subset T_{\mathbb{C}}M$.

Cite this as: AimPL: Analysis and geometry on pseudohermitian manifolds, available at http://aimpl.org/pshermitian.