Some deformations of the Hopf foliation are also Kähler

Abstract:

Fix $\alpha = ({\alpha _0}, \ldots ,{\alpha _n}) \in \mathring {\mathbf {R}}^{n + 1}$. The trajectories of the flow on ${{\mathbf {S}}^{2n + 1}} \subset {{\mathbf {C}}^{n + 1}}$ given by \[ {\phi _t}:({z_0}, \ldots ,{z_n}) \mapsto ({z_0}{e^{i{\alpha _0}t}}, \ldots ,{z_n}{e^{i{\alpha _n}t}})\] constitute the leaves of a $2n$-codimensional (nonsingular) foliation of ${{\mathbf {S}}^{2n + 1}}$. We use (locally defined) branches of the logarithm to give this foliation a (global) transverse Kähler structure.