Almost Kahler structures on complex hyperbolic space

Complex hyperbolic space has the structure of a solvable Lie group. Based on the recent classification of Riemannian left-invariant metrics we describe all left-invariant almost Kahler structures on that group. The only Kahler structure is the standard structure of the complex hyperbolic space and all others are strictly almost Kahler. We calculated the Chern connection and obtained the characterization by its torsion tensor. It is proved that the only SCF-soliton is the Kahler one.