A note on the space of Kähler metrics

To study the space C of Kähler metrics, we can consider the following two geometric objects. One is the certain complexification Y of the symplectomorphism group, and the other one is a foliation on product Kähler manifold related to the geodesic on C. In this note we study the geometry of Y and show how the properties of the associated foliation (in particular, the holomorphicity) restrict the topology of base Kähler manifold.