Symplectic reduction and the homogeneous complex Monge-Ampere equation

A Riemannian manifold (M-n, g) is said to be the center of the complex manifold X-n if M is the zero set of a smooth strictly plurisubharmonic exhaustion function nu (2) on X such that nu is plurisubharmonic and solves the Monge-Ampere equation (<<partial derivative>(partial derivative )over bar>nu)(n) = 0 off M, and g is induced by the canonical Kahler metric with fundamental two-form -root -1 <<partial derivative>(partial derivative )over bar>nu (2). Insisting that nu be unbounded puts severe restrictions on X as a complex manifold as well as on (M, g). It is an open problem to determine the class Riemannian manifolds that are centers of complex manifolds with unbounded nu. Before the present work, the list of known examples of manifolds in that class was small. In the main result of this paper we show, by means of the moment map corresponding to isometric actions and the associated bundle construction, that such class is larger than originally thought and contains many metrically and diffeomorphically 'exotic' examples.