Symplectic Reduction and the Homogeneous Complex Monge–Ampère Equation

Abstract

A Riemannian manifold ($\mathcal{M}^n$ ⁿ, g) is said to be the center of thecomplex manifold $\mathcal{X}^n$ ⁿ if $\mathcal{M}$ is the zero set of a smooth strictly plurisubharmonic exhaustion function ν² on $\mathcal{X}$ such that ν is plurisubharmonic and solves theMonge–Ampère equation (∂$\bar \partial$ν)ⁿ = 0 off $\mathcal{M}$, and g is induced by the canonical Kähler metric withfundamental two-form $- \sqrt { - 1}$ ∂$\bar \partial$ν². Insisting that ν be unbounded puts severe restrictions on $\mathcal{X}$ as acomplex manifold as well as on ($\mathcal{M}$, g). It is an open problemto determine the class Riemannian manifolds that are centers of complexmanifolds with unbounded ν. Before the present work, the list of knownexamples of manifolds in that class was small. In the main result of thispaper we show, by means of the moment map corresponding to isometric actionsand the associated bundle construction, that such class is larger than originally thought and contains many metrically and diffeomorphically`exotic' examples.