  A (bounded) manifold of circular type is a complex manifold M of dimension n
admitting a (bounded) exhaustive real function u, defined on M minus a point
x_o, so that: a) it is a smooth solution on $M\setminus {x_o}$ to the
Monge-Amp\`ere equation $(d d^c u)^n = 0$; b) x_o is a singular point for u of
logarithmic type and e^u extends smoothly on the blow up of M at x_o; c) $d d^c
(e^u) >0$ at any point of $M\setminus {x_o}$. This class of manifolds naturally
includes all smoothly bounded, strictly linearly convex domains and all
smoothly bounded, strongly pseudoconvex circular domains of $\bC^n$. The moduli
spaces of bounded manifolds of circular type are studied. In particular, for
each biholomorphic equivalence class of them it is proved the existence of an
essentially unique manifold in normal form. It is also shown that the class of
normalizing maps for an n-dimensional manifold M is a new holomorphic invariant
with the following property: it is parameterized by the points of a finite
dimensional real manifold of dimension n^2 when M is a (non-convex) circular
domain while it is of dimension $n^2 + 2 n$ when M is a strictly convex domain.
New characterizations of the circular domains and of the unit ball are also
obtained.

Complex Variables