Limits of an increasing sequence of complex manifolds

Abstract

Let M be a complex manifold which admits an exhaustion by open subsets M j each of which is biholomorphic to a fixed domain ⊂ C n . The main question addressed here is to describe M in terms of . Building on work of Fornaess-Sibony, we study two cases, namely M is Kobayashi hyperbolic and the other being the corank one case in which the Kobayashi metric degenerates along one direction. When M is Kobayashi hyperbolic, its complete description is obtained when is one of the following domains-(i) a smoothly bounded Levi corank one domain, (ii) a smoothly bounded convex domain, (iii) a strongly pseudoconvex polyhedral domain in C 2 , or (iv) a simply connected domain in C 2 with generic piecewise smooth Levi-flat boundary. With additional hypotheses, the case when is the minimal ball or the symmetrized polydisc in C n can also be handled. When the Kobayashi metric on M has corank one and is either of (i), (ii) or (iii) listed above, it is shown that M is biholomorphic to a locally trivial fibre bundle with fibre C over a holomorphic retract of or that of a limiting domain associated with it. Finally, when = × B n−1 , the product of the unit disc ⊂ C and the unit ball B n−1 ⊂ C n−1 , a complete description of holomorphic retracts is obtained. As a consequence, if M is Kobayashi hyperbolic and = ×B n−1 , it is shown that M is biholomorphic to . Further, if the Kobayashi metric on M has corank one, then M is globally a product; in fact, it is biholomorphic to Z ×C, where Z ⊂ = ×B n−1 is a holomorphic retract.