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 $$\Omega \subset \mathbb C^n$$ . The main question addressed here is to describe M in terms of $$\Omega $$ . 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 $$\Omega $$ 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 $$\mathbb C^2$$ , or (iv) a simply connected domain in $$\mathbb C^2$$ with generic piecewise smooth Levi-flat boundary. With additional hypotheses, the case when $$\Omega $$ is the minimal ball or the symmetrized polydisc in $$\mathbb C^n$$ can also be handled. When the Kobayashi metric on M has corank one and $$\Omega $$ is either of (i), (ii) or (iii) listed above, it is shown that M is biholomorphic to a locally trivial fibre bundle with fibre $$\mathbb C$$ over a holomorphic retract of $$\Omega $$ or that of a limiting domain associated with it. Finally, when $$\Omega = \Delta \times \mathbb B^{n-1}$$ , the product of the unit disc $$\Delta \subset \mathbb C$$ and the unit ball $$\mathbb B^{n-1} \subset \mathbb C^{n-1}$$ , a complete description of holomorphic retracts is obtained. As a consequence, if M is Kobayashi hyperbolic and $$\Omega = \Delta \times \mathbb B^{n-1}$$ , it is shown that M is biholomorphic to $$\Omega $$ . Further, if the Kobayashi metric on M has corank one, then M is globally a product; in fact, it is biholomorphic to $$Z \times \mathbb C$$ , where $$Z \subset \Omega = \Delta \times \mathbb B^{n-1}$$ is a holomorphic retract.