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.