Limits of an increasing sequence of complex manifolds

Abstract

Let M be a complex manifold which admits an exhaustion by open subsets Mj each of which is biholomorphic to a fixed domain Ω⊂Cn. The main question addressed here is to describe M in terms of Ω. 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 Ω 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 C2, or (iv) a simply connected domain in C2 with generic piecewise smooth Levi-flat boundary. With additional hypotheses, the case when Ω is the minimal ball or the symmetrized polydisc in Cn can also be handled. When the Kobayashi metric on M has corank one and Ω 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 Ω or that of a limiting domain associated with it. Finally, when Ω=Δ×Bn−1, the product of the unit disc Δ⊂C and the unit ball Bn−1⊂Cn−1, a complete description of holomorphic retracts is obtained. As a consequence, if M is Kobayashi hyperbolic and Ω=Δ×Bn−1, it is shown that M is biholomorphic to Ω. 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⊂Ω=Δ×Bn−1 is a holomorphic retract.