Limits of an increasing sequence of complex manifolds

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