Balakumar, GP and Borah, D and Mahajan, P and Verma, K (2022) Limits of an increasing sequence of complex manifolds. In: Annali di Matematica Pura ed Applicata .
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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.
| Item Type: | Journal Article |
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| Publication: | Annali di Matematica Pura ed Applicata |
| Publisher: | Institute for Ionics |
| Additional Information: | The copyright for this article belongs to Author(S). |
| Keywords: | Kobayashi corank one; Kobayashi hyperbolic; Levi corank one domains; Union problem |
| Department/Centre: | Division of Physical & Mathematical Sciences > Mathematics |
| Date Deposited: | 09 Jan 2023 07:10 |
| Last Modified: | 09 Jan 2023 07:10 |
| URI: | https://eprints.iisc.ac.in/id/eprint/78911 |
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