| Title: | The Oka principle for tame families of Stein manifolds |
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| Authors: | ID Forstnerič, Franc (Author)
ID Sigurðardóttir, Álfheiður Edda (Author) |
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| Files: |  PDF - Presentation file, download (537,43 KB)
MD5: 4C88BDBFB24FE0E37CC6BEE01D128968
 URL - Source URL, visit https://pubs.ams.org/journals/btran/2026-13-14/S2330-0000-2026-00257-2
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| Language: | English |
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| Typology: | 1.01 - Original Scientific Article |
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| Organization: |  IMFM - Institute of Mathematics, Physics, and Mechanics
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| Abstract: | Let $X$ be a smooth open manifold of even dimension, $T$ be a topological space, and ${\mathscr J}=\{J_t\}_{t\in T}$ be a continuous family of smooth integrable Stein structures on $X$. Under suitable additional assumptions on $T$ and ${\mathscr J}$, we prove an Oka principle for continuous families of maps from the family of Stein manifolds $(X,J_t)$, $t\in T$, to any Oka manifold, showing that every family of continuous maps is homotopic to a family of $J_t$-holomorphic maps depending continuously on $t$. We also prove the Oka-Weil theorem for sections of ${\mathscr J}$-holomorphic vector bundles on $Z = T \times X$ and the Oka principle for isomorphism classes of such bundles. The assumption on the family ${\mathscr J}$ is that the $J_t$-convex hulls of any compact set in $X$ are upper semicontinuous with respect to $t \in T$; such a family is said to be tame. For suitable parameter spaces $T$, we characterise tameness by the existence of a continuous family $\rho_t:X\to {\mathbb R}_+ = [0,+\infty)$, $t\in T$, of strongly $J_t$-plurisubharmonic exhaustion functions on $X$. Every family of complex structures on an open orientable surface is tame. We give an example of a nontame smooth family of Stein structures $J_t$ on ${\mathbb R}^{2n} (t \in {\mathbb R}, n > 1)$ such that $({\mathbb R}^{2n}, J_t)$ is biholomorphic to ${\mathbb C}^n$ for every $t\in{\mathbb R}$. We show that the Oka principle fails on any nontame family. |
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| Keywords: | Stein manifold, Oka principle, Oka manifold, vector bundle |
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| Publication status: | Published |
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| Publication version: | Version of Record |
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| Publication date: | 01.01.2026 |
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| Year of publishing: | 2026 |
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| Number of pages: | str. 477-511 |
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| Numbering: | Vol. 13 |
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| PID: | 20.500.12556/DiRROS-31171  |
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| UDC: | 517.5 |
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| ISSN on article: | 2330-0000 |
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| DOI: | 10.1090/btran/257 |
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| COBISS.SI-ID: | 285245187 |
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| Note: |
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| Publication date in DiRROS: | 17.07.2026 |
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| Views: | 31 |
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| Downloads: | 11 |
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