The Oka principle for tame families of Stein manifolds
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914123690803200 |
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| author | Forstneric, Franc Sigurdardottir, Alfheidur Edda |
| author_facet | Forstneric, Franc Sigurdardottir, Alfheidur Edda |
| contents | 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 $ρ_t:X\to \mathbb{R}_+=[0,+\infty)$, $t\in T$, 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 $\R^{2n}$ $(t\in \mathbb{R},\ n>1)$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_25887 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Oka principle for tame families of Stein manifolds Forstneric, Franc Sigurdardottir, Alfheidur Edda Complex Variables Primary 32Q56, Secondary 32E10, 32H02 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 $ρ_t:X\to \mathbb{R}_+=[0,+\infty)$, $t\in T$, 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 $\R^{2n}$ $(t\in \mathbb{R},\ n>1)$ 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. |
| title | The Oka principle for tame families of Stein manifolds |
| topic | Complex Variables Primary 32Q56, Secondary 32E10, 32H02 |
| url | https://arxiv.org/abs/2510.25887 |