The Oka principle for tame families of Stein manifoldsForstnerič, Franc (Avtor) Sigurðardóttir, Álfheiður Edda (Avtor) Stein manifoldOka principleOka manifoldvector bundleLet $X$ 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.20262026-07-17 12:19:25Neznano31171UDK: 517.5ISSN pri članku: 2330-0000DOI: 10.1090/btran/257COBISS_ID: 285245187sl