20.500.12556/DiRROS-31171 The Oka principle for tame families of Stein manifolds 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. Stein manifold Oka principle Oka manifold vector bundle Steinova mnogoterost princip Oka mnogoterost Oka vektorski sveženj true false true Angleški jezik Slovenski jezik Neznano 2026-07-17 12:19:25 2026-07-17 12:19:26 2026-07-18 03:45:03 0000-00-00 00:00:00 2026 0 0 str. 477-511 Vol. 13 2026 0000-00-00 Zaloznikova Objavljeno NiDoloceno 0000-00-00 0000-00-00 2026-01-01 517.5 2330-0000 10.1090/btran/257 285245187 S2330-0000-2026-00257-2-1.pdf S2330-0000-2026-00257-2-1.pdf 1 4C88BDBFB24FE0E37CC6BEE01D128968 c8c9f5e6d9c8918d2eb97f8902e17169f1b36588bffc15212dd3c37e5e4c3319 52c7312b-81c9-11f1-a19f-001a4af901a5 https://dirros.openscience.si/Dokument.php?lang=slv&id=48082 https://pubs.ams.org/journals/btran/2026-13-14/S2330-0000-2026-00257-2 1 fedb2e76-81c8-11f1-a19f-001a4af901a5 https://dirros.openscience.si/Dokument.php?lang=slv&id=48081 Inštitut za matematiko, fiziko in mehaniko 0 0 0