THE OKA PRINCIPLE FOR TAME FAMILIES OF STEIN MANIFOLDS

Journal article, 2026

Let X be a smooth open manifold of even dimension, T be a topological space, and J = {Jt }t∈T be a continuous family of smooth integrable Stein structures on X. Under suitable additional assumptions on T and J, we prove an Oka principle for continuous families of maps from the family of Stein manifolds (X, Jt), t ∈ T, to any Oka manifold, showing that every family of continuous maps is homotopic to a family of Jt-holomorphic maps depending continuously on t. We also prove the Oka–Weil theorem for sections of J-holomorphic vector bundles on Z = T × X and the Oka principle for isomorphism classes of such bundles. The assumption on the family J is that the Jt-convex hulls of any compact set in X are upper semicontinuous with respect to t ∈ T; such a family is said to be tame. For suitable parameter spaces T, we characterise tameness by the existence of a continuous family ρt: X → R+ = [0, +∞), t ∈ T, of strongly Jt-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 Jt on R2n (t ∈ R, n > 1) such that (R2n, Jt) is biholomorphic to Cn for every t ∈ R. We show that the Oka principle fails on any nontame family.