The Oka principle for tame families of Stein manifolds

TL;DR

This paper establishes an Oka principle for continuous families of Stein structures on open manifolds, demonstrating homotopy equivalence of continuous and holomorphic maps under tameness conditions, with applications to vector bundles.

Contribution

It introduces the concept of tameness for families of Stein structures and proves the Oka principle, Oka-Weil theorem, and classification results for such families.

Findings

Tame families allow the Oka principle to hold for continuous families of Stein structures.

Every family of complex structures on an open orientable surface is tame.

The Oka principle fails for non-tame families, exemplified by certain smooth families on n.

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 $\R^{2n}$ $(t\in \mathbb{R},\ n>1)$ 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.