The Oka principle for tame families of Stein manifolds

TL;DR

This work establishes an Oka principle for parametric families of Stein structures by introducing tameness, a condition preventing pathological hull growth. Using a parametric Hamilton (global Newlander–Nirenberg) theorem and stability of the $\overline{\partial}$-operator, the authors prove that every continuous map from a tame family of Stein manifolds to an Oka manifold can be deformed to a $\mathscr{J}$-holomorphic map with continuous dependence on the parameter, and they derive an Oka–Weil theorem for fibrewise holomorphic vector bundles. They extend these results to sections of fibre bundles and to the classification of complex vector bundles, thereby giving an Oka principle for vector-bundle isomorphisms in the family setting. The paper also provides a global analysis toolkit in this context, including a global $\overline{\partial}$-solution theory and a vector-bundle extension framework, and demonstrates tameness is necessary via explicit non-tame examples. Together, these results broaden Oka theory to deformation families and open new avenues for holomorphic mapping problems on varying complex structures.

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 $ρ_t:X\to \mathbb{R}_+=[0,+\infty)$, $t\in T$, 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.

Figures (4)

• Figure 1: An upper semicontinuous family of hulls $\widehat{K}_{\! J_t}$.
• Figure 2: The shaded areas depict the sets $\Phi_t(K_t)\subset \Phi_t(\Omega)$ for $t\in T_0$.
• Figure 3: An illustration of the choice of the set $L_2$.
• Figure 4: The set $S^2$.

Theorems & Definitions (47)

• Theorem 1.1
• Theorem 2.1
• Theorem 3.1
• Theorem 3.2
• Proposition 3.3: Sobolev embedding theorem
• proof : Proof of Theorem \ref{['th:Hamilton']}
• Theorem 4.1
• proof
• Lemma 4.2
• proof