The Oka principle in higher twisted K-theory

Abstract

The Oka principle is a heuristic in complex geometry which states that, for a wide class of complex-analytic problems concerning Stein spaces, any obstruction to finding a holomorphic solution is purely topological. A classical theorem of H.~Grauert implies that for a reduced Stein space $X$, the natural map $K^{0, \mathcal{O}}(X) \to K^{0, \mathcal{C}}(X)$ from ordinary holomorphic K-theory $K^{0, \mathcal{O}}(X)$ to ordinary topological K-theory $K^{0, \mathcal{C}}(X)$ is an isomorphism: this is the basic manifestation of the Oka principle in K-theory. In this paper, we generalise this theorem to higher twisted K-theory. For a reduced Stein space $X$ and a torsion class $\alpha \in H^3(X,\mathbb{Z})$, we prove that the natural map $K^{-n,\mathcal{O}}_\alpha(X) \to K^{-n,\mathcal{C}}_\alpha(X)$ is an isomorphism for all $n \geq 0$. We introduce the first definition of higher twisted holomorphic K-theory in the literature, defined through a simplicially enriched version of Quillen's $S^{-1}S$ construction. Our parallel construction for topological higher twisted K-theory is a new formulation which is compatible with existing theory. The proof of the main theorem employs Cartan-Grauert cohomological methods and an equivalence, which we prove, between the simplicial symmetric monoidal categories of holomorphic and topological $\alpha$-twisted vector bundles.