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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2509.24362 |
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| _version_ | 1866918150447038464 |
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| author | Sridharan, Haripriya |
| author_facet | Sridharan, Haripriya |
| contents | 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 $α\in H^3(X,\mathbb{Z})$, we prove that the natural map $K^{-n,\mathcal{O}}_α(X) \to K^{-n,\mathcal{C}}_α(X)$ 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 $α$-twisted vector bundles. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_24362 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Oka principle in higher twisted K-theory Sridharan, Haripriya K-Theory and Homology Complex Variables 19L50, 32Q56, 19D23, 32Q28 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 $α\in H^3(X,\mathbb{Z})$, we prove that the natural map $K^{-n,\mathcal{O}}_α(X) \to K^{-n,\mathcal{C}}_α(X)$ 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 $α$-twisted vector bundles. |
| title | The Oka principle in higher twisted K-theory |
| topic | K-Theory and Homology Complex Variables 19L50, 32Q56, 19D23, 32Q28 |
| url | https://arxiv.org/abs/2509.24362 |