Parametric equivariant Oka principle
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909883004092416 |
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| author | Kutzschebauch, Frank Larusson, Finnur Schwarz, Gerald W. |
| author_facet | Kutzschebauch, Frank Larusson, Finnur Schwarz, Gerald W. |
| contents | Let $G$ be a reductive complex Lie group and $K$ be a maximal compact subgroup of $G$. Let $X$ be a reduced Stein $G$-space and $Y$ be a $G$-elliptic manifold. We prove the following parametric equivariant Oka principle. The inclusion of the space of holomorphic $G$-maps $X\to Y$ into the space of continuous $K$-maps $X\to Y$ is a weak homotopy equivalence with respect to the compact-open topology. The proof is divided into a homotopy-theoretic part, which is handled by an abstract theorem of Studer, and an analytic part, for which we prove equivariant versions of the homotopy approximation theorem and the nonlinear splitting lemma that are key tools in Oka theory. The principle can be strengthened so as to allow interpolation on a $G$-invariant subvariety of $X$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_01189 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Parametric equivariant Oka principle Kutzschebauch, Frank Larusson, Finnur Schwarz, Gerald W. Complex Variables Algebraic Geometry Algebraic Topology Representation Theory Primary 32M05. Secondary 14L24, 14L30, 32E10, 32E30, 32M10, 32Q28, 32Q56 Let $G$ be a reductive complex Lie group and $K$ be a maximal compact subgroup of $G$. Let $X$ be a reduced Stein $G$-space and $Y$ be a $G$-elliptic manifold. We prove the following parametric equivariant Oka principle. The inclusion of the space of holomorphic $G$-maps $X\to Y$ into the space of continuous $K$-maps $X\to Y$ is a weak homotopy equivalence with respect to the compact-open topology. The proof is divided into a homotopy-theoretic part, which is handled by an abstract theorem of Studer, and an analytic part, for which we prove equivariant versions of the homotopy approximation theorem and the nonlinear splitting lemma that are key tools in Oka theory. The principle can be strengthened so as to allow interpolation on a $G$-invariant subvariety of $X$. |
| title | Parametric equivariant Oka principle |
| topic | Complex Variables Algebraic Geometry Algebraic Topology Representation Theory Primary 32M05. Secondary 14L24, 14L30, 32E10, 32E30, 32M10, 32Q28, 32Q56 |
| url | https://arxiv.org/abs/2511.01189 |