Holomorphic bundles framed along a real hypersurface and the Riemann-Hilbert problem
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| Format: | Preprint |
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2023
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| _version_ | 1866918076531867648 |
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| author | Teleman, Andrei |
| author_facet | Teleman, Andrei |
| contents | Let $X$ be a connected, compact complex manifold and $S\subset X$ a separating real hypersurface, so that $X$ decomposes as a union of compact complex manifolds with boundary $\bar X^\pm$. Let $\mathcal{M}$ be the moduli space of $S$-framed holomorphic bundles, i.e. of pairs $(E,θ)$ of fixed topological type consisting of a holomorphic bundle $E$ on $X$ and a trivialization $θ$ - belonging to a fixed Hölder regularity class $\mathcal{C}^{κ+1}$ - of its restriction to $S$.
The restrictions to $\bar X^\pm$ of an $S$-framed holomorphic bundle $(E,θ)$ are boundary framed formally holomorphic bundles $(E^\pm,θ^\pm)$ which induce, via $θ^\pm$, the same tangential Cauchy-Riemann operators on the trivial bundle on $S$, so one obtains a natural map from $\mathcal{M}$ into the fiber product $\mathcal{M}^-\times_\mathcal{C}\mathcal{M}^+$ over the space $\mathcal{C}$ of Cauchy-Riemann operators on the trivial bundle on $S$. Our main result states: this map is a homeomorphism for $κ\in (0,\infty]\setminus\mathbb{N}$.
The proof is based on a gluing principle for formally holomorphic bundles along a real hypersurface. This principle can also be used to give a complex geometric interpretation of the space of solutions of a large class of Riemann-Hilbert type problems. The results generalize in two directions: first one can replace the decomposition associated with a separating hypersurface by the the manifold with boundary $\widehat X_S$ obtained by cutting $X$ along any oriented hypersurface $S$. Second one can consider principal $G$ bundles for an arbitrary complex Lie group $G$. We give explicit examples of moduli spaces of (boundary) framed holomorphic bundles and explicit formulae for the homeomorphisms provided by the general results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_04341 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Holomorphic bundles framed along a real hypersurface and the Riemann-Hilbert problem Teleman, Andrei Complex Variables Differential Geometry 32L05, 32G13, 35Q15 Let $X$ be a connected, compact complex manifold and $S\subset X$ a separating real hypersurface, so that $X$ decomposes as a union of compact complex manifolds with boundary $\bar X^\pm$. Let $\mathcal{M}$ be the moduli space of $S$-framed holomorphic bundles, i.e. of pairs $(E,θ)$ of fixed topological type consisting of a holomorphic bundle $E$ on $X$ and a trivialization $θ$ - belonging to a fixed Hölder regularity class $\mathcal{C}^{κ+1}$ - of its restriction to $S$. The restrictions to $\bar X^\pm$ of an $S$-framed holomorphic bundle $(E,θ)$ are boundary framed formally holomorphic bundles $(E^\pm,θ^\pm)$ which induce, via $θ^\pm$, the same tangential Cauchy-Riemann operators on the trivial bundle on $S$, so one obtains a natural map from $\mathcal{M}$ into the fiber product $\mathcal{M}^-\times_\mathcal{C}\mathcal{M}^+$ over the space $\mathcal{C}$ of Cauchy-Riemann operators on the trivial bundle on $S$. Our main result states: this map is a homeomorphism for $κ\in (0,\infty]\setminus\mathbb{N}$. The proof is based on a gluing principle for formally holomorphic bundles along a real hypersurface. This principle can also be used to give a complex geometric interpretation of the space of solutions of a large class of Riemann-Hilbert type problems. The results generalize in two directions: first one can replace the decomposition associated with a separating hypersurface by the the manifold with boundary $\widehat X_S$ obtained by cutting $X$ along any oriented hypersurface $S$. Second one can consider principal $G$ bundles for an arbitrary complex Lie group $G$. We give explicit examples of moduli spaces of (boundary) framed holomorphic bundles and explicit formulae for the homeomorphisms provided by the general results. |
| title | Holomorphic bundles framed along a real hypersurface and the Riemann-Hilbert problem |
| topic | Complex Variables Differential Geometry 32L05, 32G13, 35Q15 |
| url | https://arxiv.org/abs/2310.04341 |