Holomorphic bundles framed along a real hypersurface and the Riemann-Hilbert problem

Abstract

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 $(E,\theta )$ of fixed topological type consisting of a holomorphic bundle $E$ on $X$ and a trivialization $\theta$ - belonging to a fixed Hölder regularity class ${\mathcal{C}}^{\kappa +1}$ - of its restriction to $S$. The restrictions to ${\bar{X}}^{\pm }$ of an $S$-framed holomorphic bundle $(E,\theta )$ are boundary framed formally holomorphic bundles $(E^{\pm },{\theta }^{\pm })$ which induce, via ${\theta }^{\pm }$, 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 $\kappa \in (0,\infty ]\setminus \mathbb{N}$. 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 ${\hat{X}}_S$ 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.

53 pages, minor corrections in the revised version. To appear in the "Annales de la Faculté des Sciences de Toulouse". Second revision: minor corrections