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

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 $\kappa \in (0,\mathrm{\infty }]\setminus \mathbb{N}$. 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.