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 H\"older 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 $\widehat X_S$ 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.