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

Let X be a connected, compact complex manifold, S⊂X a separating real hypersurface, so X decomposes as a union of compact complex manifolds with boundary X ¯ ± with X ¯ + ∩X ¯ - =S. Let 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 𝒞 κ+1 – of its restriction to S. Our problem: compare, via the obvious restriction maps, the moduli space ℳ to the corresponding Donaldson’s moduli spaces ℳ ± of boundary framed formally holomorphic bundles on X ¯ ±. The restrictions to X ¯ ± of an S-framed holomorphic bundle (E, θ) are boundary framed formally holomorphic bundles (E ±, θ ±) which induce, via θ ±, the same tangential Cauchy–Riemann operator on the trivial bundle on S. Therefore one obtains a natural map from ℳ into the fiber product ℳ - × 𝒞 ℳ + over the space 𝒞 of Cauchy–Riemann operators on the trivial bundle on S. Our main result states: this map is a homeomorphism for κ∈ (0, ∞]∖ℕ. Note that, by theorems due to S. Donaldson and Z. Xi, the moduli spaces ℳ ± can be further identified with moduli spaces of boundary framed Hermitian Yang-Mills connections. The proof of our isomorphism theorem is based on a gluing principle for formally holomorphic bundles along a real hypersurface. The same gluing theorem can be used to give a complex geometric interpretation of the space of solutions of a large class of Riemann–Hilbert type problems. We generalize these results in two directions: first, we will replace the decomposition X=X ¯ - ∪X ¯ + associated with a separating hypersurface by the manifold with boundary X ^ S obtained by cutting X along any (not necessarily separating) oriented hypersurface S. Second, instead of vector bundles, we will 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.