Andrei Teleman
Aix-Marseille Université, Marseille, France

Let $X$ be a connected, compact complex manifold, and $S\subset X$ be a separating real hypersurface. $X$ decomposes as a union of compact complex manifolds with boundary $\bar X^\pm$ with $\bar X^+\cap \bar X^-=S$. Let $\mathcal{M}$ be the moduli space of $S$-framed holomorphic bundles on $X$, i.e. of pairs $(E,\theta)$ (of fixed topological type) consisting of a holomorphic bundle $E$ on $X$ endowed with a differentiable trivialization $\theta$ on $S$. This moduli space is the main object of a joint research project with Matei Toma.

The problem addressed in my talk: compare, via the obvious restriction maps, the moduli space $\mathcal{M}$ with the corresponding Donaldson moduli spaces $\mathcal{M}^\pm$ of boundary framed holomorphic bundles on $\bar X^\pm$. 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 $S$. Therefore 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 result states: this map is bijective. Note that, by theorems due to S. Donaldson and Z. Xi, the moduli spaces $\mathcal{M}^\pm$ can be identified with moduli spaces of boundary framed Hermitian Yang-Mills connections.