Moduli spaces of holomorphic bundles framed along a real hypersurface

Andrei Teleman, Aix-Marseille University

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.

This seminar will be held both online and in person:

• Room: MC 5417
• Zoom link: https://uwaterloo.zoom.us/j/96883292635?pwd=KytGYnEvRmhyTTV1NC9Gc2dnT05oQT09