Holomorphic bundles on complex manifolds with boundary

TL;DR

This paper proves that under strict pseudoconvexity, a formally integrable bundle almost complex structure on a complex manifold with boundary extends holomorphically, generalizing a problem posed by Donaldson and exploring boundary trivializations.

Contribution

It establishes extension results for bundle almost complex structures on manifolds with boundary and provides a gauge theoretical interpretation of boundary quotient spaces.

Findings

Extension of bundle almost complex structures under pseudoconvex boundary conditions.

A gauge theoretical interpretation of boundary quotient spaces for Stein manifolds.

Examples showing the density and non-existence of holomorphic trivializations at boundary points.

Abstract

Let $\Omega$ be a complex manifold, and let $X\subset \Omega$ be an open submanifold whose closure $\bar X$ is a (not necessarily compact) submanifold with smooth boundary. Let $G$ be a complex Lie group, $\Pi$ be a differentiable principal $G$-bundle on $\Omega$ and $J$ a formally integrable bundle almost complex structure on the restriction $\bar P:= \Pi|_{\bar X}$. We prove that, if the boundary of $\bar X$ is strictly pseudoconvex, $J$ extends to a holomorphic structure on the restriction of $\Pi$ to a neighborhood of $\bar X$ in $\Omega$. This answers positively and generalizes a problem stated in the article "Boundary value problems for Yang-Mills fields" by S. Donaldson. We obtain a gauge theoretical interpretation of the quotient $\mathcal{C}^\infty(\partial \bar X,G)/\mathcal{O}^\infty(\bar X,G)$ associated with any compact Stein manifold with boundary $\bar X$ endowed with a Hermitian metric. For a fixed differentiable $G$-bundle $\bar P$ on a complex manifold $\bar X$ with non-pseudoconvex boundary, we study the set of formally integrable almost complex structures on $\bar P$ which admit formally holomorphic local trivializations at boundary points. We give an example where a "generic" formally integrable almost complex on $\bar P$ admits formally holomorphic local trivializations at no boundary point, whereas the set of formally integrable almost complex structures which admit formally holomorphic local trivializations at all boundary points is dense.