An extension theorem for bundles with respect to strictly pseudoconvex extensions

Abstract

Let $\mathcal{X}$ be a complex manifold and $X$ a compact complex manifold with boundary in $\mathcal{X}$. For a complex Lie group $G$ and a regularity class $$\mathfrak{r}\in \{{\mathcal{C}}^k|k\in \mathbb{N}\cup \{\infty \}\}\cup \{{\Lambda }_{\mathrm{loc}}^r|r\in (0,\infty )\}$$ we define the sheaf of groups ${\mathcal{O}}_X^{\mathfrak{r}G}$ on $X$ by \begin{align*} \mathcal{O}^{\mathfrak{r}\,G}_X(V):=\{u\in \mathcal{C}(V,G)|\ &u \hbox{ has regularity class $\mathfrak{r}$ on $V$}, \\ &u|_{V\cap\mathrm{int}(X)} \hbox{ is holomorphic}\}. \end{align*} Let $X_0⋐Z_0⋐\mathcal{X}$ be a strictly pseudoconvex extension in $\mathcal{X}$, and let $X:={\bar{X}}_0$, $Z:={\bar{Z}}_0$ be the corresponding compact manifolds with boundary in $\mathcal{X}$. Let $K\subset X_0$ be a compact set and $\mathcal{P}$ a holomorphic principal $G$-bundle of class $\mathfrak{r}$ (i.e. an ${\mathcal{O}}_X^{\mathfrak{r}G}$-torsor) on $X\setminus K$. Assuming that (H1) the given strictly pseudoconvex extension $X_0⋐Z_0$ is non-critical, or that (H2) the underlying topological bundle of $\mathcal{P}$ extends to $Z\setminus K$, we prove that $\mathcal{P}$ admits an extension to $Z\setminus K$. This gives a new proof and a new generalisation of the extension problem stated in the article S. Donaldson, Boundary value problems for Yang-Mills fields, Journal of Geometry and Physics 8, (1992) and studied with different methods in a previous article of the author.