AI 中文总结
该研究针对复流形上的全纯主G丛，在严格伪凸延拓非临界或拓扑丛可延拓的假设下，证明其可延拓，为Donaldson提出的延拓问题提供新证明与推广。
AI 中文摘要
设$\mathcal{X}$为复流形，$X$为$\mathcal{X}$中带边界的紧复流形。对于复李群$G$与正则性类$$\mathfrak{r}\in \big\{\mathcal{C}^k|\ k\in\mathbb{N}\cup\{\infty\}\big\}\cup\big\{Λ^r_{\rm loc}|\ r\in (0,\infty)\big\} $$，我们通过下式定义$X$上的群层$\mathcal{O}^{\mathfrak{r}\,G}_X$：\n\begin{align*}\n\mathcal{O}^{\mathfrak{r}\\,G}_X(V):=\{u\in \mathcal{C}(V,G)|\\ &u \text{ 在$V$上具有正则性类$\mathfrak{r}$}, \n&u|_{V\cap\mathrm{int}(X)} \text{ 是全纯的}\}.\n\end{align*}\n设$X_0\Subset Z_0\Subset \mathcal{X}$为$\mathcal{X}$中的严格伪凸延拓，$X:= \bar X_0$、$Z:= \bar Z_0$为$\mathcal{X}$中对应的带边界紧流形。设$K\subset X_0$为紧集，$\mathcal{P}$为$X\setminus K$上的$\mathfrak{r}$类全纯主$G$-丛（即$\mathcal{O}^{\mathfrak{r}\,G}_X$- torsor）。假设\n（H1）给定的严格伪凸延拓$X_0\Subset Z_0$是非临界的，\n或者\n（H2）$\mathcal{P}$的底层拓扑丛可延拓至$Z\setminus K$，\n我们证明$\mathcal{P}$可延拓至$Z\setminus K$。\n这为S. Donaldson发表于《Journal of Geometry and Physics》第8卷（1992年）的《Yang-Mills场的边值问题》一文中提出、且作者此前一篇文章用不同方法研究过的延拓问题，提供了一种新证明与新推广。
英文摘要
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 $$\mathfrak{r}\in \big\{\mathcal{C}^k|\ k\in\mathbb{N}\cup\{\infty\}\big\}\cup\big\{Λ^r_{\rm loc}|\ r\in (0,\infty)\big\} $$ we define the sheaf of groups $\mathcal{O}^{\mathfrak{r}\,G}_X$ 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 $X_0\Subset Z_0\Subset \mathcal{X}$ 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 $\mathcal{O}^{\mathfrak{r}\,G}_X$-torsor) on $X\setminus K$. Assuming that (H1) the given strictly pseudoconvex extension $X_0\Subset Z_0$ 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.
