论文标题
与边界的复杂歧管上的全态束
Holomorphic bundles on complex manifolds with boundary
论文作者
论文摘要
令$ω$为一个复杂的歧管，让$ x \ subsetω$为一个开放的子手法，其闭合$ \ bar x $是（不一定是紧凑的）子曼if，并具有光滑的边界。令$ g $为一个复杂的谎言组，$π$是$ω$上的可区分的主体$ g $ - 捆绑物，$ j $正式集成的捆绑包几乎是限制$ \ bar p：=π| _ {\ bar x} $。我们证明，如果$ \ bar x $的边界严格为伪元，则$ j $扩展到$ \ bar x $ in $ω$的$ \ bar x $的限制的全态结构。这是积极的答案，并概括了S. Donaldson的文章“ Yang-Mills Fields的边界价值问题”中所述的问题。我们获得了商的理论解释$ \ MATHCAL {C}^\ infty（\ partial \ bar x，g）/\ Mathcal {o}^\ infty（\ bar x，g）$与与边界$ \ bar x $ enderian n e hermitian nmitian metric相关的任何紧凑型Stein流形。 对于一个固定的$ g $ -Bundle $ \ bar p $，在复杂的歧管$ \ bar x $带有非peudoconvex边界上，我们研究了$ \ bar p $上正式可正式整合的几乎复杂结构的集合，这些结构正式承认在边界点正式正式置换了荷兰的本地琐事。我们举了一个例子，其中一个“通用”在$ \ bar p $上几乎可以正式整合复杂的复合物，在无边界点处正式地构成了塑形的局部琐碎化，而正式集成的几乎可以集成的几乎复杂的结构集合正式承认在所有边界点处正式构成全体形态的局部琐碎。
Let $Ω$ be a complex manifold, and let $X\subset Ω$ be an open submanifold whose closure $\bar X$ is a (not necessarily compact) submanifold with smooth boundary. Let $G$ be a complex Lie group, $Π$ be a differentiable principal $G$-bundle on $Ω$ and $J$ a formally integrable bundle almost complex structure on the restriction $\bar P:= Π|_{\bar X}$. We prove that, if the boundary of $\bar X$ is strictly pseudoconvex, $J$ extends to a holomorphic structure on the restriction of $Π$ to a neighborhood of $\bar X$ in $Ω$. 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.
