论文标题
通过捆绑模型的一类分析希尔伯特模块的一类的几何不变剂
Geometric invariants for a class of submodules of analytic Hilbert modules via the sheaf model
论文作者
论文摘要
令$ω\ subseteq \ mathbb c^m $为有界连接的开放式集合，$ \ MATHCAL H \ subseteq \ Mathcal O（ω）$是一个分析性的希尔伯特模块，即希尔伯特太空$ \ Mathcal H $具有重现的Kernel $ k $ k $ $ k $ ythe $ knomial mathbb c [\ boldsymbol {z}] \ subseteq \ mathcal h $是密集的，并且由$ p \ in \ mathbb c [\ boldsymbol {z}] $在$ \ mathcal h $上界限。我们修复了由$ p_1，\ ldots，p_t $生成的理想$ \ mathcal i \ subseteq \ mathbb c [\ boldsymbol {z}] $，让$ [\ mathcal i] $表示完成$ \ mathcal i $ in $ \ \ \ \ mathcal h $。与分析性希尔伯特模块相关的捆绑$ \ MATHCAL S^\ MATHCAL H $ MATHCAL H $是$ω$上的sheaf $ \ Mathcal o（ω）$（ω）$，因此是免费的。但是，与$ [\ Mathcal i] $关联的subsheaf $ \ mathcal s^{\ Mathcal [\ Mathcal I]} $是连贯的，不一定是本地免费的。在\ cite {bmp}的早期作品的基础上，我们为连贯的捆式杂色的遗产结构开了一个遗产结构，并使用它来查找可拖动的不变性。此外，我们证明，如果零设置$ v _ {[\ Mathcal I]} $是Codimension $ t $的子手机，那么沿零套件的kernel $ k _ {[\ MATHCAL I]} $沿零集可以用作holomorphic brine for vector bunce $ v y的零集合，将有独特的本地分解。该矢量束的复杂几何不变性也是subsodule $ [\ mathcal i] \ subseteq \ Mathcal H $的单一不变性。
Let $Ω\subseteq \mathbb C^m$ be a bounded connected open set and $\mathcal H \subseteq \mathcal O(Ω)$ be an analytic Hilbert module, i.e., the Hilbert space $\mathcal H$ possesses a reproducing kernel $K$, the polynomial ring $\mathbb C[\boldsymbol{z}]\subseteq \mathcal H$ is dense and the point-wise multiplication induced by $p\in \mathbb C[\boldsymbol{z}]$ is bounded on $\mathcal H$. We fix an ideal $\mathcal I \subseteq \mathbb C[\boldsymbol{z}]$ generated by $p_1,\ldots,p_t$ and let $[\mathcal I]$ denote the completion of $\mathcal I$ in $\mathcal H$. The sheaf $\mathcal S^\mathcal H$ associated to analytic Hilbert module $\mathcal H$ is the sheaf $\mathcal O(Ω)$ of holomorphic functions on $Ω$ and hence is free. However, the subsheaf $\mathcal S^{\mathcal [\mathcal I]}$ associated to $[\mathcal I]$ is coherent and not necessarily locally free. Building on the earlier work of \cite{BMP}, we prescribe a hermitian structure for a coherent sheaf and use it to find tractable invariants. Moreover, we prove that if the zero set $V_{[\mathcal I]}$ is a submanifold of codimension $t$, then there is a unique local decomposition for the kernel $K_{[\mathcal I]}$ along the zero set that serves as a holomorphic frame for a vector bundle on $V_{[\mathcal I]}$. The complex geometric invariants of this vector bundle are also unitary invariants for the submodule $[\mathcal I] \subseteq \mathcal H$.
