Geometric invariants for a class of submodules of analytic Hilbert modules

Abstract

Let $\Omega \subseteq \mathbb C^m$ be a bounded connected open set and $\mathcal H \subseteq \mathcal O(\Omega)$ be an analytic Hilbert module, i.e., the Hilbert space $\mathcal H$ possesses a reproducing kernel $K$, the polynomial ring $\mathbb C[\underline{z}]\subseteq \mathcal H$ is dense and the point-wise multiplication induced by $p\in \mathbb C[\underline{z}]$ is bounded on $\mathcal H$. We fix an ideal $\mathcal I \subseteq \mathbb C[\underline{z}]$ generated by $p_1,\ldots,p_t$ and let $[\mathcal I]$ denote the completion of $\mathcal I$ in $\mathcal H$. Let $X:[\mathcal I] \to \mathcal H$ be the inclusion map. Thus we have a short exact sequence of Hilbert modules \begin{tikzcd} 0 \arrow{r} and\mbox{[} \mathcal I \mbox{]} \arrow{r}{X} and {\mathcal H} \arrow{r}{\pi} and \mathcal Q \arrow{r}and 0 , \end{tikzcd} where the module multiplication in the quotient $\mathcal Q:=[\mathcal I]^\perp$ is given by the formula $m_p f = P_{[\mathcal I]^\perp} (p f),$ $p\in \mathbb C[\underline{z}],\,f\in \mathcal Q$. The analytic Hilbert module $\mathcal H$ defines a subsheaf $\mathcal S^\mathcal H$ of the sheaf $\mathcal O(\Omega)$ of holomorphic functions defined on $\Omega$. For any open $U \subset \Omega$, it is obtained by setting $$\mathcal S^\mathcal H(U) := \Big \{\, \sum_{i=1}^n ({f_i|}_U) h_i : f_i \in \mathcal H, h_i \in \mathcal O(U), n\in\mathbb N\,\Big \}.$$ This is locally free and naturally gives rise to a holomorphic line bundle on $\Omega$. However, in general, the sheaf corresponding to the sub-module $[\mathcal I]$ is not locally free but only coherent. Building on the earlier work of S. Biswas, a decomposition theorem is obtained for the kernel $K_{[\mathcal I]}$ along the zero set $V_{[\mathcal I]}:=\big\{z\in \mathbb C^m: f(z) = 0, f\in [\mathcal I]\big\}$ which is assumed to be a submanifold of codimension $t$: There exists anti-holomorphic maps $F_1, \ldots, F_t: V_{[\mathcal I]}\to [\mathcal I]$ such that $$ K_{[\mathcal I]}(\cdot, u) = \overline{p_1(u)} F^1_w(u) + \cdots \overline{p_t(u)} F_w^t(u)..