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} &\mbox{[} \mathcal I \mbox{]} \arrow{r}{X} & {\mathcal H} \arrow{r}{\pi} & \mathcal Q \arrow{r}& 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 $$ K_{[\mathcal I]}(\cdot, u) = \overline{p_1(u)} F^1_w(u) + \cdots \overline{p_t(u)} F_w^t(u),\, u\in \Omega_w,$$ in some neighbourhood $\Omega_w$ of each fixed but arbitrary $w\in V_{[\mathcal I]}$ for some anti-holomorphic maps $F_w^1, \ldots, F^t_w: \Omega_w \to [\mathcal I]$ extending $F_1, \ldots,F_t$. The anti-holomorphic maps $F_1, \ldots,F_t$ are linearly independent on $V_{[\mathcal I]}$, defining a rank $t$ anti-holomorphic Hermitian vector bundle on it. This gives rise to complex geometric invariants for the pair $([\mathcal I], \mathcal H)$.

Next, using a decomposition formula obtained from an earlier work of Douglas, Misra and Varughese, the maps $F_1, \ldots, F_t: V_{[\mathcal I]}\to [\mathcal I]$ are explicitly determined with the additional assumption that $p_{i},p_{j}$ are relatively prime for $i\neq j$. Using this, a line bundle on $V_{[\mathcal I]}\times\mathbb{P}^{t-1}$ is constructed via the monoidal transformation around $V_{[\mathcal I]}$ which provides useful invariants for $([\mathcal I], \mathcal H)$.

Localising the modules $[\mathcal I]$ and $\mathcal H$ at $w\in \Omega$, we obtain the localization $X(w)$ of the module map $X$. The localizations are nothing but the quotient modules $[\mathcal I]/{[\mathcal I]_w}$ and $\mathcal H/{\mathcal H_w}$, where $[\mathcal I]_w$ and $\mathcal H_w$ are the maximal sub-modules of functions vanishing at $w$. These clearly define anti-holomorphic line bundles $E_{[\mathcal I]}$ and $E_\mathcal H$, respectively, on $\Omega\setminus V_{[\mathcal I]}$. However, there is a third line bundle, namely, ${\rm Hom}(E_\mathcal H, E_{[\mathcal I]})$ defined by the anti-holomorphic map $X(w)^*$. The curvature of a holomorphic line bundle $\mathcal L$ on $\Omega$, computed with respect to a holomorphic frame $\gamma$ is given by the formula $$\mathcal K_\mathcal L(z) = \sum_{i,j=1}^{m}\tfrac{\partial^2}{\partial z_i \partial \bar{z}_j}\log\|\gamma(z)\|^2 dz_i \wedge d\bar{z}_j.$$ It is a complete invariant for the line bundle $\mathcal L$. The alternating sum $$ \mathcal A_{[\mathcal I], \mathcal H}(w):=\mathcal K_X(w) - \mathcal K_{[\mathcal{I}]}(w) + \mathcal K_{\mathcal{H}}(w) = 0,\,\, w\in \Omega \setminus V_{[\mathcal I]}, $$ where $\mathcal K_X$, $\mathcal K_{[\mathcal{I}]}$ and $\mathcal K_{\mathcal{H}}$ denote the curvature $(1,1)$ form of the line bundles $E_X$, $E_{[\mathcal{I}]}$ and $E_{\mathcal{H}}$, respectively. Thus it is an invariant for the pair $([\mathcal I], \mathcal H)$. However, when $\mathcal I$ is principal, by taking distributional derivatives, $\mathcal A_{[\mathcal I], \mathcal H}(w)$ extends to all of $\Omega$ as a $(1,1)$ current. Consider the following diagram of short exact sequences of Hilbert modules: $$(1)\,\,\,\,\,\,\,\,\,\,\, \begin{tikzcd} 0\arrow{r} &\mbox{[}\mathcal I\mbox{]} \arrow{d} \arrow{r} {X} & {\mathcal H}\arrow{d}{L} \arrow{r}{\pi} & \mathcal Q\arrow{d} \arrow{r}& 0\\ 0\arrow{r} &\mbox{[}\widetilde{\mathcal I}\mbox{]} \arrow{r}{\widetilde{X}} &\widetilde{\mathcal H} \arrow{r}{\tilde{\pi}}& \widetilde{\mathcal Q} \arrow{r}& 0, \end{tikzcd} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, (2)\,\,\,\,\,\,\,\,\,\,\, \begin{tikzcd} \mbox{[}\mathcal I\mbox{]} \arrow{d} \arrow{r} {X} & {\mathcal H}\arrow{d}{L}\\ \mbox{[}\widetilde{\mathcal I}\mbox{]} \arrow{r}{\widetilde{X}} &\widetilde{\mathcal H} \end{tikzcd}$$ It is shown that if $\mathcal A_{[\mathcal I], \mathcal H}(w)=\mathcal A_{[\widetilde{\mathcal I}], \widetilde{\mathcal H}}(w)$, then $L|_{[\mathcal I]}$ makes the second diagram commute. Hence, if $L$ is bijective, then $[\mathcal I]$ and $[\widetilde{\mathcal I]}$ are equivalent as Hilbert modules. It follows that the alternating sum is an invariant for the ``rigidity'' phenomenon.