  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 $\mathbb C[\boldsymbol{z}]\subseteq \mathcal H$ 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 $\mathcal
O(\Omega)$ of holomorphic functions on $\Omega$ 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 $[\mathcal I] \subseteq \mathcal H$.
