  We show that any Dolbeault cohomology group $H^{p,q}(D)$, $p\ge0$, $q\ge1$,
of an open subset $D$ of a closed finite codimensional complex Hilbert
submanifold of $\ell_2$ is either zero or infinite dimensional. We also show
that any continuous character of the algebra of holomorphic functions of a
closed complex Hilbert submanifold $M$ of $\ell_2$ is induced by evaluation at
a point of $M$. Lastly, we prove that any closed split infinite dimensional
complex Banach submanifold of $\ell_1$ admits a nowhere critical holomorphic
function.
