On the annihilator of a Dolbeault group

We show that any Dolbeault cohomology group $H^{p,q}(D)$, $p\ge 0$, $q\ge 1$, of an open subset $D$ of a closed finite codimensional complex Hilbert submanifold of ${\mathrm{\ell }}_2$ 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 ${\mathrm{\ell }}_2$ is induced by evaluation at a point of $M$. Lastly, we prove that any closed split infinite dimensional complex Banach submanifold of ${\mathrm{\ell }}_1$ admits a nowhere critical holomorphic function.