ON THE ANNIHILATOR OF A DOLBEAULT GROUP

We show that any Dolbeault cohomology group Hp,q(D), p ≥ 0, q ≥ 1, of an open subset D of a closed finite codimensional complex Hilbert submanifold of ℓ2 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 ℓ2 is induced by its evaluation at a point of M. Lastly, we prove that any closed split infinite dimensional complex Banach submanifold of ℓ2 admits a nowhere critical holomorphic function.