On the annihilator of a Dolbeault group

TL;DR

This paper investigates properties of Dolbeault cohomology groups and holomorphic functions on complex Hilbert submanifolds of ll_2, revealing their infinite-dimensionality, character representations, and existence of nowhere critical functions.

Contribution

It establishes the zero or infinite dimensionality of Dolbeault cohomology groups and characterizes holomorphic function evaluation, advancing understanding of complex structures in infinite dimensions.

Findings

Dolbeault cohomology groups are either zero or infinite dimensional.

Continuous characters of holomorphic function algebras are point evaluations.

Existence of nowhere critical holomorphic functions on certain Banach submanifolds.

Abstract

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.