On the Hausdorff property of some Dolbeault cohomology groups

Abstract

Let X be a complex manifold. The study of the closed-range property of the Cauchy–Riemann equations is of fundamental importance both from the sheaf theoretic point of view and the PDE point of view. In terms of the associated cohomology, it means that the corresponding cohomology is Hausdorff, hence separated. There are many known results for the Hausdorff property of such cohomologies in complex manifolds (see in particular [16–18]). For instance, it is well-known that for a bounded pseudoconvex domain D inCn , the Dolbeault cohomology H p,q(D) in the Frechet space C∞p,q(D) vanishes for all q > 0. It is also known that the L2 cohomology also vanishes. Much less is known about the cohomologies whose topology does not have the Hausdorff property, even for domains in Cn (an example is given in [13], section 14). In this paper, we study duality of the Cauchy–Riemann complex in various function spaces and the Hausdorff property of the corresponding cohomologies. Such duality is classical if