An L 2 version of the Serre duality on domains in complex manifolds involving duality of Hilbert space realizations of the ∂-operator is established. This duality is used to study the solution of the ∂-equation with prescribed support. Applications are given to ∂-closed extension of forms, as well to Bochner-Hartogs type extension of CR functions.2000 Mathematics Subject Classification. 32C37, 35N15, 32W05.
Let Ω = Ω \ D where Ω is a bounded domain with connected complement in C n (or more generally in a Stein manifold) and D is relatively compact open subset of Ω with connected complement in Ω. We obtain characterizations of pseudoconvexity of Ω and D through the vanishing or Hausdorff property of the Dolbeault cohomology groups on various function spaces. In particular, we show that if the boundaries of Ω and D are Lipschitz and C 2 -smooth respectively, then both Ω and D are pseudoconvex if and only if 0 is not in the spectrum of the ∂-Neumann Laplacian on (0, q)-forms for 1 ≤ q ≤ n − 2 when n ≥ 3; or 0 is not a limit point of the spectrum of the ∂-Neumannn Laplacian on (0, 1)-forms when n = 2. (2000): 32C35, 32C37, 32W05.
Mathematics Subject Classification
We give an example of a pseudoconvex domain in a complex manifold whose L 2 -Dolbeault cohomology is non-Hausdorff, yet the domain is Stein. The domain is a smoothly bounded Levi-flat domain in a two complex-dimensional compact complex manifold. The domain is biholomorphic to a product domain in C 2 , hence Stein. This implies that for q > 0, the usual Dolbeault cohomology with respect to smooth forms vanishes in degree (p, q). But the L 2 -Cauchy-Riemann operator on the domain does not have closed range on (2, 1)-forms and consequently its L 2 -Dolbeault cohomology is not Hausdorff. * c are again closed, densely defined, unbounded operators on L 2 * , * (Ω). The Complex Laplacian is the operator = ∂∂ * + ∂ * ∂, and its inverse (modulo kernel) is the ∂-Neumann
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