Holomorphic Approximation via Dolbeault Cohomology

The purpose of this paper is to study holomorphic approximation and approximation of $\bar\partial$-closed forms in complex manifolds of complex dimension $n\geq 1$. We consider extensions of the classical Runge theorem and the Mergelyan property to domains in complex manifolds for the smooth and the $L^2$ topology. We characterize the Runge or Mergelyan property in terms of certain Dolbeault cohomology groups and some geometric sufficient conditions are given

HOLOMORPHIC APPROXIMATION VIA DOLBEAULT COHOMOLOGY

International audienceThe purpose of this paper is to study holomorphic approximation and approximation of $\overline\partial$-closed forms in complex manifolds of complex dimension $n\geq 1$. We consider extensions of the classical Runge theorem and the Mergelyan property to domains in complex manifolds for the smooth and the $L^2$ topology. We characterize the Runge or Mergelyan property in terms of certain Dolbeault cohomology groups and some geometric sufficient conditions are given

HOLOMORPHIC APPROXIMATION AND MIXED BOUNDARY VALUE PROBLEMS FOR ∂

International audienceIn this paper we study holomorphic approximation using boundary value problems for $\overline\partial$ on an annulus in the Hilbert space setting. The associated boundary conditions for $\overline\partial$ are the mixed boundary problems on an annulus. We characterize pseudoconvexity and Runge type property of the domain by the vanishing of related $L^2$ cohomology groups