Holomorphic approximation via Dolbeault cohomology
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Mathematische Zeitschrift
Holomorphic approximation via Dolbeault cohomology Christine Laurent-Thiébaut1,2 · Mei-Chi Shaw3 Received: 7 May 2019 / Accepted: 23 November 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract The purpose of this paper is to study holomorphic approximation and approximation of ∂closed forms in complex manifolds of complex dimension n ≥ 1. We consider extensions of the classical Runge theorem and the Mergelyan property to domains in complex manifolds for the C ∞ -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. Keywords Runge’s theorem · Mergelyan property · Dolbeault cohomology Mathematics Subject Classification 32E30 · 32W05
1 Introduction Holomorphic approximation is a fundamental subject in complex analysis. The Runge theorem asserts that, if K is a compact subset of an open Riemann surface X such that X \K has no relatively compact connected components, then every holomorphic function on a neighborhood of K can be approximated uniformly on K by holomorphic functions on X . If K is a compact subset of an open Riemann surface X , we denote by A(K ) the space of continuous functions on K , which are holomorphic in the interior of K . Then the Mergelyan theorem asserts that, if K is such that X \K has no relatively compact connected components, then every function in A(K ) can be approximated uniformly on K by holomorphic functions on X .
The first author would like to thank the university of Notre Dame for its support during her stay in April 2019. The second author was partially supported by National Science Foundation Grant DMS-1700003.
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Christine Laurent-Thiébaut [email protected] Mei-Chi Shaw [email protected]
1
Institut Fourier, Université Grenoble-Alpes, CS 40700, 38058 Grenoble cedex 9, France
2
Institut Fourier, CNRS UMR 5582, 38402 Saint-Martin d’Hères, France
3
Department of Mathematics, University of Notre Dame, Notre Dame, IN 46656, USA
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C. Laurent-Thiébaut, M.-C. Shaw
Holomorphic approximation in one complex variable has been studied and well understood, while the analogous problems in several variables are much less understood with many open questions. An up-to-date account of the history and recent development of holomorphic approximation in one and several variables can be found in the paper by Fornaess et al. [4]. In this paper we will consider holomorphic approximation in complex manifolds of higher complex dimension and also approximation of ∂-closed forms for different topologies like the uniform or the smooth topology on compact subsets or the L 2 topology. The aim is to characterize different types of holomorphic or ∂-closed approximation in a subdomain of a complex manifold using properties of the Dolbeault cohomology with compact or prescribed support in the domain or using properties of the Dolbeault cohomology of the complement of the domain with respect to some famil
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