Holomorphic approximation via Dolbeault cohomology

Abstract

The purpose of this paper is to study holomorphic approximation and approximation of \(\overline{\partial }\)-closed forms in complex manifolds of complex dimension \(n\ge 1\). We consider extensions of the classical Runge theorem and the Mergelyan property to domains in complex manifolds for the \({{\mathcal {C}}}^\infty \)-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.

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Correspondence to Christine Laurent-Thiébaut.

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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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Laurent-Thiébaut, C., Shaw, MC. Holomorphic approximation via Dolbeault cohomology. Math. Z. 296, 1027–1047 (2020). https://doi.org/10.1007/s00209-020-02470-3

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Keywords

  • Runge’s theorem
  • Mergelyan property
  • Dolbeault cohomology

Mathematics Subject Classification

  • 32E30
  • 32W05