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VI Domains of holomorphy and pseudoconvexity

In: Holomorphic Function Theory in Several Variables

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  • Christine Laurent-Thiébaut

    (Université Joseph Fourier, Institut Fourier)

Abstract

At the end of Chapter I and in Chapter III we met open sets in ℂ n on which any holomorphic function can be extended to a larger open set. The open sets which do not have this property are called domains of holomorphy: in this chapter we study such open sets. We start by giving a characterisation of domains of holomorphy in terms of holomorphic convexity (the Cartan–Thullen theorem). We then introduce the notion of pseudoconvexity in order to get a more analytic characterisation of domains of holomorphy. This requires us to define plurisubharmonic functions. We then prove that every domain of holomorphy is pseudoconvex: the converse, which is known as the Levi problem, is studied in Chapter VII.

Suggested Citation

  • Christine Laurent-Thiébaut, 2011. "VI Domains of holomorphy and pseudoconvexity," Springer Books, in: Holomorphic Function Theory in Several Variables, pages 113-145, Springer.
    • Handle: RePEc:spr:sprchp:978-0-85729-030-4_6
    DOI: 10.1007/978-0-85729-030-4_6
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