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Cartan-Thullen theorem and Levi problem in context of generalised convexity

Krzysztof J. Ciosmak

TL;DR

This work develops a generalised convexity framework in which the Cartan--Thullen theorem and the Levi problem are derived from classical functional-analytic results, notably the Banach--Alaoglu and Banach--Steinhaus theorems. It introduces abstract analogues of domains of holomorphy and domain of existence, along with notions of completeness and exhaustibility by polytopes, and proves that holomorphic completeness characterises Stein spaces. Through a Gelfand transform, holomorphic convexity is linearised and linked to convex geometry in a dual space, enabling a unified treatment of holomorphic, plurisubharmonic, subharmonic, and convex settings. The Bremermann--Lelong lemma is shown to be equivalent to the positive resolution of the Levi problem, tying together several classical results under the generalized convexity umbrella and highlighting broad applications in complex-analytic and functional-analytic contexts.

Abstract

We demonstrate that the Cartan-Thullen theorem and its generalisation to the context of generalised convexity, which we establish herein, can be regarded as consequences of the classical theorems of functional analysis: the Banach-Steinhaus theorem and the Banach-Alaoglu theorem. Furthermore, we characterise the domains of holomorphy, and their generalisations, as the spaces that are complete, or as the spaces exhaustible by suitably defined polytopes. We also provide an abstract analogue of the Levi problem and its elementary resolution. Our results allow for a novel characterisation of Stein spaces as the holomorphically complete spaces, as well as a proof that the Bremermann-Lelong lemma is equivalent to the positive answer to the Levi problem. Another contribution of ours is the introduction of the analogues of the notions of the complex analysis to the setting of generalised convexity.

Cartan-Thullen theorem and Levi problem in context of generalised convexity

TL;DR

This work develops a generalised convexity framework in which the Cartan--Thullen theorem and the Levi problem are derived from classical functional-analytic results, notably the Banach--Alaoglu and Banach--Steinhaus theorems. It introduces abstract analogues of domains of holomorphy and domain of existence, along with notions of completeness and exhaustibility by polytopes, and proves that holomorphic completeness characterises Stein spaces. Through a Gelfand transform, holomorphic convexity is linearised and linked to convex geometry in a dual space, enabling a unified treatment of holomorphic, plurisubharmonic, subharmonic, and convex settings. The Bremermann--Lelong lemma is shown to be equivalent to the positive resolution of the Levi problem, tying together several classical results under the generalized convexity umbrella and highlighting broad applications in complex-analytic and functional-analytic contexts.

Abstract

We demonstrate that the Cartan-Thullen theorem and its generalisation to the context of generalised convexity, which we establish herein, can be regarded as consequences of the classical theorems of functional analysis: the Banach-Steinhaus theorem and the Banach-Alaoglu theorem. Furthermore, we characterise the domains of holomorphy, and their generalisations, as the spaces that are complete, or as the spaces exhaustible by suitably defined polytopes. We also provide an abstract analogue of the Levi problem and its elementary resolution. Our results allow for a novel characterisation of Stein spaces as the holomorphically complete spaces, as well as a proof that the Bremermann-Lelong lemma is equivalent to the positive answer to the Levi problem. Another contribution of ours is the introduction of the analogues of the notions of the complex analysis to the setting of generalised convexity.
Paper Structure (27 sections, 19 theorems, 119 equations)

This paper contains 27 sections, 19 theorems, 119 equations.

Table of Contents

  1. Introduction
  2. The Cartan--Thullen theorem and the Levi problem
  3. Generalised convexity
  4. Examples
  5. Domains of holomorphy and domains of existence
  6. Main results
  7. Cones satisfying the maximum principle
  8. Equivalence of the Bremermann--Lelong lemma and the positive resolution of the Levi problem
  9. Examples
  10. Holomorphic convexity
  11. Stein spaces
  12. Plurisubharmonic functions
  13. Subharmonic functions
  14. Convex functions
  15. Gelfand transform and relation to convexity
  16. ...and 12 more sections

Key Result

Theorem 1.2

Suppose that $\Omega\subset\mathbb{C}^n$ is an open set. Then the following conditions are equivalent:

Theorems & Definitions (74)

  • Definition 1.1
  • Theorem 1.2
  • Definition 1.3
  • Remark 1.4
  • Definition 1.5
  • Definition 1.6
  • Remark 1.7
  • Example 1.8
  • Definition 1.9
  • Definition 1.10
  • ...and 64 more