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Global embeddings of weakly pseudoconvex complex spaces and refined approximation theorems

Yuta Watanabe

TL;DR

The paper develops a global embedding framework for non-compact weakly pseudoconvex complex spaces with positive line bundles by refining approximation theorems for holomorphic sections via exhaustively singular-positive metrics and a canonical desingularization. It proves that the regular locus embeds into a projective space, while adjoint bundles on the desingularized space become ample or big on the complement of the exceptional set, enabling global embedding via suitable Kodaira-like maps. A refined approximation theorem ensures density of global sections on sublevel sets, enabling globalization of local embedding data. As an application, the Union Problem is solved for weakly pseudoconvex manifolds, showing that unions of Stein subdomains remain Stein under the appropriate hypotheses. The approach extends Grauert–Grauert and Takayama-type results to spaces with singularities and provides a cohesive analytic-algebraic toolkit for global embedding and cohomology questions.

Abstract

In this paper, by refining approximation theorems for holomorphic sections of adjoint line bundles, it is proved that the regular locus of a weakly pseudoconvex complex space admitting a positive line bundle can be embedded into a complex projective space. As an application of approximation theorems, it is shown that the Union problem can be solved for weakly pseudoconvex complex manifolds.

Global embeddings of weakly pseudoconvex complex spaces and refined approximation theorems

TL;DR

The paper develops a global embedding framework for non-compact weakly pseudoconvex complex spaces with positive line bundles by refining approximation theorems for holomorphic sections via exhaustively singular-positive metrics and a canonical desingularization. It proves that the regular locus embeds into a projective space, while adjoint bundles on the desingularized space become ample or big on the complement of the exceptional set, enabling global embedding via suitable Kodaira-like maps. A refined approximation theorem ensures density of global sections on sublevel sets, enabling globalization of local embedding data. As an application, the Union Problem is solved for weakly pseudoconvex manifolds, showing that unions of Stein subdomains remain Stein under the appropriate hypotheses. The approach extends Grauert–Grauert and Takayama-type results to spaces with singularities and provides a cohesive analytic-algebraic toolkit for global embedding and cohomology questions.

Abstract

In this paper, by refining approximation theorems for holomorphic sections of adjoint line bundles, it is proved that the regular locus of a weakly pseudoconvex complex space admitting a positive line bundle can be embedded into a complex projective space. As an application of approximation theorems, it is shown that the Union problem can be solved for weakly pseudoconvex complex manifolds.

Paper Structure

This paper contains 10 sections, 17 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Canonical desingularization of complex spaces
  4. Plurisubharmonicity and positivity on complex spaces
  5. Singular Hermitian metric and exhaustive positivity
  6. Algebro-geometric positivity
  7. Exhaustive positivity
  8. Approximation theorems
  9. Points separation and Embedding theorems
  10. Union Problem

Key Result

Theorem 1.1

Let $X$ be a non-compact weakly pseudoconvex complex space of pure dimension $n$ and $L\longrightarrow X$ be a holomorphic line bundle. If $L$ is positive, then the regular locus $X_{reg}$ of $X$ can be holomorphically embedded into $\mathbb{P}^{2n+1}$. Furthermore, let $\pi:\widetilde{X}\longrighta

Theorems & Definitions (44)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 2.1
  • Lemma 2.2
  • Definition 2.3
  • Definition 2.4
  • ...and 34 more