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The Density Property for Vector Bundles

Riccardo Ugolini, Joerg Winkelmann

Abstract

We prove that holomorphic vector bundles over Stein manifolds with the density property also satisfy the density property, provided that the total space is holomorphically flexible. We apply this result to provide a new class of Stein manifolds with the density property.

The Density Property for Vector Bundles

Abstract

We prove that holomorphic vector bundles over Stein manifolds with the density property also satisfy the density property, provided that the total space is holomorphically flexible. We apply this result to provide a new class of Stein manifolds with the density property.
Paper Structure (15 sections, 44 theorems, 99 equations)

This paper contains 15 sections, 44 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Connections and horizontal vector fields
  3. The Euler vector field
  4. Vector fields (in)compatible with the vector bundle structure
  5. Preparations
  6. Existence of certain vector fields
  7. Runge Theory
  8. Fréchet Spaces
  9. Vertical vector fields
  10. Supporting Lemmas
  11. Generalities
  12. Main result for rank at least two
  13. The line bundle case
  14. Flexibility
  15. Tame sets

Key Result

Theorem 1.2

Let $X$ be a Stein manifold with density property, $\pi:E\to X$ a holomorphic vector bundle. Assume that there exists an automorphism $\phi$ of the total space $E$ and points $p,q\in E$ such that $\pi(p)=\pi(q)$, but $\pi(\phi(p))\ne\pi(\phi(q))$. Then $E$ satisfies the density property.

Theorems & Definitions (95)

  • Definition 1.1: Varolin2001
  • Theorem 1.2: Main theorem
  • Definition 1.3
  • Theorem 1.4
  • Corollary 1.5
  • proof
  • Theorem 1.6: =Theorem \ref{['vb-flexible']}
  • Corollary 1.7
  • proof : Proof of Corollary \ref{['quasi-hom']}
  • Theorem 1.8
  • ...and 85 more