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The Levi problem over generalized Hirzebruch manifolds

S. Ivashkovych, C. Miebach, V. Shevchishin

Abstract

We review classical methods to solve the Levi problem in the presence of symmetries, established by Hirschowitz and by Grauert-Remmert-Ueda. We then illustrate these methods by solving the Levi problem in some new situations, namely generalized Hirzebruch manifolds and primary Hopf surfaces of non-diagonal type.

The Levi problem over generalized Hirzebruch manifolds

Abstract

We review classical methods to solve the Levi problem in the presence of symmetries, established by Hirschowitz and by Grauert-Remmert-Ueda. We then illustrate these methods by solving the Levi problem in some new situations, namely generalized Hirzebruch manifolds and primary Hopf surfaces of non-diagonal type.
Paper Structure (13 sections, 8 theorems, 14 equations)

This paper contains 13 sections, 8 theorems, 14 equations.

Table of Contents

  1. Introduction
  2. The Levi problem over generalized Hirzebruch manifolds
  3. Ueda's generalization of methods by Grauert and Remmert
  4. Generalized Hirzebruch manifolds
  5. Locally Stein domains over $X_k$
  6. Case 1
  7. Case 2
  8. Case 3
  9. Subcase 3a
  10. Subcase 3b
  11. The Levi problem in non-diagonal primary Hopf surfaces
  12. Hirschowitz' techniques
  13. Locally Stein domain in non-diagonal Hopf surfaces

Key Result

Theorem 1

Let $D$ be a locally Stein domain over a generalized Hirzebruch manifold $X_k$ for $k\geqslant 0$. If $D$ is not Stein, then one of the following possibilities occurs:

Theorems & Definitions (11)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 2.1: Grauert-Remmert, Ueda
  • Example
  • Remark
  • Remark
  • Lemma 2.2
  • Theorem 2.3
  • Proposition 3.1
  • ...and 1 more