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On some properties of Hopf manifolds

Nicolina Istrati, Alexandra Otiman

TL;DR

This survey analyzes Hopf manifolds as canonical non-Kähler models, focusing on both their analytic invariants and metric geometry. It leverages Poincaré-Dulac normal forms to classify contractions up to conjugacy, distinguishing diagonal and non-diagonal Hopf manifolds, and it describes the automorphism group and deformation theory that governs their moduli. Cohomological computations reveal precise Dolbeault and Bott-Chern structures, Kodaira dimension, and the algebraic dimension in diagonal cases, while metric results center on locally conformally Kähler structures, the existence of Vaisman metrics, and deformation-based extensions. The work also clarifies the limitations on other distinguished metrics, showing, for example, the nonexistence of pluriclosed and strongly Gauduchon metrics on higher-dimensional Hopf manifolds, highlighting the intricate interplay between complex structure, deformation theory, and Hermitian geometry.

Abstract

We review old and new properties of Hopf manifolds from the point of view of their analytic and metric structure.

On some properties of Hopf manifolds

TL;DR

This survey analyzes Hopf manifolds as canonical non-Kähler models, focusing on both their analytic invariants and metric geometry. It leverages Poincaré-Dulac normal forms to classify contractions up to conjugacy, distinguishing diagonal and non-diagonal Hopf manifolds, and it describes the automorphism group and deformation theory that governs their moduli. Cohomological computations reveal precise Dolbeault and Bott-Chern structures, Kodaira dimension, and the algebraic dimension in diagonal cases, while metric results center on locally conformally Kähler structures, the existence of Vaisman metrics, and deformation-based extensions. The work also clarifies the limitations on other distinguished metrics, showing, for example, the nonexistence of pluriclosed and strongly Gauduchon metrics on higher-dimensional Hopf manifolds, highlighting the intricate interplay between complex structure, deformation theory, and Hermitian geometry.

Abstract

We review old and new properties of Hopf manifolds from the point of view of their analytic and metric structure.

Paper Structure

This paper contains 8 sections, 13 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Normal forms of contractions and the classification of Hopf manifolds
  3. Moduli spaces and deformations
  4. Automorphisms
  5. Cohomology and analytic invariants
  6. Hermitian metrics
  7. lcK and Vaisman metrics
  8. Non-existence of other distinguished metrics

Key Result

theorem Theorem 2.1

(see k in dimension $2$, ha in general dimension) Let $X$ be a Hopf manifold. Then $\pi_1(X)$ is a finite extension of an infinite cyclic subgroup generated by a contraction. In particular, $X$ admits a primary Hopf surface as a finite unramified normal cover.

Theorems & Definitions (35)

  • definition Definition 2.1
  • theorem Theorem 2.1
  • definition Definition 2.2
  • definition Definition 2.3
  • remark Remark 2.1
  • definition Definition 2.4
  • remark Remark 2.2
  • remark Remark 2.3
  • theorem Theorem 2.2
  • definition Definition 2.5
  • ...and 25 more