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On the Classification of Vaisman Manifolds with Vanishing First Basic Chern Class

Lucas H. S. Gomes

Abstract

We show that every Vaisman manifold with high first Betti number and vanishing first basic Chern class is diffeomorphic to a Kodaira-Thurston manifold. Furthermore, its complex structure is left-invariant, the characteristic foliation is regular, and the associated fibration is given by the Albanese map. Under the additional assumption that the LCK rank is $1$, the Vaisman structure is also left-invariant. We further prove that if all basic harmonic $1$-forms have constant length, then the Vaisman manifold with high first Betti number is diffeomorphic to a Kodaira-Thurston manifold and its complex structure is the standard complex structure. Finally, we discuss the relationship of this condition with transverse geometric formality in this setting.

On the Classification of Vaisman Manifolds with Vanishing First Basic Chern Class

Abstract

We show that every Vaisman manifold with high first Betti number and vanishing first basic Chern class is diffeomorphic to a Kodaira-Thurston manifold. Furthermore, its complex structure is left-invariant, the characteristic foliation is regular, and the associated fibration is given by the Albanese map. Under the additional assumption that the LCK rank is , the Vaisman structure is also left-invariant. We further prove that if all basic harmonic -forms have constant length, then the Vaisman manifold with high first Betti number is diffeomorphic to a Kodaira-Thurston manifold and its complex structure is the standard complex structure. Finally, we discuss the relationship of this condition with transverse geometric formality in this setting.

Paper Structure

This paper contains 11 sections, 35 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Riemannian Foliations
  4. Vaisman Manifolds
  5. Basic Vector Bundles and the Basic Chern Classes
  6. The Albanese Torus and the Albanese Map of Vaisman Manifolds
  7. Deformation of Vaisman Structures
  8. Vaisman Manifolds with Basic Harmonic 1-Forms of Constant Length
  9. The Main Theorem
  10. Transversely Geometrically Formal Vaisman Manifolds
  11. The Mapping Torus and Vaisman Structures of LCK Rank 1

Key Result

Theorem 1.1

Let $(M^{2n+2}, J, \omega, \theta)$ be a Vaisman manifold with $b_1(M) = 2n+1$ and $c_{1, B}(M) = 0$. Then, $M$ is diffeomorphic to a Kodaira-Thurston manifold and $J$ is left-invariant. Moreover, the characteristic foliation $\Sigma$ is regular, $M$ is a fiber bundle over the Albanese torus where $\alpha$ is the Albanese map, whose fibers are the leaves of $\Sigma$, which are biholomorphic to a

Theorems & Definitions (70)

  • Theorem 1.1: Theorem \ref{['MainTheorem']}
  • Theorem 1.2: Theorem \ref{['Constant Length']}
  • Theorem 1.3
  • Definition 2.1
  • Theorem 2.2: ElKacimi, Tondeur
  • Remark 2.3
  • Definition 2.4
  • Theorem 2.5: VaismanGHopfMfd
  • Remark 2.6
  • Proposition 2.7: VaismanGHopfMfd, TsukadaHolomorphicForms
  • ...and 60 more