Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Conformal foliations, Kähler twists and the Weinstein construction

Paul-Andi Nagy, Liviu Ornea

TL;DR

This work classifies local and global Kähler geometries supporting totally geodesic, holomorphic and homothetic foliations with complex leaves, showing that Swann's twist machinery and Weinstein's construction exhaust the main building blocks. The twist framework relates invariant metrics and complex structures on a base S to those on a twisted space Z, yielding a robust link between Hamiltonian circle actions, Picard groups, and Kähler geometry. A central result is that, when the twist class is integral, either the foliation is Riemannian or the universal cover of the manifold arises from the Weinstein construction; in the simply connected case, Z is obtained by Weinstein from a base M and a fibre N. The authors derive a twist correspondence for Hamiltonian forms, describe the Kähler cone under twisting, and show how multiple TGHH foliations can coexist, producing several compatible complex structures. Applications include holomorphic harmonic morphisms with arbitrary fibre dimensions, conformal submersions with complex fibres, and novel Hermitian geometries on twist manifolds, unifying diverse aspects of Kähler, Riemannian, and Hermitian geometry with concrete structural and cohomological data.

Abstract

We classify both local and global Kähler structures admitting totally geodesic homothetic foliations with complex leaves. The main building blocks are related to Swann's twists and are obtained by applying Weinstein's method of constructing symplectic bundles to Kähler data. As a byproduct we obtain new classes of: holomorphic harmonic morphisms with fibres of arbitrary dimension from compact Kähler manifolds; non-Kähler balanced metrics conformal to Kähler ones (but compatible with different complex structures). Some classes of non-Einstein constant scalar curvature Kähler metrics are also obtained in this way.

Conformal foliations, Kähler twists and the Weinstein construction

TL;DR

This work classifies local and global Kähler geometries supporting totally geodesic, holomorphic and homothetic foliations with complex leaves, showing that Swann's twist machinery and Weinstein's construction exhaust the main building blocks. The twist framework relates invariant metrics and complex structures on a base S to those on a twisted space Z, yielding a robust link between Hamiltonian circle actions, Picard groups, and Kähler geometry. A central result is that, when the twist class is integral, either the foliation is Riemannian or the universal cover of the manifold arises from the Weinstein construction; in the simply connected case, Z is obtained by Weinstein from a base M and a fibre N. The authors derive a twist correspondence for Hamiltonian forms, describe the Kähler cone under twisting, and show how multiple TGHH foliations can coexist, producing several compatible complex structures. Applications include holomorphic harmonic morphisms with arbitrary fibre dimensions, conformal submersions with complex fibres, and novel Hermitian geometries on twist manifolds, unifying diverse aspects of Kähler, Riemannian, and Hermitian geometry with concrete structural and cohomological data.

Abstract

We classify both local and global Kähler structures admitting totally geodesic homothetic foliations with complex leaves. The main building blocks are related to Swann's twists and are obtained by applying Weinstein's method of constructing symplectic bundles to Kähler data. As a byproduct we obtain new classes of: holomorphic harmonic morphisms with fibres of arbitrary dimension from compact Kähler manifolds; non-Kähler balanced metrics conformal to Kähler ones (but compatible with different complex structures). Some classes of non-Einstein constant scalar curvature Kähler metrics are also obtained in this way.

Paper Structure

This paper contains 30 sections, 61 theorems, 185 equations.

Table of Contents

  1. Introduction
  2. Kähler structures of Weinstein type
  3. Classification results
  4. Swann's twist machinery
  5. Relations to other geometries
  6. Riemannian properties
  7. Hermitian properties
  8. Complex homothetic foliations
  9. General observations
  10. Holomorphic and conformal foliations
  11. Global aspects
  12. Factorisation
  13. The Kähler geometry of twists
  14. Hamiltonian forms
  15. Lifting circle actions and twists
  16. ...and 15 more sections

Key Result

Theorem 1.1

Let $\mathcal{F}$ be a holomorphic and conformal foliation, $\mathrm{codim}_{\mathbb{R}} \mathcal{F} \geq 4$, on a Kähler manifold $Z$. Then

Theorems & Definitions (150)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • Proposition 2.2
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Definition 2.5
  • ...and 140 more