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Special non-Kähler metrics -- old and new

Liviu Ornea, Miron Stanciu

TL;DR

This survey maps the landscape of non-Kähler Hermitian metrics on compact complex manifolds, detailing how diverse structures such as $d\omega=\theta\wedge\omega$ (LCK), $dd^c\omega=0$ (pluriclosed), $dd^c\omega^k\wedge\omega^{n-k-1}=0$ (k-Gauduchon), and related locally conformal, balanced, and Hermitian-symplectic classes interact, coexist, and deform. It synthesizes construction methods (notably left-invariant structures on nilmanifolds and Sasakian-product constructions) and documents robust incompatibility results that constrain when two non-Kähler metrics can live on the same manifold or within the same conformal class. Stability under blow-up and deformation is compiled, with precise positive results (e.g., preservation of pluriclosed and LCK-with-potential under certain operations) and notable obstructions (e.g., general LCK stability under deformation fails). Collectively, the work provides both practical recipes for generating explicit examples and a consolidated framework for understanding the rigidity and flexibility of non-Kähler geometry, guiding future exploration of metric coexistence and deformation behavior.

Abstract

We give an account of old and new results concerning many types of non-Kähler metrics, with focus on the problem of their coexistence on compact complex manifolds, and their behaviour at deformations and blow-up. We also describe a mechanism that several authors have used to construct examples of nilmanifolds admitting metrics with certain properties.

Special non-Kähler metrics -- old and new

TL;DR

This survey maps the landscape of non-Kähler Hermitian metrics on compact complex manifolds, detailing how diverse structures such as (LCK), (pluriclosed), (k-Gauduchon), and related locally conformal, balanced, and Hermitian-symplectic classes interact, coexist, and deform. It synthesizes construction methods (notably left-invariant structures on nilmanifolds and Sasakian-product constructions) and documents robust incompatibility results that constrain when two non-Kähler metrics can live on the same manifold or within the same conformal class. Stability under blow-up and deformation is compiled, with precise positive results (e.g., preservation of pluriclosed and LCK-with-potential under certain operations) and notable obstructions (e.g., general LCK stability under deformation fails). Collectively, the work provides both practical recipes for generating explicit examples and a consolidated framework for understanding the rigidity and flexibility of non-Kähler geometry, guiding future exploration of metric coexistence and deformation behavior.

Abstract

We give an account of old and new results concerning many types of non-Kähler metrics, with focus on the problem of their coexistence on compact complex manifolds, and their behaviour at deformations and blow-up. We also describe a mechanism that several authors have used to construct examples of nilmanifolds admitting metrics with certain properties.
Paper Structure (24 sections, 12 theorems, 5 equations)

This paper contains 24 sections, 12 theorems, 5 equations.

Table of Contents

  1. Introduction
  2. A wealth of Hermitian metrics
  3. Locally conformally Kähler metrics
  4. Vaisman manifolds
  5. LCK manifolds with potential
  6. Kato manifolds
  7. Oeljeklaus--Toma manifolds
  8. Gauduchon metrics
  9. Strongly Gauduchon metrics
  10. $k$-Gauduchon metrics
  11. Endo--Pajitnov manifolds
  12. Pluriclosed metrics
  13. Astheno-Kähler metrics
  14. Balanced metrics
  15. Locally conformally balanced metrics
  16. ...and 9 more sections

Key Result

proposition Proposition 2.1

For any Hermitian metric $\omega$, there exists a unique $1$-form $\theta$ (called the Lee form of $\omega$) such that $d \omega^{n-1} = (n-1) \theta \wedge \omega^{n-1}$.

Theorems & Definitions (26)

  • proposition Proposition 2.1
  • proof
  • remark Remark 2.1
  • remark Remark 2.2
  • definition Definition 3.1
  • definition Definition 3.2
  • definition Definition 3.3
  • definition Definition 3.4
  • proposition Proposition 3.2
  • remark Remark 3.3
  • ...and 16 more