[Paper Review] Quasi-Kähler manifolds with trivial Chern Holonomy

Antonio J. Di Scala, Luigi Vezzoni|arXiv (Cornell University)|Jul 10, 2008

Geometry and complex manifolds8 references4 citations

TL;DR

This paper classifies compact quasi-Kähler manifolds with trivial Chern holonomy (Chern-flat), proving they are all 2-step nilmanifolds with specific almost complex structures. It further shows that such structures can be tamed by a symplectic form only if the manifold is a complex torus, resolving a conjectural problem in Hermitian geometry via Lie algebra techniques and cohomological analysis of forms on nilmanifolds.

ABSTRACT

In this paper we study almost complex manifolds admitting a quasi-Kähler Chern-flat metric (Chern-flat means that the holonomy of the Chern connection is trivial). We prove that in the compact case such manifolds are all nilmanifolds. Some partial classification results are established and we prove that a quasi-Kähler Chern-flat structure can be tamed by a symplectic form if and only if the ambient space is isomorphic to a flat torus.

Motivation & Objective

• To classify compact almost complex manifolds admitting a quasi-Kähler metric with trivial Chern holonomy (Chern-flat).
• To determine when such structures can be tamed by a symplectic form, extending results on the Iwasawa manifold.
• To characterize the underlying Lie algebras of these manifolds via algebraic conditions on their complexified decomposition.
• To study infinitesimal deformations of quasi-Kähler Chern-flat Lie algebras.
• To establish a link between Chern-flatness, integrability, and Kählerity under symplectic taming.

Proposed method

• Utilizes Palais' theorem to reduce classification of compact Chern-flat quasi-Kähler manifolds to classifying 2-step nilpotent Lie algebras with specific complex structures.
• Applies the Chern connection's simple formula in the quasi-Kähler setting, leveraging its torsion and curvature properties.
• Employs a coframe formalism on the universal cover to express forms and derive differential equations involving ∂̄ and A operators.
• Analyzes the condition ∂̄β + Aβ̄ = 0 for (2,0)-forms β to prove closedness, using compactness and invariance.
• Uses the structure of the Lie algebra decomposition 𝔤 = 𝔛 ⊕ 𝔠, with 𝔠 the center, to derive cohomological constraints.
• Applies results from [4] and [13] to connect Chern-flatness with integrability and Kähler metrics under symplectic taming.

Experimental results

Research questions

• RQ1 Which compact almost complex manifolds admit a quasi-Kähler metric with trivial Chern holonomy?
• RQ2 Under what conditions can a quasi-Kähler Chern-flat structure be tamed by a symplectic form?
• RQ3 What algebraic conditions define the Lie algebras of such manifolds?
• RQ4 How do infinitesimal deformations of these Lie algebras behave?
• RQ5 Does Chern-flatness in the quasi-Kähler setting imply integrability of the almost complex structure?

Key findings

• All compact quasi-Kähler Chern-flat manifolds are isomorphic to 2-step nilmanifolds with a left-invariant almost complex structure satisfying [𝔤^{1,0}, 𝔤^{0,1}] = 0 and [𝔤^{1,0}, 𝔤^{1,0}] ⊆ 𝔤^{0,1}.
• The vector space of infinitesimal deformations of a quasi-Kähler Chern-flat Lie algebra is trivial, indicating rigidity of the structure.
• A (2,0)-form β satisfying ∂̄β + Aβ̄ = 0 on a compact Chern-flat quasi-Kähler manifold must be closed, i.e., dβ = 0.
• The existence of a symplectic form taming a quasi-Kähler Chern-flat structure implies the metric is Kähler and the manifold is a complex torus.
• The Chern connection's holonomy is trivial if and only if the underlying Lie algebra satisfies the specified integrability and bracket conditions.
• The Iwasawa manifold with J₃ is a canonical example of a compact quasi-Kähler Chern-flat manifold, and it cannot be tamed by a symplectic form.