[Paper Review] Cohomologically symplectic solvmanifolds are symplectic

Hisashi Kasuya|arXiv (Cornell University)|May 7, 2010

Geometry and complex manifolds5 references3 citations

TL;DR

This paper proves that cohomologically symplectic aspherical manifolds with torsion-free virtually polycyclic fundamental groups are symplectic, extending known results for nilmanifolds to a broader class of solvmanifolds via algebraic hulls and invariant forms. The key result is that cohomological symplecticity implies genuine symplecticity for such solvmanifolds.

ABSTRACT

We consider aspherical manifolds with torsion-free virtually polycyclic fundamental groups, constructed by Baues. We prove that if those manifolds are cohomologically symplectic then they are symplectic. As a corollary we show that cohomologically symplectic solvmanifolds are symplectic.

Motivation & Objective

• To determine whether cohomologically symplectic solvmanifolds are genuinely symplectic, extending the known equivalence for nilmanifolds.
• To analyze the cohomological structure of aspherical manifolds with torsion-free virtually polycyclic fundamental groups via Baues' construction.
• To establish a link between cohomological symplecticity and the existence of invariant symplectic forms on the associated Lie group.
• To resolve the open problem of whether cohomological symplecticity implies symplecticity in the general solvmanifold setting.

Proposed method

• Constructs a standard Γ-manifold MΓ = α(Γ)\UΓ using the real algebraic hull HΓ of a torsion-free virtually polycyclic group Γ.
• Utilizes the decomposition HΓ = T ⋉ UΓ, where T is a maximal reductive subgroup and UΓ is the unipotent radical, with UΓ ≅ ℝ^n via the exponential map.
• Applies Theorem 3.6 to show that the inclusion (⋀𝔲*人均)^T ⊂ A*(MΓ) induces an isomorphism on cohomology.
• Translates the cohomological condition ω^n ≠ 0 in H^2(Γ, ℝ) to a non-vanishing pointwise condition (ω^n)Γp ≠ 0 on MΓ.
• Uses the invariance of ω under the action of T and the structure of the algebraic hull to show that ω is a closed, non-degenerate 2-form.
• Establishes that the non-degeneracy of ω^n on the manifold implies ω is a symplectic form on MΓ.

Experimental results

Research questions

• RQ1 Are cohomologically symplectic solvmanifolds necessarily symplectic?
• RQ2 Does the existence of a non-vanishing top power of a closed 2-form in cohomology imply the existence of a symplectic structure?
• RQ3 Can the cohomological symplecticity condition be lifted to a genuine symplectic form via the structure of the algebraic hull?
• RQ4 What is the role of the unipotent radical and reductive part in realizing symplectic forms on solvmanifolds?
• RQ5 Is there a counterexample where a solvmanifold has an invariant symplectic form on the Lie group but not on the quotient manifold?

Key findings

• Cohomologically symplectic aspherical manifolds with torsion-free virtually polycyclic fundamental groups are symplectic.
• The standard Γ-manifold MΓ constructed via Baues' method is symplectic if H^2(Γ, ℝ) contains a class ω with ω^n ≠ 0.
• The isomorphism between H*(MΓ, ℝ) and H*((⋀𝔲*)^T) ensures that cohomological data lifts to differential forms.
• The non-vanishing of (ω^n)Γp at every point p ∈ MΓ confirms that ω is a symplectic form.
• The example of G/D with Γ = D/D₀ shows a non-symplectic solvmanifold with trivial H^2((⋀𝔲*)^{±1}), confirming the necessity of the cohomological condition.
• There exist symplectic solvmanifolds (e.g., H/Δ) that are not covered by any solvable Lie group with an invariant symplectic form, showing the failure of the converse of Proposition 4.1.