Co-Kähler manifolds may be thought of as odd-dimensional versions of Kähler manifolds. The authors study some properties of co-Kähler manifolds by showing how such properties are inherited from those of the Kähler manifolds that construct them. First, the authors recall the definition of an almost contact metric structure \((J,\xi, \eta, g)\) on a manifold \(M^{2n+1}\). After presenting the local adapted \(J\)-basis for \(TM\), \(\{X_1,\dots, X_n, JX_1, \dots, JX_n,\xi\}\), the fundamental \(2\) form is given by \(\omega(X,Y)=g(JX,Y)\), and some classes of almost contact metric structures are recalled: co-symplectic (when \(d\omega =0=d\eta)\), normal (when \([J,J]+ 2d\eta\otimes \xi=0\)), co-Kähler when the given almost contact metric structure is both co-symplectic and normal; equivalently, \(J\) is parallel with respect to the metric \(g\). In several works, the co-symplectic structures are interpreted as corank 1 Poisson structures. The Sasakian structures have been studied intensively in the last years.
In the present paper the authors study how the properties of the co-Kähler manifolds are intimately tied up with those of the corresponding Kähler manifolds. First, they examine the cohomology algebra of a co-Kähler manifold and its effect on the manifold’s rational homotopy structure. Then they consider the structures of the minimal models (in the sense of Sullivan) of co-Kähler manifolds in terms of the decomposition obtained by H. Li [Asian J. Math. 12, No. 4, 527–544 (2008; Zbl 1170.53014)]. Next, the authors go beyond algebraic considerations in showing that co-Kähler manifolds satisfy the so-called Toral Rank Conjecture: \(\dim (H^*(M,\mathbb{Q}))\geq 2^r\) for any \(r\)-torus \(T^r\) which acts almost freely on \(M\). This result connects the geometry of the co-Kähler manifold to the size of its cohomology. The paper contains the following sections: The Lefschetz property and associated algebraic models, toral rank of co-Kähler manifolds.