Examples of compact non-Kähler almost Kähler manifolds. (English) Zbl 0575.53015

The main result of this paper is the construction of a family of compact almost Kähler manifolds \(M^{2(r+1)}\), \(r\geq 1\), which are not Kählerian. These manifolds are compact quotients of \(H(1,r)\times S^ 1\) by a discrete subgroup (H(1,r) being a generalized Heisenberg group). Each of the manifolds considered is an \((r+1)\)-torus bundle over an \((r+1)\)-torus and in this way, the examples are generalizations of Thurston’s example \((r=1)\). Using the first Betti number, it is shown that, for r odd, \(M^{2(r+1)}\) cannot admit any Kähler structure. For r even, the authors show that the given metric on the example is also not Kählerian. In a forthcoming paper, the first two authors and Gray will show that also for r even \(M^{2(r+1)}\) can have no Kähler structure.
Reviewer: L.Vanhecke


53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C55 Global differential geometry of Hermitian and Kählerian manifolds
53C30 Differential geometry of homogeneous manifolds
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