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

MSC:

 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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