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Symplectic manifolds with no Kähler structure. (English) Zbl 0596.53030

The purpose of the paper is to construct several examples of compact symplectic manifolds having no Kähler structure. W. P. Thurston gave in 1976 such an example [Proc. Am. Math. Soc. 55, 467-468 (1976; Zbl 0324.53031)]. The authors define what they call generalized Iwasawa manifolds, I(p), of real dimensions \(4p+2\). The reason of the name is that for \(p=1\), I(1) is the well known Iwasawa manifold obtained as quotient of the group of all the matrices of the form \[ \begin{pmatrix} 1& z_ 1& z_ 2 \\ 0& 1& z_ 3 \\ 0& 0& 1 \end{pmatrix} \] with \(z_ 1,z_ 2,z_ 3\in {\mathbb{C}}\) by the discrete subgroup of those matrices for which \(z_ 1,z_ 2,z_ 3\) are Gaussian integers. They construct a symplectic form on I(p) and prove (using the theory of minimal models) that I(p) has no Kähler structure. In the same spirit they also give another family of compact symplectic manifolds \(M(p,q)\), of real dimensions \(2p+2q+2\), having no Kähler structures.
Reviewer: J.Girbau

MSC:

53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)

Citations:

Zbl 0324.53031
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