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Compact parallelizable four dimensional symplectic and complex manifolds. (English) Zbl 0656.53034

A. Van de Ven [Proc. Natl. Acad. Sci. USA 55, 1624-1627 (1966; Zbl 0144.210)] and S. T. Yau [Topology 15, 51-53 (1976; Zbl 0331.32013)] have given examples of compact 4-dimensional almost complex manifolds with no complex structures, and N. Brotherton [Bull. Lond. Math. Soc. 10, 303-304 (1978; Zbl 0409.53031)] has proven the nonexistence of complex structures on certain parallelizable 4- dimensional manifolds.
In this paper, the authors define a real parallelizable, compact 4- dimensional manifold \(E^ 4\) to be a principal circle bundle over a principal circle bundle \(E^ 3\) over a torus \(T^ 2\) with the first Betti number \(b_ 1(E^ 4)\) satisfying \(2\leq b_ 1(E^ 4)\leq 4\), and prove the following: if \(b_ 1(E^ 4)=2\) then \(E^ 4\) has symplectic but no complex structures, and if \(b_ 1(E^ 4)=3\) then \(E^ 4\) has both symplectic and complex structures but no positive definite Kaehler metrics, though it carries indefinite Kaehler metrics. Moreover, \(b_ 1(E^ 4)=4\) if and only if \(E^ 4\) is a 4-torus \(T^ 4\).
Reviewer: A.Bucki

MSC:

53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C55 Global differential geometry of Hermitian and Kählerian manifolds
32Q99 Complex manifolds
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