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Symplectic manifolds with no Kähler structure. (English) Zbl 0891.53001
Lecture Notes in Mathematics. 1661. Berlin: Springer. viii, 207 p. (1997).
During the last few years, a substantial contribution to the development of various aspects of symplectic geometry has been made by applying the methods of rational homotopy theory (especially Sullivan’s minimal model). In particular, these techniques have proved useful in attacking the Thurston-Weinstein problem of constructing compact symplectic manifolds not admitting (positive definite) Kähler metrics [see, for example, L. A. Cordero, M. Fernández and A. Gray, Topology 25, 375-380 (1986; Zbl 0596.53030); M. Fernández, M. J. Gotay and A. Gray, Proc. Am. Math. Soc. 103, 1209-1212 (1988; Zbl 0656.53034); K. Hasegawa, ibid. 106, 65-71 (1989; Zbl 0691.53040); M. Fernández, A. Gray and J. W. Morgan, Mich. Math. J. 38, 271-283 (1991; Zbl 0726.53028)], certain cases of Arnold’s conjecture [see Ch. McCord and J. Oprea, Topology 32, 701-717 (1993; Zbl 0798.58017)], Brylinski’s problems as well as some other geometric problems.
In this book the authors collect the majority of known results on the problem of constructing symplectic manifolds with no (positive definite) Kähler metrics, and they discuss such a problem for nilmanifolds, solvmanifolds and fiber bundles.
Furthermore, the authors present some conjectures and problems that have been formulated in recent years due to the influx of homotopical ideas, for example the Lupton-Oprea conjecture and the Benson-Gordon conjecture, which are in the spirit of some older and still unsolved problems, namely Thurston’s conjecture and Sullivan’s problem. These problems and conjectures are presented, in a unified way, stressing geometric techniques.

MSC:
53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
55P62 Rational homotopy theory
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