Examples of simply-connected symplectic non-Kählerian manifolds. (English) Zbl 0567.53031

McDuff, Dusa

J. Differ. Geom. 20, 267-277 (1984).

Let \((M,\sigma)\) be a symplectic manifold with symplectic structure \(\sigma\). Examples of closed non-Kähler \((M,\sigma)\) are not so many. W. Thurston showed a 4-dimensional closed non-Kähler \((M,\sigma)\), which is not simply connected. His \((M,\sigma)\) can be embedded in \((CP^ 5,\omega_ 0)\), where \(\omega_ 0\) is the standard Kähler (and so symplectic) form of \(CP^ 5\). Let \((\tilde X,{\tilde \omega}\)) be the symplectic manifold obtained by blowing up \(CP^ 5\) along M. With the support of some propositions, the author proves the main theorem: \((\tilde X,{\tilde \omega}\)) is a simply connected, symplectic closed manifold with \(\beta_ 3(\tilde X)=\beta_ 1(M)=3\) where the \(\beta_ i\) are the Betti numbers of respective dimension. Hence \(\tilde X\) is not Kähler.

Reviewer: H.Wakakuwa

MSC:

53C15

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

Keywords:

Lefschetz theorem; symplectic manifold; Betti numbers