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0656.53034
Fernández,
Marisa; Gotay,
Mark J.; Gray,
Alfred
Compact parallelizable four dimensional symplectic and complex manifolds.
(English)
[J] Proc.
Am. Math. Soc. 103, No.4, 1209-1212 (1988). [ISSN 0002-9939]
{\it A. Van de Ven} [Proc. Natl. Acad. Sci. USA 55, 1624-1627 (1966; Zbl
0144.210)] and {\it 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 {\it 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. \par In this paper, the authors
define a real parallelizable, compact 4- dimensional manifold $E\sp 4$ to
be a principal circle bundle over a principal circle bundle $E\sp 3$ over
a torus $T\sp 2$ with the first Betti number $b\sb 1(E\sp 4)$ satisfying
$2\le b\sb 1(E\sp 4)\le 4$, and prove the following: if $b\sb 1(E\sp 4)=2$
then $E\sp 4$ has symplectic but no complex structures, and if $b\sb 1(E\sp
4)=3$ then $E\sp 4$ has both symplectic and complex structures but no positive
definite Kaehler metrics, though it carries indefinite Kaehler metrics.
Moreover, $b\sb 1(E\sp 4)=4$ if and only if $E\sp 4$ is a 4-torus $T\sp
4$.
[ A.Bucki
]
Keywords: symplectic structure; principal circle bundle; torus;
Betti number; complex structures; indefinite Kaehler metrics
Citations:
Zbl 0144.210; Zbl
0331.32013; Zbl
0409.53031