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Compact manifolds with indefinite Kähler metrics. (English) Zbl 0696.53037

Differential geometry, Proc. 6th Int. Colloq., Santiago de Compostela/Spain 1988, Cursos Congr. Univ. Santiago de Compostela 61, 25-50 (1989).
[For the entire collection see Zbl 0682.00012.]
Some of the properties of compact manifolds with indefinite Kähler metrics are discussed. The theory of indefinite Kähler metrics has its similarities and its differences with the theory of positive definite Kähler metrics. After showing that the formalism associated with the covariant derivative and curvature operator is almost identical, the authors prove that there are a large number of compact nonflat and Ricci flat indefinite Kähler manifolds. In particular they prove the following theorem: “Let M be a complex manifold of complex dimension n with an indefinite Kähler metric and a nonzero holomorphic parallel form \(\omega\) of bidegree (n,0); then the Ricci curvature of M vanishes.”
There are many compact complex nilmanifolds that satisfy the hypotheses of this theorem; thus, it is quite easy to find compact manifolds with indefinite Kähler metrics that are Ricci flat. The existence of flat indefinite Kähler metrics is shown by considering the Kodaira-Thurston manifold [see W. P. Thurston, Proc. Am. Math. Soc. 55, 467-468 (1976; Zbl 0324.53031), or the reviewer, M. Fernández and A. Gray, Topology 25, 375-380 (1986; Zbl 0596.53030)] and proving the following theorem: “The Kodaira-Thurston manifold with its natural left invariant complex structure has a real 4-parameter family of left invariant indefinite Kähler metrics; furthermore, any left invariant indefinite Kähler metric on it is flat”. Finally, the authors consider the Iwasawa manifold \(I_ 3\) and prove the following theorem: “The manifold \(I_ 3\) (with respect to a nonstandard left invariant complex structure) possesses a Ricci flat indefinite Kähler metric that is non flat.”
Reviewer: L.A.Cordero

MSC:

53C55 Global differential geometry of Hermitian and Kählerian manifolds