Coeffective-Dolbeault cohomology of compact indefinite Kähler manifolds (Q1378314)

\textit{T. Bouché} [Bull. Sci. Math., II. Ser. 114, 115-122 (1990; Zbl 0714.58001)] defined the so-called coeffective cohomology of a compact symplectic manifold \(M\); in particular, he showed that a certain relation between the coeffective cohomology groups and the de Rham groups of \(M\) follows when the symplectic structure on \(M\) is in fact a Kähler structure. Such a relation is no longer true if the Kähler structure on \(M\) is indefinite, as has been shown by \textit{L. C. de Andrés}, \textit{M. Fernández}, \textit{R. Ibañez}, \textit{M. de Léon} and \textit{J. J. Mencía} [C. R. Acad. Sci., Paris, Sér. I 318, 231-236 (1994; Zbl 0814.57020)]. In the paper under review, the author considers, for an indefinite Kähler manifold \(M\), a certain differential subcomplex of the Dolbeault complex of \(M\) whose cohomology is called the coeffective-Dolbeault cohomology of \(M\), and then he states some significant results about it. In particular, if \(M\) is a compact (positive definite) Kähler manifold, then: (1) there is a certain relation between the coeffective-Dolbeault cohomology of \(M\) and its Dolbeault cohomology; (2) a Hodge decomposition theorem for the coeffective cohomology of \(M\) is proved, thus relating this cohomology with the coeffective-Dolbeault cohomology of \(M\). These two relations are shown to be no longer true if the Kähler metric on \(M\) is indefinite; in fact, two examples which are compact nilmanifolds with indefinite Kähler structures are constructed and the corresponding cohomologies on them are computed.