Hodge-Bott-Chern decompositions of mixed type forms on foliated Kähler manifolds (Q2925535)

In 2006, \textit{C. Zhong} and \textit{T. Zhong} [Sci. China, Ser. A 49, No. 11, 1696--1714 (2006; Zbl 1107.53049)] proved a Hodge decomposition theorem for Kähler manifolds. Next, in 2007, \textit{M. Schweitzer} [``Autour de la cohomologie de Bott-Chern'', Preprint, \url{arXiv:0709.3528}] developed a Bott-Chern cohomology theory on complex manifolds and obtained a Hodge type decomposition with respect to the assoiated Bott-Chern Laplacian. Recently, \textit{L.-S. Tseng} and \textit{S.-T. Yau} [J. Differ. Geom. 91, No. 3, 383--416 (2012; Zbl 1275.53079); ibid. 91, No. 3, 417--443 (2012; Zbl 1275.53080)] studied cohomology and Hodge theory of Bott-Chern type on symplectic manifolds in 2012. In this paper, using and developing the methods of \textit{I. Vaisman} [J. Differ. Geom. 12, 119--131 (1977; Zbl 0346.32027)] and Tseng and Yau [loc. cit.], the author studies a Bott-Chern cohomology theory and Hodge-type decompositions for differential forms of mixed type on a compact foliated Kähler manifold. Namely, they present the definition of the Bott-Chern and Aeppli cohomology groups of differential forms of mixed type and prove two Hodge decomposition theorems of Bott-Chern type for these forms (Therems 3.1, 3.2 in Section 3). In the last section, the author remark that Hodge-Bott-Chern decompositions are also valid for differential forms of mixed type on the projectivized tangent bundle of a complex Finsler manifold.