  For a compact complex manifold, we introduce holomorphic foliations
associated with certain abelian subgroups of the automorphism group. Such
foliations are generalizations of holomorphic principal torus bundles. If there
exists a transverse K\"ahler structure on such a foliation, then we obtain a
nice differential graded algebra which is quasi-isomorphic to the de Rham
complex and a nice differential bigraded algebra which is quasi-isomorphic to
the Dolbeault complex like the formality of compact K\"ahler manifolds.
Moreover, under certain additional condition, we can develop Morgan's theory of
mixed Hodge structures as similar to the study on smooth algebraic varieties.
