Transverse Kähler structures on central foliations of complex manifolds

Abstract

For a compact complex manifold, we introduce holomorphic foliations associated with certain abelian subgroups of the automorphism group. If there exists a transverse Kahler 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 bi-graded algebra which is quasi-isomorphic to the Dolbeault complex like the formality of compact Kahler 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.