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Transverse Kähler structures on central foliations of complex manifolds. (English) Zbl 1412.32018
Summary: For a compact complex manifold, we introduce holomorphic foliations associated with certain abelian subgroups of the automorphism group. If there exists a transverse Kähler 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 Kähler 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.
MSC:
32M25 Complex vector fields, holomorphic foliations, \(\mathbb{C}\)-actions
37F75 Dynamical aspects of holomorphic foliations and vector fields
32Q55 Topological aspects of complex manifolds
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