Transverse K\"ahler structures on central foliations of complex manifolds

TL;DR

This paper explores the existence of transverse K"ahler structures on holomorphic foliations of compact complex manifolds, revealing algebraic formality properties and extending Morgan's mixed Hodge theory under specific conditions.

Contribution

It introduces a new class of holomorphic foliations linked to abelian automorphism subgroups and establishes their algebraic and Hodge-theoretic properties when equipped with transverse K"ahler structures.

Findings

Existence of differential graded algebra quasi-isomorphic to de Rham complex.

Development of a differential bigraded algebra akin to Dolbeault complex.

Extension of Morgan's mixed Hodge structures to these foliations.

Abstract

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.