On deformations of transversely homogeneous foliations (Q5956084)

The study of deformations of geometric structures has its roots in the work of \textit{K. Kodaira} and \textit{D. C. Spencer} [Ann. Math. (2) 67, 328-466 (1958; Zbl 0128.16901)] and \textit{M. Kuranishi} [Deformations of compact complex manifolds (1971; Zbl 0256.32014, Zbl 0382.32014)] on deformations of complex structures. Subsequently \textit{P. A. Griffiths} [Math. Ann. 155, 292-315 (1964; Zbl 0133.15503); 158, 326-351 (1965; Zbl 0156.42703)] developed a deformation theory for a large class of G-structures, and \textit{J. Girbau, A. Haefliger}, and \textit{D. Sundararaman} [ J. Reine Angew. Math. 345, 122-147 (1983; Zbl 0538.32015)] proved the existence of a versal space for deformations of a given transversely holomorphic foliation. We should stress that in each case the local model of the geometric structure under consideration is unique and remains fixed under the deformation.NEWLINENEWLINENEWLINEIn the paper the authors give an affirmative answer to the question posed by E. Ghys in an appendix to [\textit{P. Molino}, Riemannian foliations (1988; Zbl 0633.53001)], concerning the existence of a versal space of deformations for any Riemannian foliation on a compact manifold. They study deformations of a wide class of transversely homogeneous foliations called \(g/h\)-foliations, which includes Lie foliations, moreover they permit the local model to vary. Under some cohomological restrictions the authors show the existence of a versal space for deformations on a compact manifold. For Lie foliations they prove the existence of a weakly versal family of deformations without any cohomological restriction. The proof is an adaptation of the Kodaira-Spencer-Kuranishi method. The main novelty is a ``twisted Hodge theory'' for a class of differential complexes which are not elliptic but very closely related to elliptic ones. The paper is supplemented by several well-chosen examples illustrating the theory.