Kuranishi-type moduli spaces for proper CR-submersions fibering over the circle

Kuranishi’s fundamental result (1962) associates to any compact complex manifold a finite-dimensional analytic space which has to be thought of as a local moduli space of complex structures close to . In this paper, we give an analogous statement for Levi-flat CR-manifolds fibering properly over the circle by associating to any such the loop space of a finite-dimensional analytic space which serves as a local moduli space of CR-structures close to . We then develop in this context a Kodaira–Spencer deformation theory making clear the likenesses as well as the differences with the classical case. The article ends with applications and examples.