Pushing down the Rumin complex to conformally symplectic quotients

Given a contact manifold M#M# together with a transversal infinitesimal automorphism ξ, we show that any local leaf space M for the foliation determined by ξ   naturally carries a conformally symplectic (cs-) structure. Then we show that the Rumin complex on M#M# descends to a complex of differential operators on M, whose cohomology can be computed. Applying this construction locally, one obtains a complex intrinsically associated with any manifold endowed with a cs-structure, which recovers the generalization of the so-called Rumin–Seshadri complex to the conformally symplectic setting. The cohomology of this more general complex can be computed using the push-down construction.