Pushing down the Rumin complex to conformally symplectic quotients

Abstract

Given a contact manifold $M_#$ together with a transversal infinitesimal automorphism $\xi$, we show that any local leaf space $M$ for the foliation determined by $\xi$ naturally carries a conformally symplectic (cs-) structure. Then we show that the Rumin complex on $M_#$ descends to a complex of differential operators on $M$, whose cohomology can be computed. Applying this construction locally, one obtains a complex intrinsically associated to 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.Comment: 13 pages v2: added reference [7] to earlier construction of the Rumin-Seshadri complex in a conformally symplectic setting; changed terminonolgy from "locally conformally symplectic" to "conformally symplectic"; some other minor changes; finaly version to appear in Differential Geom. App

Parabolic conformally symplectic structures II: parabolic contactification

“…In Sect. 3 of this article, we show that this is indeed the case and extend the above-mentioned results on structures on quotients and contactifications fromČap and Salač [6] also to this setting. This will allow us to include the C n -types into a uniform treatment in the last part of this series.…”

“…Then we describe the relation between the two kinds of structures, generalizing the analogous results fromČap and Salač [6], which relate conformally symplectic structures to contact structures.…”

“…The corresponding result for conformally symplectic structures has been proved in Proposition 3.1 of [6], so we only have to prove compatibility with the additional structures. Proof Since ϕ # is a contactomorphism, (ϕ # ) * ξ ∈ X(M # ) is a transverse infinitesimal automorphism of the contact structure H ⊂ T M # .…”

“…Next, there is a unique contact form α on M # such that α(ξ ) ≡ 1 and by Proposition 2.2 of [6], there is a unique symplectic form ω on M which is a section of and satisfies q * ω = dα. By assumption, the conformally symplectic structure induced by the reduction p 0 : G 0 → M of structure group is ⊂ Λ 2 T * M. Given a point u ∈ G 0 with p 0 (u) = x, we have ω(x) ∈ x so there is a linear isomorphism ψ(u) : g −2 → R such that, viewing θ(u) as a map T x M → g −1 , we get…”

“…Any transversal infinitesimal contactomorphism defines a foliation with one-dimensional leaves. In [6], we have shown that any local leaf space for such a foliation naturally inherits a conformally symplectic structure. Further we have shown that locally any conformally symplectic structure can be realized in this way and that these realizations are locally unique up to contactomorphism.…”