Parabolic conformally symplectic structures II: parabolic contactification

Abstract

Parabolic almost conformally symplectic structures were introduced in the first part of this series of articles as a class of geometric structures which have an underlying almost conformally symplectic structure. If this underlying structure is conformally symplectic, then one obtains a PCS-structure. In the current article, we relate PCS-structures to parabolic contact structures. Starting from a parabolic contact structure with a transversal infinitesimal automorphism, we first construct a natural PCS-structure on any local leaf space of the corresponding foliation. Then we develop a parabolic version of contactification to show that any PCS-structure can be locally realized (uniquely up to isomorphism) in this way. In the second part of the paper, these results are extended to an analogous correspondence between contact projective structures and so-called conformally Fedosov structures. The developments in this article provide the technical background for a construction of sequences and complexes of differential operators which are naturally associated to PCS-structures by pushing down BGG sequences on parabolic contact structures. This is the topic of the third part of this series of articles.

Calculus on symplectic manifolds

“…This results in fewer technicalities and in this article we include more detail, especially in constructing the BGG-like complexes in §5. Further indications justifying the shape of our complexes can be found in [3,4,5,6,7].…”

“…The following lemma is also the key step in [11]. (5). Then Θ has constant rank and the bundles ker Θ and coker Θ acquire from ∇, flat connections defining locally constant sheaves ker Θ and coker Θ, respectively.…”

“…. It is part of the curvature of ∇ and if this is the only curvature, then (5) (∇ a ∇ b − ∇ b ∇ a )Σ = 2J ab ΘΣ, and we shall say that ∇ is symplectically flat. Looking back at (1), it is easy to see that there are coupled operators…”

Resolution of the $${\varvec{k}}$$ k -Dirac operator

“…As we mentioned above, the resolution of (1.1) is obtained by descending the k-Dirac complex. The descending of differential operators which are natural to parabolic structures was developed in the recent series of papers [6], [7] and [8] with preliminary paper [5] in full generality for parabolic contact structures. The parabolic geometry of type (G, P) is contact if, and only if k = 2.…”

Parabolic conformally symplectic structures I; definition and distinguished connections

“…Hence we exclude them from the discussion this part of the series. We will discuss conformally Fedosov structures in the framework of contactification in the second part [5] of the series.…”