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Parabolic conformally symplectic structures. III: Invariant differential operators and complexes. (English) Zbl 1430.53083

Summary: This is the last part of a series of articles on a family of geometric structures (PACS-structures) which all have an underlying almost conformally symplectic structure. While the first part of the series was devoted to the general study of these structures, the second part focused on the case that the underlying structure is conformally symplectic (PCS-structures). In that case, we obtained a close relation to parabolic contact structures via a concept of parabolic contactification. It was also shown that special symplectic connections (and thus all connections of exotic symplectic holonomy) arise as the canonical connection of such a structure.
In this last part, we use parabolic contactifications and constructions related to Bernstein-Gelfand-Gelfand (BGG) sequences for parabolic contact structures, to construct sequences of differential operators naturally associated to a PCS-structure. In particular, this gives rise to a large family of complexes of differential operators associated to a special symplectic connection. In some cases, large families of complexes for more general instances of PCS-structures are obtained.
For Parts I and II, see [the authors, Forum Math. 30, No. 3, 733–751 (2018; Zbl 1405.53042); Ann. Mat. Pura Appl. (4) 197, No. 4, 1175–1199 (2018; Zbl 1410.53078)].

MSC:

53D05 Symplectic manifolds (general theory)
53D10 Contact manifolds (general theory)
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
58J10 Differential complexes
53C10 \(G\)-structures
53C55 Global differential geometry of Hermitian and Kählerian manifolds
58A10 Differential forms in global analysis
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