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Infinitesimal automorphisms and deformations of parabolic geometries. (English) Zbl 1161.32020
The aim of the paper is to study infinitesimal deformations and the closely related deformations for parabolic geometries. Such deformations are nicely described in terms of the twisted de Rham sequences associated to a certain linear connection on the tractor bundle. The machinery of Bernstein-Gelfand-Gelfand sequences (in short: BGG sequences) is used. For locally flat geometries this leads to a deformation complex, which generalizes the well known complex for locally conformally flat manifolds. The theory of BGG sequences is also applied to certain types of torsion free parabolic geometries including quaternionic structures, quaternionic contact structures and CR structures. It is shown that for these structures one of the subcomplexes in the adjoint BGG sequence leads to a complex governing deformations in the subcategory of torsion free geometries.

MSC:
32V05 CR structures, CR operators, and generalizations
53A40 Other special differential geometries
53B15 Other connections
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53D10 Contact manifolds (general theory)
58H15 Deformations of general structures on manifolds
58J10 Differential complexes
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