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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.

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
Full Text: DOI arXiv
[1] Akahori, T.: The new estimate for the subbundles Ej and its application to the deforma- tion of the boundaries of strongly pseudoconvex domains. Invent. Math. 63, 311-334 (1981) · Zbl 0496.32015
[2] Akahori, T., Garfield, P. M., Lee, J. M.: Deformation theory of 5-dimensional CR struc- tures and the Rumin complex. Michigan Math. J. 50, 517-549 (2002) · Zbl 1065.32018
[3] Biquard, O.: Métriques d’Einstein asymptotiquement symétriques. Astérisque 265 (2000) · Zbl 0967.53030
[4] Biquard, O.: Quaternionic contact structures. In: Quaternionic Structures in Mathematics and Physics (Rome, 1999) (electronic), Univ. Studi Roma, “La Sapienza”, 23-30 (1999) · Zbl 0993.53017
[5] Calderbank, D. M. J., Diemer, T.: Differential invariants and curved Bernstein-Gelfand- Gelfand sequences. J. Reine Angew. Math. 537, 67-103 (2001) · Zbl 0985.58002
[6] \check Cap, A.: Correspondence spaces and twistor spaces for parabolic geometries, J. Reine Angew. Math. 582, 143-172 (2005) · Zbl 1075.53022
[7] \check Cap, A.: Two constructions with parabolic geometries. Rend. Circ. Mat. Palermo Suppl. 79, 11-37 (2006) · Zbl 1120.53013
[8] \check Cap, A., Gover, A. R.: Tractor calculi for parabolic geometries. Trans. Amer. Math. Soc. 354, 1511-1548 (2002), · Zbl 0997.53016
[9] \check Cap, A., Gover, A. R.: Standard tractors and the conformal ambient metric construction. Ann. Global Anal. Geom. 24, 231-259 (2003) · Zbl 1039.53021
[10] \check Cap, A., Schichl, H.: Parabolic geometries and canonical Cartan connections. Hokkaido Math. J. 29, 453-505 (2000), · Zbl 0996.53023
[11] \check Cap, A., Slovák, J.: Weyl structures for parabolic geometries. Math. Scand. 93, 53-90 (2003) · Zbl 1076.53029
[12] \check Cap, A., Slovák, J.: Parabolic Geometries I: Background and General Theory. Book in prepa- ration · Zbl 1183.53002
[13] \check Cap, A., Slovák, J., Sou\check cek, V.: Bernstein-Gelfand-Gelfand sequences. Ann. of Math. 154, 97-113 (2001), · Zbl 1159.58309
[14] \check Cap, A., Sou\check cek, V.: Subcomplexes in curved BGG sequences. Preprint ESI 1683 (2005); http://www.esi.ac.at · Zbl 1318.32044
[15] Gasqui, J., Goldschmidt, H.: Déformations infinitésimales des structures conformes plates. Progr. Math. 52, Birkhäuser (1984) · Zbl 0585.53001
[16] Itoh, M.: Moduli of half conformally flat structures. Math. Ann. 296, 687-708 (1993) · Zbl 0788.58011
[17] King, A. D., Kotschick, D.: The deformation theory of anti-self-dual conformal structures. Math. Ann. 294, 591-609 (1992) · Zbl 0765.58005
[18] Kostant, B.: Lie algebra cohomology and the generalized Borel-Weil theorem. Ann. of Math. 74, 329-387 (1961), · Zbl 0134.03501
[19] Yamaguchi, K.: Differential systems associated with simple graded Lie algebras. Adv. Stud. Pure Math. 22, 413-494 (1993) · Zbl 0812.17018
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