Čap, Andreas; Salač, Tomáš

Parabolic Conformally Symplectic Structures. III: Invariant Differential Operators and Complexes

Doc. Math. 24, 2203-2240 (2019)
DOI: 10.25537/dm.2019v24.2203-2240
Communicated by Christian Bär


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. \par 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.

Mathematics Subject Classification

53D05, 53D10, 53C15, 58J10, 53C10, 53C55, 58A10


parabolic geometry, conformally symplectic structure, invariant differential operator, differential complex, BGG sequence


  • 1. Gover, A. Rod; Eastwood, Michael; Čap, Andreas; Branson, Thomas, Prolongations of geometric overdetermined systems, Internat. J. Math., 17, 6, 641-664 (2006); DOI 10.1142/S0129167X06003655; zbl 1101.35060; MR2246885; arxiv math/0402100.
  • 2. Griffiths, P. A.; Goldschmidt, H. L.; Gardner, R. B.; Chern, S. S.; Bryant, R. L., Exterior differential systems, Mathematical Sciences Research Institute Publications, 18, viii+475 pp. (1991), New York: Springer-Verlag; zbl 0726.58002; MR1083148.
  • 3. Schwachhöfer, Lorenz J.; Cahen, Michel, Special symplectic connections, J. Differential Geom., 83, 2, 229-271 (2009); DOI 10.4310/jdg/1261495331; zbl 1190.53019; MR2577468; arxiv math/0402221.
  • 4. Diemer, Tammo; Calderbank, David M. J., Differential invariants and curved Bernstein-Gelfand-Gelfand sequences, J. Reine Angew. Math., 537, 67-103 (2001); DOI 10.1515/crll.2001.059; zbl 0985.58002; MR1856258; arxiv math/0001158.
  • 5. Čap, Andreas, Infinitesimal automorphisms and deformations of parabolic geometries, J. Eur. Math. Soc. (JEMS), 10, 2, 415-437 (2008); DOI 10.4171/JEMS/116; zbl 1161.32020; MR2390330; arxiv math/0508535.
  • 6. Hammerl, M.; Gover, A. R.; Čap, A., Holonomy reductions of Cartan geometries and curved orbit decompositions, Duke Math. J., 163, 5, 1035-1070 (2014); DOI 10.1215/00127094-2644793; zbl 1298.53042; MR3189437; arxiv 1103.4497.
  • 7. Salač, Tomáš; Čap, Andreas, Pushing down the Rumin complex to conformally symplectic quotients, Differential Geom. Appl., 35, suppl., 255-265 (2014); DOI 10.1016/j.difgeo.2014.05.004; zbl 1319.58019; MR3254307; arxiv 1312.2712.
  • 8. Salač, Tomáš; Čap, Andreas, Parabolic conformally symplectic structures. I: Definition and distinguished connections, Forum Math., 30, 3, 733-751 (2018); DOI 10.1515/forum-2017-0018; zbl 1405.53042; MR3794908; arxiv 1605.01161.
  • 9. Salač, Tomáš; Čap, Andreas, Parabolic conformally symplectic structures. II: Parabolic contactification, Ann. Mat. Pura Appl. (4), 197, 4, 1175-1199 (2018); DOI 10.1007/s10231-017-0719-3; zbl 1410.53078; MR3829565; arxiv 1605.01897.
  • 10. Slovák, Jan; Čap, Andreas, Parabolic geometries. I: Background and general theory, Mathematical Surveys and Monographs, 154, x+628 pp. (2009), Providence, RI: American Mathematical Society; zbl 1183.53002; MR2532439.
  • 11. Souček, Vladimír; Slovák, Jan; Čap, Andreas, Bernstein-Gelfand-Gelfand sequences, Ann. of Math., 154, 1, 97-113 (2001); DOI 10.2307/3062111; zbl 1159.58309; MR1847589; arxiv math.DG/0001164.
  • 12. Souček, Vladimír; Čap, Andreas, Subcomplexes in curved BGG-sequences, Math. Ann., 354, 1, 111-136 (2012); DOI 10.1007/s00208-011-0726-4; zbl 1318.32044; MR2957620; arxiv math/0508534.
  • 13. Souček, Vladimir; Čap, Andreas, Relative BGG sequences. I: Algebra, J. Algebra, 463, 188-210 (2016); DOI 10.1016/j.jalgebra.2016.06.007; zbl 1380.17009; MR3527545; arxiv 1510.03331.
  • 14. Souček, Vladimír.; Čap, Andreas, Relative BGG sequences. II: BGG machinery and invariant operators, Adv. Math., 320, 1009-1062 (2017); DOI 10.1016/j.aim.2017.09.016; zbl 1418.17022; MR3709128; arxiv 1510.03986.
  • 15. Gauduchon, Paul; David, Liana, The Bochner-flat geometry of weighted projective spaces. In: CRM Proc. Lecture Notes, 40, Amer. Math. Soc., Providence, RI, Perspectives in Riemannian geometry; zbl 1109.32019; MR2237108.
  • 16. Goldschmidt, Hubert; Eastwood, Michael, Zero-energy fields on complex projective space, J. Differential Geom., 94, 1, 129-157 (2013); zbl 1276.53080; MR3031862; arxiv 1108.1602.
  • 17. Slovák, Jan; Eastwood, Michael, Conformally Fedosov manifolds, Adv. Math., 349, 839-868 (2019); DOI 10.1016/j.aim.2019.04.004; zbl 1419.53018; MRnumber=3943783; arxiv 1210.5597.
  • 18. Fox, Daniel J. F., Contact projective structures, Indiana Univ. Math. J., 54, 6, 1547-1598 (2005); DOI 10.1512/iumj.2005.54.2603; zbl 1093.53083; MR2189678; arxiv math/0402332.
  • 19. Kostant, Bertram, Lie algebra cohomology and the generalized Borel-Weil theorem, Ann. of Math. (2), 74, 329-387 (1961); DOI 10.2307/1970237; zbl 0134.03501; MR0142696.
  • 20. Lepowsky, J., A generalization of the Bernstein-Gelfand-Gelfand resolution, J. Algebra, 49, 2, 496-511 (1977); DOI 10.1016/0021-8693(77)90254-X; zbl 0381.17006; MR0476813.
  • 21. Sternberg, Shlomo, Lectures on differential geometry, xv+390 pp. (1964), Prentice-Hall, Inc., Englewood Cliffs, N.J; zbl 0129.13102; MR0193578.
  • 22. Webster, Sidney M., On the pseudo-conformal geometry of a Kähler manifold, Math. Z., 157, 3, 265-270 (1977); DOI 10.1007/BF01214356; zbl 0354.53022; MR477122.


Čap, Andreas
Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, A-1090 Wien, Austria
Salač, Tomáš
Mathematical Institute, Charles University, Prague, Czech Republic