Previous |  Up |  Next


symplectic structure; Kähler structure; tractor calculus; exact complex; BGG machinery
On a symplectic manifold, there is a natural elliptic complex replacing the de Rham complex. It can be coupled to a vector bundle with connection and, when the curvature of this connection is constrained to be a multiple of the symplectic form, we find a new complex. In particular, on complex projective space with its Fubini–Study form and connection, we can build a series of differential complexes akin to the Bernstein–Gelfand–Gelfand complexes from parabolic differential geometry.
[1] Bailey, T.N., Eastwood, M.G., Gover, A.R.: Thomas’s structure bundle for conformal, projective and related structures. Rocky Mountain J. Math. 24 (1994), 1191–1217. DOI 10.1216/rmjm/1181072333 | MR 1322223 | Zbl 0828.53012
[2] Bryant, R.L., Eastwood, M.G., Gover, A.R., Neusser, K.: Some differential complexes within and beyond parabolic geometry. arXiv:1112.2142.
[3] Cahen, M., Schwachofer, L. J.: Special symplectic connections. J. Differential Geom. 83 (2009), 229–271. DOI 10.4310/jdg/1261495331 | MR 2577468
[4] Čap, A., Salač, T.: Parabolic conformally symplectic structures I; definition and distinguished connections. Forum Math., to appear, arXiv:1605.01161. MR 3794908
[5] Čap, A., Salač, T.: Parabolic conformally symplectic structures II; parabolic contactization. Ann. Mat. Pura Appl., to appear, arXiv:1605.01897. MR 3829565
[6] Čap, A., Salač, T.: Parabolic conformally symplectic structures III; invariant differential operators and complexes. arXiv:1701.01306. MR 3829565
[7] Čap, A., Salač, T.: Pushing down the Rumin complex to conformally symplectic quotients. Differential Geom. Appl. 35 (2014), 255–265, arXiv:1312.2712. DOI 10.1016/j.difgeo.2014.05.004 | MR 3254307
[8] Čap, A., Slovák, J.: Parabolic Geometries I: Background and General Theory. Math. Surveys Monogr. 154 (209). MR 2532439
[9] Eastwood, M.G.: Extensions of the coeffective complex. Illinois J. Math. 57 (2013), 373–381. MR 3263038
[10] Eastwood, M.G., Goldschmidt, H.: Zero-energy fields on complex projective space. J. Differential Geom. 94 (2013), 129–157. DOI 10.4310/jdg/1361889063 | MR 3031862
[11] Eastwood, M.G., Slovák, J.: Conformally Fedosov manifolds. arXiv:1210. 5597.
[12] Gelfand, I.M., Retakh, V.S., Shubin, M.A.: Fedosov manifolds. Adv. Math. 136 (1998), 104–140. DOI 10.1006/aima.1998.1727 | MR 1623673 | Zbl 0945.53047
[13] Knapp, A.W.: Lie Groups, Lie Algebras, and Cohomology. Princeton University Press, 1988. MR 0938524
[14] Kostant, B.: Lie algebra cohomology and the generalized Borel–Weil theorem. Ann. of Math. (2) 74 (1961), 329–387. DOI 10.2307/1970237 | MR 0142696 | Zbl 0134.03501
[15] Penrose, R., Rindler, W.: Spinors and Space-time. vol. 1, Cambridge University Press, 1984. MR 0776784
[16] Seshadri, N.:
[17] Smith, R.T.: Examples of elliptic complexes. Bull. Amer. Math. Soc. (N.S.) 82 (1976), 294–299. DOI 10.1090/S0002-9904-1976-14028-1 | MR 0397796
[18] Tseng, L.-S., Yau, S.-T.: Cohomology and Hodge theory on symplectic manifolds: I and II. J. Differential Geom. 91 (2012), 383–416, 417–443. DOI 10.4310/jdg/1349292670 | MR 2981843
Partner of
EuDML logo