Calculus on symplectic manifolds

Michael Eastwood; Jan Slovák

Archivum Mathematicum (2018)

  • Volume: 054, Issue: 5, page 265-280
  • ISSN: 0044-8753

Abstract

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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.

How to cite

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Eastwood, Michael, and Slovák, Jan. "Calculus on symplectic manifolds." Archivum Mathematicum 054.5 (2018): 265-280. <http://eudml.org/doc/294700>.

@article{Eastwood2018,
abstract = {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.},
author = {Eastwood, Michael, Slovák, Jan},
journal = {Archivum Mathematicum},
keywords = {symplectic structure; Kähler structure; tractor calculus; exact complex; BGG machinery},
language = {eng},
number = {5},
pages = {265-280},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Calculus on symplectic manifolds},
url = {http://eudml.org/doc/294700},
volume = {054},
year = {2018},
}

TY - JOUR
AU - Eastwood, Michael
AU - Slovák, Jan
TI - Calculus on symplectic manifolds
JO - Archivum Mathematicum
PY - 2018
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 054
IS - 5
SP - 265
EP - 280
AB - 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.
LA - eng
KW - symplectic structure; Kähler structure; tractor calculus; exact complex; BGG machinery
UR - http://eudml.org/doc/294700
ER -

References

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  1. Bailey, T.N., Eastwood, M.G., Gover, A.R., 10.1216/rmjm/1181072333, Rocky Mountain J. Math. 24 (1994), 1191–1217. (1994) Zbl0828.53012MR1322223DOI10.1216/rmjm/1181072333
  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., 10.4310/jdg/1261495331, J. Differential Geom. 83 (2009), 229–271. (2009) MR2577468DOI10.4310/jdg/1261495331
  4. Čap, A., Salač, T., Parabolic conformally symplectic structures I; definition and distinguished connections, Forum Math., to appear, arXiv:1605.01161. MR3794908
  5. Čap, A., Salač, T., Parabolic conformally symplectic structures II; parabolic contactization, Ann. Mat. Pura Appl., to appear, arXiv:1605.01897. MR3829565
  6. Čap, A., Salač, T., Parabolic conformally symplectic structures III; invariant differential operators and complexes, arXiv:1701.01306. MR3829565
  7. Čap, A., Salač, T., 10.1016/j.difgeo.2014.05.004, Differential Geom. Appl. 35 (2014), 255–265, arXiv:1312.2712. (2014) MR3254307DOI10.1016/j.difgeo.2014.05.004
  8. Čap, A., Slovák, J., Parabolic Geometries I: Background and General Theory, Math. Surveys Monogr. 154 (209). (209) MR2532439
  9. Eastwood, M.G., 10.1215/ijm/1408453587, Illinois J. Math. 57 (2013), 373–381. (2013) MR3263038DOI10.1215/ijm/1408453587
  10. Eastwood, M.G., Goldschmidt, H., 10.4310/jdg/1361889063, J. Differential Geom. 94 (2013), 129–157. (2013) MR3031862DOI10.4310/jdg/1361889063
  11. Eastwood, M.G., Slovák, J., Conformally Fedosov manifolds, arXiv:1210. 5597. 
  12. Gelfand, I.M., Retakh, V.S., Shubin, M.A., 10.1006/aima.1998.1727, Adv. Math. 136 (1998), 104–140. (1998) Zbl0945.53047MR1623673DOI10.1006/aima.1998.1727
  13. Knapp, A.W., Lie Groups, Lie Algebras, and Cohomology, Princeton University Press, 1988. (1988) MR0938524
  14. Kostant, B., 10.2307/1970237, Ann. of Math. (2) 74 (1961), 329–387. (1961) Zbl0134.03501MR0142696DOI10.2307/1970237
  15. Penrose, R., Rindler, W., Spinors and Space-time, vol. 1, Cambridge University Press, 1984. (1984) MR0776784
  16. Seshadri, N., [unknown] 
  17. Smith, R.T., 10.1090/S0002-9904-1976-14028-1, Bull. Amer. Math. Soc. (N.S.) 82 (1976), 294–299. (1976) MR0397796DOI10.1090/S0002-9904-1976-14028-1
  18. Tseng, L.-S., Yau, S.-T., 10.4310/jdg/1349292670, J. Differential Geom. 91 (2012), 383–416, 417–443. (2012) MR2981843DOI10.4310/jdg/1349292670

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