Natural operations on holomorphic forms

A. Navarro; J. Navarro; C. Tejero Prieto

Archivum Mathematicum (2018)

  • Volume: 054, Issue: 4, page 239-254
  • ISSN: 0044-8753

Abstract

top
We prove that the only natural differential operations between holomorphic forms on a complex manifold are those obtained using linear combinations, the exterior product and the exterior differential. In order to accomplish this task we first develop the basics of the theory of natural holomorphic bundles over a fixed manifold, making explicit its Galoisian structure by proving a categorical equivalence à la Galois.

How to cite

top

Navarro, A., Navarro, J., and Tejero Prieto, C.. "Natural operations on holomorphic forms." Archivum Mathematicum 054.4 (2018): 239-254. <http://eudml.org/doc/294155>.

@article{Navarro2018,
abstract = {We prove that the only natural differential operations between holomorphic forms on a complex manifold are those obtained using linear combinations, the exterior product and the exterior differential. In order to accomplish this task we first develop the basics of the theory of natural holomorphic bundles over a fixed manifold, making explicit its Galoisian structure by proving a categorical equivalence à la Galois.},
author = {Navarro, A., Navarro, J., Tejero Prieto, C.},
journal = {Archivum Mathematicum},
keywords = {natural bundles; natural operations},
language = {eng},
number = {4},
pages = {239-254},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Natural operations on holomorphic forms},
url = {http://eudml.org/doc/294155},
volume = {054},
year = {2018},
}

TY - JOUR
AU - Navarro, A.
AU - Navarro, J.
AU - Tejero Prieto, C.
TI - Natural operations on holomorphic forms
JO - Archivum Mathematicum
PY - 2018
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 054
IS - 4
SP - 239
EP - 254
AB - We prove that the only natural differential operations between holomorphic forms on a complex manifold are those obtained using linear combinations, the exterior product and the exterior differential. In order to accomplish this task we first develop the basics of the theory of natural holomorphic bundles over a fixed manifold, making explicit its Galoisian structure by proving a categorical equivalence à la Galois.
LA - eng
KW - natural bundles; natural operations
UR - http://eudml.org/doc/294155
ER -

References

top
  1. Atiyah, M., 10.1090/S0002-9947-1957-0086359-5, Trans. Amer. Math. Soc. 85 (1957), 181–207. (1957) MR0086359DOI10.1090/S0002-9947-1957-0086359-5
  2. Atiyah, M., Bott, R., Patodi, V.K., 10.1007/BF01425417, Invent. Math. 19 (1973), 279–330. (1973) MR0650828DOI10.1007/BF01425417
  3. Bernig, A., 10.1016/j.difgeo.2016.10.005, Differential Geom. Appl. 50 (2017), 34–51. (2017) MR3588639DOI10.1016/j.difgeo.2016.10.005
  4. Epstein, D.B.A., Thurston, W.P., Transformation groups and natural bundles, Proc. Lond. Math. Soc. 38 (1976), 219–236. (1976) MR0531161
  5. Freed, D.S., Hopkins, M.J., 10.1090/S0273-0979-2013-01415-0, Bull. Amer. Math. Soc. 50 (2013), 431–468. (2013) MR3049871DOI10.1090/S0273-0979-2013-01415-0
  6. Goodman, R., Wallach, N., Representation and Invariants of the Classical Groups, Cambridge University Press, 1998. (1998) MR1606831
  7. Gordillo, A., Navarro, J., Sancho, P., A remark on the invariant theory of real Lie groups, Colloq. Math., to appear. 
  8. Katsylo, P.I., Timashev, D.A., 10.1070/SM2008v199n10ABEH003969, Sbornik: Mathematics 199 (2008), 1481–1503. (2008) MR2473812DOI10.1070/SM2008v199n10ABEH003969
  9. Kolář, I., Michor, P.W., Slovák, J., Natural Operations in Differential Geometry, Springer-Verlag, Berlin, 1993. (1993) MR1202431
  10. Krupka, D., Mikolášová, V., On the uniqueness of some differential invariants: d , [ , ] , , Czechoslovak Math. J. 34 (1984), 588–597. (1984) MR0764440
  11. Mason-Brown, L., Natural structures in differential geometry, private communication. 
  12. Navarro, J., Sancho, J.B., Peetre-Slovák’s theorem revisited, arXiv: 1411.7499. 
  13. Navarro, J., Sancho, J.B., 10.1016/j.difgeo.2014.12.003, Differential Geom. Appl. 38 (2015), 159–174. (2015) MR3304675DOI10.1016/j.difgeo.2014.12.003
  14. Nijenhuis, A., Natural bundles and their general properties, Differential Geometry in honor of K. Yano, Kinokuniya, Tokyo, 1972, pp. 317–334. (1972) Zbl0246.53018MR0380862
  15. Palais, R.S., 10.1090/S0002-9947-1959-0116352-7, Trans. Amer. Math. Soc. 92 (1959), 125–141. (1959) MR0116352DOI10.1090/S0002-9947-1959-0116352-7
  16. Terng, C.L., 10.2307/2373910, Amer. J. Math. 100 (1978), 775–828. (1978) Zbl0422.58001MR0509074DOI10.2307/2373910
  17. Timashev, D.A., On differential characteristic classes of metrics and connections, Fundam. Priklad. Mat. 20 (2015), 167–183. (2015) MR3472276

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.