Geodesic
Natural operations on holomorphic forms
Archivum mathematicum, Tome 54 (2018) no. 4, pp. 239-254 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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.
DOI : 10.5817/AM2018-4-239
Classification : 32L05, 58A32
Keywords: natural bundles; natural operations 
@article{10_5817_AM2018_4_239,
     author = {Navarro, A. and Navarro, J. and Tejero Prieto, C.},
     title = {Natural operations on holomorphic forms},
     journal = {Archivum mathematicum},
     pages = {239--254},
     year = {2018},
     volume = {54},
     number = {4},
     doi = {10.5817/AM2018-4-239},
     mrnumber = {3887363},
     zbl = {06997353},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.5817/AM2018-4-239/}
}
TY  - JOUR
AU  - Navarro, A.
AU  - Navarro, J.
AU  - Tejero Prieto, C.
TI  - Natural operations on holomorphic forms
JO  - Archivum mathematicum
PY  - 2018
SP  - 239
EP  - 254
VL  - 54
IS  - 4
UR  - http://geodesic.mathdoc.fr/articles/10.5817/AM2018-4-239/
DO  - 10.5817/AM2018-4-239
LA  - en
ID  - 10_5817_AM2018_4_239
ER  - 
%0 Journal Article
%A Navarro, A.
%A Navarro, J.
%A Tejero Prieto, C.
%T Natural operations on holomorphic forms
%J Archivum mathematicum
%D 2018
%P 239-254
%V 54
%N 4
%U http://geodesic.mathdoc.fr/articles/10.5817/AM2018-4-239/
%R 10.5817/AM2018-4-239
%G en
%F 10_5817_AM2018_4_239
Navarro, A.; Navarro, J.; Tejero Prieto, C. Natural operations on holomorphic forms. Archivum mathematicum, Tome 54 (2018) no. 4, pp. 239-254. doi: 10.5817/AM2018-4-239

[1] Atiyah, M.: Complex analytic connections in fibre bundles. Trans. Amer. Math. Soc. 85 (1957), 181–207. | DOI | MR

[2] Atiyah, M., Bott, R., Patodi, V.K.: On the heat equation and the index theorem. Invent. Math. 19 (1973), 279–330. | DOI | MR

[3] Bernig, A.: Natural operations on differential forms on contact manifolds. Differential Geom. Appl. 50 (2017), 34–51. | DOI | MR

[4] Epstein, D.B.A., Thurston, W.P.: Transformation groups and natural bundles. Proc. Lond. Math. Soc. 38 (1976), 219–236. | MR

[5] Freed, D.S., Hopkins, M.J.: Chern-Weil forms and abstract homotopy theory. Bull. Amer. Math. Soc. 50 (2013), 431–468. | DOI | MR

[6] Goodman, R., Wallach, N.: Representation and Invariants of the Classical Groups. Cambridge University Press, 1998. | MR

[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.: Natural differential operations on manifolds: an algebraic approach. Sbornik: Mathematics 199 (2008), 1481–1503. | DOI | MR

[9] Kolář, I., Michor, P.W., Slovák, J.: Natural Operations in Differential Geometry. Springer-Verlag, Berlin, 1993. | MR

[10] Krupka, D., Mikolášová, V.: On the uniqueness of some differential invariants: $\,d, [,], \nabla $. Czechoslovak Math. J. 34 (1984), 588–597. | MR

[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.: Natural operations on differential forms. Differential Geom. Appl. 38 (2015), 159–174. | DOI | MR

[14] Nijenhuis, A.: Natural bundles and their general properties. Differential Geometry in honor of K. Yano, Kinokuniya, Tokyo, 1972, pp. 317–334. | MR | Zbl

[15] Palais, R.S.: Natural operations on differential forms. Trans. Amer. Math. Soc. 92 (1959), 125–141. | DOI | MR

[16] Terng, C.L.: Natural vector bundles and natural differential operators. Amer. J. Math. 100 (1978), 775–828. | DOI | MR | Zbl

[17] Timashev, D.A.: On differential characteristic classes of metrics and connections. Fundam. Priklad. Mat. 20 (2015), 167–183. | MR

Cité par Sources :