On the complex analytic Gel'fand-Fuks cohomology of open Riemann surfaces
Annales de l'Institut Fourier, Tome 43 (1993) no. 3, pp. 655-712.

Dans cet article nous étudions la cohomologie continue de l’algèbre de Lie L(M) des champs de vecteurs analytiques complexes sur une surface de Riemann ouverte M. Nous montrons que le groupe de cohomologie à coefficients dans le L(M)-module des germes de champs de tenseurs analytiques complexes sur le produit M n se décompose en la partie globale dérivée de l’homologie de M et la partie locale provenant des coefficients.

The continuous cohomology theory of the Lie algebra L(M) of complex analytic vector fields on an open Riemann surface M is studied. We show that the cohomology group with coefficients in the L(M)-module of germs of complex analytic tensor fields on the product space M n decomposes into the global part derived from the homology of M and the local part coming from the coefficients.

@article{AIF_1993__43_3_655_0,
     author = {Kawazumi, Nariya},
     title = {On the complex analytic {Gel'fand-Fuks} cohomology of open {Riemann} surfaces},
     journal = {Annales de l'Institut Fourier},
     pages = {655--712},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {43},
     number = {3},
     year = {1993},
     doi = {10.5802/aif.1351},
     mrnumber = {94i:58211},
     zbl = {0782.57019},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.1351/}
}
TY  - JOUR
AU  - Kawazumi, Nariya
TI  - On the complex analytic Gel'fand-Fuks cohomology of open Riemann surfaces
JO  - Annales de l'Institut Fourier
PY  - 1993
SP  - 655
EP  - 712
VL  - 43
IS  - 3
PB  - Institut Fourier
PP  - Grenoble
UR  - http://www.numdam.org/articles/10.5802/aif.1351/
DO  - 10.5802/aif.1351
LA  - en
ID  - AIF_1993__43_3_655_0
ER  - 
%0 Journal Article
%A Kawazumi, Nariya
%T On the complex analytic Gel'fand-Fuks cohomology of open Riemann surfaces
%J Annales de l'Institut Fourier
%D 1993
%P 655-712
%V 43
%N 3
%I Institut Fourier
%C Grenoble
%U http://www.numdam.org/articles/10.5802/aif.1351/
%R 10.5802/aif.1351
%G en
%F AIF_1993__43_3_655_0
Kawazumi, Nariya. On the complex analytic Gel'fand-Fuks cohomology of open Riemann surfaces. Annales de l'Institut Fourier, Tome 43 (1993) no. 3, pp. 655-712. doi : 10.5802/aif.1351. http://www.numdam.org/articles/10.5802/aif.1351/

[A] V.I. Arnol'D, The cohomology ring of the colored braid group, Math. notes (Mat. Zametki), 5 (1969), 138-140. | MR | Zbl

[ADKP] E. Arbarello, C. Decontini, V.G. Kac, and C. Procesi, Moduli spaces of curves and representation theory, Commun. Math. Phys., 117 (1988), 1-36. | MR | Zbl

[B] G.E. Bredon, Sheaf theory, McGraw-Hill, 1967. | MR | Zbl

[BeSt] H. Behnke und K. Stein, Entwicklung analytischer Funktionen auf Riemannschen Flächen, Math. Ann., 120 (1948), 430-461. | MR | Zbl

[BS] R. Bott and G. Segal, The cohomology of the vector fields on a manifold, Topology, 16 (1977) 285-298. | MR | Zbl

[C] H. Cartan, Séminaire H. Cartan 1951/1952, Fonctions analytiques de plusieurs variables. | Zbl

[F] B.L. Feigin, The plenary lecture at ICM, Kyoto 1990.

[FF] B.L. Feigin and D.B. Fuks, Homology of the Lie algebra of vector fields on the line, Functional Anal. Appl., 14 (1980), 201-212. | Zbl

[Go] L.V. Goncharova, Cohomology of Lie algebra of formal vector fields on the line, Functional Anal. Appl., 7 (2) (1973), 6-14. | MR | Zbl

[G,TVS] A. Grothendieck, Topological Vector Spaces, Gordon and Breach, New York, London, Paris, 1973.

[G,PTT] A. Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc., 16, (1955). | MR | Zbl

[G,DF] A. Grothendieck, Sur les espaces (F) et (DF), Summa Brasil. Math., 3 (1954), 57-122. | MR | Zbl

[GF] I.M. Gel'Fand and D.B. Fuks, The cohomologies of the Lie algebra of the vector fields in a circle, Functional Anal. Appl., 2 (1968), 342-343. | MR | Zbl

[GF1] I.M. Gel'Fand and D.B. Fuks, Cohomologies of the Lie algebra of tangential vector fields of a smooth manifold. I Functional Anal. Appl., 3 (1969), 194-210. II, 4 (1970), 110-116. | Zbl

[H] L. Hörmander, An introduction to complex analysis in several variables, 2nd. ed., van Nostrand, 1966. | Zbl

[Ha] A. Haefliger, Sur la cohomologie de l'algèbre de Lie des champs de vecteurs, Ann. scient. Éc. Norm. Sup., 9 (1976), 503-532. | Numdam | MR | Zbl

[HS] G. Hochschild and J.-P. Serre, Cohomology of Lie algebras, Ann. Math., 57 (1953), 591-603. | MR | Zbl

[K] H. Komatsu, Theory of locally convex spaces, Dept. Math., Univ. of Tokyo, 1974.

[K1] H. Komatsu, Projective and injective limits of weakly compact sequences of locally convex spaces, J. Math. Soc. Japan, 19 (1955), 366-383. | MR | Zbl

[M] J. Milnor, On axiomatic homology theory, Pacific J. Math, 12 (1962), 337-341. | MR | Zbl

[P] V.P. Palamodov, The projective limit functor in the category of linear topological spaces, Math. USSR-Sbornik, 4 (1968), 529-559. | Zbl

[R] V.N. Rešetnikov, On the cohomology of the Lie algebra of vector fields on a manifold with non trivial coefficients, Soviet Math. Dokl., 14 (1) (1973), 234-240. | Zbl

[R1] V.N. Rešetnikov, On the cohomology of two Lie algebras of vector fields on a circle, Uspehi Mat. Nauk, 26 (1) (1971), 231-232 (Russian). | Zbl

[RF] V.S. Retakh and B.L. Feigin, On the cohomology of certain Lie algebras and superalgebras of vector fields, Russian Math. Surveys, 37 (2) (1982), 251-252. | MR | Zbl

[S] L. Schwartz, Séminaire L. Schwartz 1953/1954, Produits tensoriels topologiques d'espaces vectoriels topologiques, 1954.

[T] F. Treves, Topological vector spaces, distributions and kernels, Academic Press, 1967. | MR | Zbl

[Ts] T. Tsujishita, Continuous cohomology of the Lie algebra if vector fields, Mem. Amer. Math. Soc., 253, (1981). | MR | Zbl

[V] F.V. Vainshtein, Filtering bases, cohomology of infinite dimensional Lie algebras and Laplace operators, Functional Anal. Appl., 19 (1985), 259-269. | Zbl

Cité par Sources :