On the complex analytic Gel'fand-Fuks cohomology of open Riemann surfaces

Nariya Kawazumi

Annales de l'institut Fourier (1993)

  • Volume: 43, Issue: 3, page 655-712
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Kawazumi, Nariya. "On the complex analytic Gel'fand-Fuks cohomology of open Riemann surfaces." Annales de l'institut Fourier 43.3 (1993): 655-712. <http://eudml.org/doc/75015>.

@article{Kawazumi1993,
abstract = {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.},
author = {Kawazumi, Nariya},
journal = {Annales de l'institut Fourier},
keywords = {Gel'fand-Fuks cohomology; spaces of holomorphic functions; complex analytic vector fields; Riemann surface},
language = {eng},
number = {3},
pages = {655-712},
publisher = {Association des Annales de l'Institut Fourier},
title = {On the complex analytic Gel'fand-Fuks cohomology of open Riemann surfaces},
url = {http://eudml.org/doc/75015},
volume = {43},
year = {1993},
}

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
PB - Association des Annales de l'Institut Fourier
VL - 43
IS - 3
SP - 655
EP - 712
AB - 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.
LA - eng
KW - Gel'fand-Fuks cohomology; spaces of holomorphic functions; complex analytic vector fields; Riemann surface
UR - http://eudml.org/doc/75015
ER -

References

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  1. [A] V.I. ARNOL'D, The cohomology ring of the colored braid group, Math. notes (Mat. Zametki), 5 (1969), 138-140. Zbl0277.55002MR39 #3529
  2. [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. Zbl0647.17010MR89i:14019
  3. [B] G.E. BREDON, Sheaf theory, McGraw-Hill, 1967. Zbl0158.20505MR36 #4552
  4. [BeSt] H. BEHNKE und K. STEIN, Entwicklung analytischer Funktionen auf Riemannschen Flächen, Math. Ann., 120 (1948), 430-461. Zbl0038.23502MR10,696c
  5. [BS] R. BOTT and G. SEGAL, The cohomology of the vector fields on a manifold, Topology, 16 (1977) 285-298. Zbl0387.57012MR58 #31102
  6. [C] H. CARTAN, Séminaire H. Cartan 1951/1952, Fonctions analytiques de plusieurs variables. Zbl0049.06404
  7. [F] B.L. FEIGIN, The plenary lecture at ICM, Kyoto 1990. 
  8. [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. Zbl0487.57011
  9. [Go] L.V. GONCHAROVA, Cohomology of Lie algebra of formal vector fields on the line, Functional Anal. Appl., 7 (2) (1973), 6-14. Zbl0284.17006MR49 #4058a
  10. [G,TVS] A. GROTHENDIECK, Topological Vector Spaces, Gordon and Breach, New York, London, Paris, 1973. 
  11. [G,PTT] A. GROTHENDIECK, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc., 16, (1955). Zbl0064.35501MR17,763c
  12. [G,DF] A. GROTHENDIECK, Sur les espaces (F) et (DF), Summa Brasil. Math., 3 (1954), 57-122. Zbl0058.09803MR17,765b
  13. [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. Zbl0176.11501MR39 #6348a
  14. [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. Zbl0216.20301
  15. [H] L. HÖRMANDER, An introduction to complex analysis in several variables, 2nd. ed., van Nostrand, 1966. Zbl0138.06203
  16. [Ha] A. HAEFLIGER, Sur la cohomologie de l'algèbre de Lie des champs de vecteurs, Ann. scient. Éc. Norm. Sup., 9 (1976), 503-532. Zbl0342.57014MR56 #6674
  17. [HS] G. HOCHSCHILD and J.-P. SERRE, Cohomology of Lie algebras, Ann. Math., 57 (1953), 591-603. Zbl0053.01402MR14,943c
  18. [K] H. KOMATSU, Theory of locally convex spaces, Dept. Math., Univ. of Tokyo, 1974. 
  19. [K1] H. KOMATSU, Projective and injective limits of weakly compact sequences of locally convex spaces, J. Math. Soc. Japan, 19 (1955), 366-383. Zbl0168.10603MR36 #646
  20. [M] J. MILNOR, On axiomatic homology theory, Pacific J. Math, 12 (1962), 337-341. Zbl0114.39604MR28 #2544
  21. [P] V.P. PALAMODOV, The projective limit functor in the category of linear topological spaces, Math. USSR-Sbornik, 4 (1968), 529-559. Zbl0175.41801
  22. [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. Zbl0295.57009
  23. [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). Zbl0225.57025
  24. [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. Zbl0505.58040MR83m:58083
  25. [S] L. SCHWARTZ, Séminaire L. Schwartz 1953/1954, Produits tensoriels topologiques d'espaces vectoriels topologiques, 1954. 
  26. [T] F. TREVES, Topological vector spaces, distributions and kernels, Academic Press, 1967. Zbl0171.10402MR37 #726
  27. [Ts] T. TSUJISHITA, Continuous cohomology of the Lie algebra if vector fields, Mem. Amer. Math. Soc., 253, (1981). Zbl0482.58036MR84b:17015
  28. [V] F.V. VAINSHTEIN, Filtering bases, cohomology of infinite dimensional Lie algebras and Laplace operators, Functional Anal. Appl., 19 (1985), 259-269. Zbl0593.17010

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