Cohomology

  • Th. Peternell
Part of the Encyclopaedia of Mathematical Sciences book series (EMS, volume 74)

Abstract

Cohomology (with values in a sheaf) — attaching to every sheaf ℒ of abelian groups “cohomology groups” Hq(X, ℒ), q ≥ 0 — was invented in the late 1940s but implicitly it has been present since the 19th century. We want to explain this and demonstrate the necessity for a cohomology theory at the hand of several classical or “basic” problems.

Keywords

Filtration Manifold Hull Stein Tral 

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References

  1. [AG62]
    Andreotti, A.; Grauert, H.: Théorèmes de finitude pour la cohomologie des espaces complexes. Bull. Soc. Math. Fr. 90, 193–259 (1962) Zb1. 106, 55.Google Scholar
  2. [AtSi63]
    Atiyah, M.F.; Singer, I.M.: The index of elliptic operators on compact manifolds. Bull. Am. Math. Soc. 69, 422–433 (1963) Zb1. 118, 312.Google Scholar
  3. [BeSt49]
    Behnke, H.; Stein, K.: Entwicklung analytischer Funktionen auf Riemannschen Flächen. Math. Ann. 120, 430–461 (1949) Zb1. 38, 235.Google Scholar
  4. [BaSt76]
    Banica, C.; Stanasila, O.: Algebraic Methods in the Global Theory of Complex Spaces. Wiley 1976. Zb1. 284. 32006.Google Scholar
  5. [Bo57]
    Bott, R.: Homogeneous vector bundles. Ann. Math., H. Ser. 66, 203–248 (1957) Zb1. 94, 357.Google Scholar
  6. [BoSe59]
    Bord, A.; Serre, J-P.: Le théorème de Riemann-Roch. Bull. Soc. Math. Fr. 86, 97–136 (1959) Zb1. 91, 330.Google Scholar
  7. [BPV84]
    Barth, W.; Peters, C.; van de Ven, A.: Compact Complex Surfaces. Erg. Math. 4, Springer 1984, Zb1. 718. 14023.Google Scholar
  8. [CaEi56]
    Cartan, H.; Eilenberg, S.: Homological Algebra. Princeton Univ. Press 1956, Zb1. 75, 243.Google Scholar
  9. [CaSe53]
    Cartan, H.; Serre, J.P.: Un théorème de finitude concernant les variétés analytiques compactes. C.R. Acad. Sci. Paris 237, 128–130 (1953)MathSciNetMATHGoogle Scholar
  10. [Dem85]
    Demailly, J.P.: Champs magnétiques et inégalites de Morse pour la d“-cohomologie. Ann. Inst. Fourier 35, No. 4, 185–229 (1985) Zb1. 565. 58017.Google Scholar
  11. [DV74]
    Douady, A.; Verdier, J.P. (ed.): Différents aspects de la positivité. Astérisque 17. Paris 1974.Google Scholar
  12. [FoKn71]
    Forster, O.; Knorr, K.: Ein Beweis des Grauertschen Bildgarbensatzes nach Ideen von B. Malgrange. Manuscr. Math. 5, 19–44 (1971) Zb1. 242. 32008.Google Scholar
  13. [Ful84]
    Fulton, W.: Intersection theory. Erg. d. Math., 3 Folge, Bd 2. Springer 1984. Zb1. 541. 14005.Google Scholar
  14. [GH78]
    Griffiths, Ph.; Harris, J.: Principles of Algebraic Geometry. Wiley 1978, Zb1. 408. 14001.Google Scholar
  15. [God58]
    Godement, R.: Topologie algébrique et théorie des faisceaux. Herman, Paris 1958, Zb1. 80, 162.Google Scholar
  16. [Gr55]
    Grauert, H.: Charakterisierung der holomorph-vollständigen Räume. Math. Ann. 129, 233–259 (1955) Zb1. 64, 326.Google Scholar
  17. [Gr58]
    Grauert, H.: On Levi’s problem and the embedding of real-analytic manifolds. Ann. Math., II. Ser. 68, 460–472 (1958) Zb1. 108, 78.Google Scholar
  18. [Gr60]
    Grauert, H.: Ein Theorem der analytischen Garbentheorie und die Modulräume komplexer Strukturen. Publ. Math., Inst. Hauter Etud. Sci. 5, 5–64 (1960) Zb1. 100, 80.Google Scholar
  19. GrRe77] Grauert, H.; Remmert, R.: Theorie der Steinschen Räume. Grundl. 227, Springer Math. Wiss. 1977, Zb1. 379. 32001.Google Scholar
  20. [GrRe84]
    Grauert, H.; Remmert, R.: Coherent Analytic Sheaves. Grundl. 265, Springer 1984, Zb1. 537. 32001.Google Scholar
  21. [Ha83]
    Hamm, H.A.: Zum Homotopietyp Steinscher Räume. J. Reine Augew. Math. 338, 121–135 (1983) Zb1. 491. 32010.Google Scholar
  22. [Ha66]
    Hartshorne, R.: Residues and duality. Lect. Notes Math. 20, Springer 1966, Zb1. 212, 261.Google Scholar
  23. [Hir56]
    Hirzebruch, F.: Neue Topologische Methoden in der Algebraischen Geometrie. Grundl. Math. Wiss. 131, Springer 1956, Zb1. 70, 163.Google Scholar
  24. [HiSt71]
    Hilton, P J.; Stammbach, U.: A Course in Homological Algebra. Graduate Texts Math. 4, Springer 1971, Zbl. 238. 18006.Google Scholar
  25. [Kun75]
    Kunz, E.: Holomorphe Differentialformen auf algebraischen Varietäten mit Singularitäten I. Manuscr. Math. 15, 91–108 (1975) Zb1. 299. 14013.Google Scholar
  26. [Kun77]
    Kunz, E.: Residuen von Differentialformen auf Cohen-Macaulay-Varietäten. Math. Z. 152, 165–189 (1977) Zb1. 342. 14022.Google Scholar
  27. [Kun78]
    Kunz, E.: Differentialformen auf algebraischen Varietäten mit Singularitäten II. Abh. Math. Semin. Univ. Hamb. 47, 42–70 (1978) Zb1. 379. 14005.Google Scholar
  28. [Lei90]
    Leiterer, J.: Holomorphic vector bundles and the Oka-Grauert principle. In: Encycl. Math. Sci. 10, 63–103, Springer 1990, Zb1. 639. 00015.Google Scholar
  29. [Lip84]
    Lipman, J.: Dualizing sheaves, differentials and residues of algebraic varieties. Astérisque 117, 1984, Zb1. 562. 14008.Google Scholar
  30. [Na67]
    Narasimham, R.: On the homology groups of Stein spaces. Invent. Math. 2, 377–385 (1967) Zb1. 148, 322.Google Scholar
  31. [Pe91]
    Peternell, Th.: Hodge-Kohomologie und Steinsche Mannigfaltigkeiten. In: Complex Analysis, Wuppetal, Ed. K. Diederich. Vieweg 1991.Google Scholar
  32. [Re57]
    Remmert, R.: Holomorphe und meromorphe Abbildungen komplexer Räume. Math. Ann. 133, 328–370 (1957) Zb1. 79, 102.Google Scholar
  33. [RR70]
    Ramis, J.P.; Ruget, G.: Complexe dualisant et théorèmes de dualité en géométrie analytique complexe. Publ. Math. Inst. Hautes Etud. Sci. 38, 77–91 (1970) Zbl. 206, 250.Google Scholar
  34. RR74] Ramis, J.P.; Ruget, G.: Résidus et dualité. Invent. Math. 26, 89–131 (1974) Zb1. 304. 32007.Google Scholar
  35. SC52] Séminaire Cartan. Théorie des fonctions de plusieurs variables. Paris 1951/52.Google Scholar
  36. [Ser55]
    Serre, J.-P.: Faisceaux algébriques cohérents. Ann. Math., H. Ser. 61, 197–278 (1955) Zb1. 67, 162.Google Scholar
  37. [Ser55-2]
    Serre, J-P.: Un théorème de dualité. Comment. Math. Helv. 29, 9–26 (1955) Zb1. 67, 161.Google Scholar
  38. SGA2] Grothendieck, A.: Séminaire de géometrie algébrique 2. Cohomologie locale des fais- ceaux cohérents. North Holland 1968, Zb1. 197, 472.Google Scholar
  39. [SiTr71]
    Siu, Y.T.; Trautmann, G.: Gap-sheaves and extensions of coherent analytic subsheaves. Lect. Notes Math. 172, Springer 1971, Zb1. 208, 104.Google Scholar
  40. [Sn86]
    Snow, D.: Cohomology of twisted holomorphic forms on Grassmann manifolds and quadric hypersurfaces. Math. Ann. 276, 159–176 (1986) Zb1. 596. 32016Google Scholar
  41. [St51]
    Stein, K.: Analytische Funktionen mehrerer komplexer Veränderlichen zu vorgegebenen Periodizitätsmoduln und das zweite Cousinsche Problem. Math. Ann. 123, 201222 (1951) Zb1. 42, 87.Google Scholar
  42. [ToTo76]
    Toledo, D.; Tong, Y.L.L.: A parametrix ford and Riemann-Roch in tech theory. Topology 15, 273–301 (1976) Zb1. 355. 58014.Google Scholar
  43. [Ue75]
    Ueno, K.: Classification theory of algebraic varieties and compact complex spaces. Lect. Notes Math. 439, Springer 1975, Zbl. 299. 14007.Google Scholar
  44. [We80]
    Wells, R.O.: Differential analysis on complex manifolds. 2nd ed. Springer 1980, Zb1.435.32004; Zb1. 262. 32005.Google Scholar
  45. [Weh85]
    Wehler, J.: Der relative Dualitätssatz für Cohen-Macaulay-Räume. Schriftenr. Math. Inst. Univ. Münster, 2. Ser. 35, Zb1. 625. 32010.Google Scholar

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© Springer-Verlag Berlin Heidelberg 1994

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  • Th. Peternell

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