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Sheaves and cohomology

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Part of the Lecture Notes in Physics book series (LNP, volume 97)

Keywords

Exact Sequence Line Bundle Holomorphic Function Algebraic Variety Short Exact Sequence 
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Notes

  1. 1.
    For a description of the limiting procedure involved here see Gunning, 1966, p. 30.. Also, see pp. 44–47 in the same reference for a discussion of “Leray's theorem” which gives a set of conditions sufficient to ensure that a covering Ui is general enough to calculate the cohomology of a space M.Google Scholar
  2. 2.
    Strictly speaking in order to establish this result we need to know that a covering of Pl by two open sets suffices to compute its cohomology.Google Scholar
  3. 3.
    Cross-sections of the sheaf O(n) are often referred to as “twisted functions”; and O(n) itself is called the “sheaf of germs of holomorphic functions, twisted by n”.Google Scholar
  4. 4.
    For further discussion of the long exact cohomology sequence, see, for example, Gunning, 1966, pp. 32-34.Google Scholar
  5. 5.
    Holomorphic line bundles and holomorphic vector bundles—built over suitable regions of projective twistor space—can be used to describe self-dual solutions of Maxwell's equations and the Yang-Mills equations (without sources). See Ward (1977a and 1977b), Atiyah and Ward (1977), Hartshorne (1978), and Ward (1979) for various details of the procedure. Also see Burnett-Stuart (1978) and Moore (1978).Google Scholar
  6. 6.
    Note that for sequence (9.7.8) we have an isomorphism between O(−2) and IV. In the case of sequence (9.7.11) we have the following isomorphism: IV = Oij(−2)/Image [OA′(−3)]Google Scholar

Copyright information

© Springer-Verlag 1979

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