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# Sheaves and cohomology

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## Keywords

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

- 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 U
_{i}is general enough to calculate the cohomology of a space M.Google Scholar - 2.Strictly speaking in order to establish this result we need to know that a covering of P
^{l}by two open sets suffices to compute its cohomology.Google Scholar - 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.For further discussion of the long exact cohomology sequence, see, for example, Gunning, 1966, pp. 32-34.Google Scholar
- 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.Note that for sequence (9.7.8) we have an isomorphism between O(−2) and I
_{V}. In the case of sequence (9.7.11) we have the following isomorphism: I_{V}= O^{ij}(−2)/Image [O^{A′}(−3)]Google Scholar

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© Springer-Verlag 1979