Differential Analysis on Complex Manifolds by R. O. Wells Jr. (auth.)

By R. O. Wells Jr. (auth.)

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Resolutions 0/ Sheaves Sec. 4, since the presheaf of quotients of sections of Z by sections of d violates Axiom S2' Following the terminology of homological algebra for modules, we make the following definitions where sheaf means sheaf of abelian groups or sheaf of modules. 2). Aresolution of a sheaf 5' is an exact sequenee of sheaves of the form 0--5' __ 5'0 __ 5'1 __ .. • , which we also denote symbolically by 0--5' - - 5'*, the maps being understood. We shall see later that various types of information for a given sheaf 5' can be obtained from knowledge of agiven resolution.

Therefore sections can be thought of as "twisted" vector-valued functions. Remarks: (a) A section s is often identified with its image s(X) c E; for example, the term zero section is used to refer to the section 0: X -+ E given by O(x) = 0 E E" and is often identified with its image, which is, in fact, S,-isomorphic with the base space X. o. X and E' ~ X we can identify the set of s,bundle homomorphisms of E into E', with S,(X, Hom(E, E'». , = s(n(e» for e E E. /lh on U/I (") U/I =f. 0. , and 1/1 as column vectors.

Namely, let U be a trivializing neighborhood for E so that h: Elu ~ U X Kn, and thus we have an isomorphism h*: S(U, Elu) ~ S(U, U x K"). Consider the vector-valued functions el = (1,0, ... ,0), ez = (0, 1, ... ,0), ... ,e" = (0, ... ,0, I), Sec. 3 A/most Comp/ex Manifolds and the ä-Operator 33 which cIearly form a (constant) frame for V x K', and thus {(h'*tl(e l ), ... )} forms a frame for E lu, since the bundle mapping h is an isomorphism on fibres, carrying a basis to a basis. Therefore we see that having a frame is equivalent to having a trivialization and that the existence of a global frame (defined over X) is equivalent to the bundle being trivial.

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