1. — Bundles and sheaf cohomology.
The theory of holomorphic bundles is intimately related to the cohomology theory of sheaves. Although this cohomology theory arises in various contexts, and can be developed independently of bundle theory, it is rather natural from our point of view to explain it in terms of bundles. The main thing to emphasize just now is that cohomology is a linear or abelian theory and it first arises from abelian gauge groups and, at a later stage, from solvable groups.
We shall begin, as in Chapter IV, § 3, by considering holomorphic bundles over -Pi(O). This simple case provides a good illustration of the general theory and ties up with the classical function theory of one complex variable. In addition this special case occupies a key role in our applications, since Pi(0) appears as the fibre of our basic fibration P3(C) -> S1.
In Chapter IY, § 3, we explained how a holomorphic line-bundle over Pi(0) given by the holomorphic gauge-transformation f(z), near the equator, could be reduced, as in (3.1), to the standard form zk. In particular when 1c = 0 we could take g(z) = log f(z) and using the Laurent expansion
we could write g(z) in the form
with g0(z) holomorphic for |«| < R and ga(z) holomorphic for r < \z\. Suppose now we ask instead whether we can write g(z) in the form
where m is some fixed integer, and g0, are as before holomorphic near 0, oo respectively. Clearly if m<l this is possible while for m>2 it is possible if and only if the coefficients of (1.1) satisfy:
In other words the functions z, ..., z'"-1 in general provide obstructions to the solubility of (1.2). Alternatively the space of all holomorphic functions g(z), modulo those which can be written in the form (1.3), has dimension m — 1 and a basis is represented by z,..., This is our first encounter with a sheaf cohomology group: it is usually denoted by EX(P1, Lm) or H1(P1, 0(— m)). The reasons for these notations will be explained shortly.