3. - Bundles over Pi(G).
In order to familiarize ourselves with holomorphic vector bundles we shall now describe the simplest case, namely over the complex projective line.
The basic example of a complex line bundle (fibre 0l) is, as we have seen, given by the family i(z) of complex lines in 02, parametrized by the corresponding point (zjePjfC). This standard bundle is topologically non- trivial, and can be described by two holomorphic gauges near z = 0 and z = oo on Pt (z being a non-homogeneous coordinate), with the gauge transformation from one to the other being multiplication by z. More generally the bundle Ln = L ®... ® L (n times) is similarly given by the gauge transformation zn. Taking L 1 = L* (the dual) we can also allow n to be a negative integer.
the integral taken over \z\ = 1. Suppose first fc = 0, then we have a well- defined holomorphic function
in some annulus v < \z\ < R. Take the Laurent expansion
One can now ask if there are any further holomorphic line bundles not isomorphic to some Ln. Suppose for example we construct a bundle by using as a gauge transformation any function f(z) which is holomorphic and nonzero near the equator \z\ = 1. First of all we can define the topological invariant where g0 is holomorphic in \z\ < B and g„ is holomorphic in r < \z\, i.e. g0 involves terms with n < 0 and gm those with «>0. This decomposition is therefore unique up to additive constants. Exponentiating we get
which we can interpret as saying that the holomorphic bundle given by f(z) is holomorphically trivial. The terms fg(z) and fjz) allow us to change our original holomorphic gauges so that they now coincide near the equator and so provide a global holomorphic gauge. Thus if fc = 0 the bundle is holomorphically trivial and a similar argument tells us in general that
and so the bundle is holomorphically equivalent to L . A slight generalization of this argument, allowing more than two local gauges, leads to the conclusion that every holomorphic line-bundle on P^G) is isomorphic to some Lh.
Passing now to vector bundles with fibre G" we can construct obvious examples as direct sums:
A much subtler theorem, proved in various versions over the years by Hilbert, G. D. Birkhoff, Grothendieck  and others asserts that every holomorphic vector bundle over P1(0) is isomorphic to such a direct sum of Lkl and the integers fc„) are unique up to permutation. In more
concrete terms this implies in particular a matrix version of (3.1), namely if f(z) is a holomorphic function defined in the annulus r < \z\< B, taking values in GL(n, G), then we can write f(z) in the form
where A(z) is the diagonal matrix with entries zk< and f„(z), /„(«) are holomorphic functions with values in GL(n, G) defined for \z\ < B and r < \z\ respectively.
and write it in the form
It is important to note that although the exponents lc( in (3.2) are holomorphic invariants of the bundle E they are not topological invariants, onlythe sum h = ^ki is a topological invariant. Again, in more concrete terms, if in (3.3) f(z) is holomorphic in z and depends continuously on some other parameter t, then the factorization (3.3) cannot in general be made continuous in t. The integers lct will depend on t in a semi-continuous fashion. Namely as the differences — kj(t)j can suddenly increase. In
particular this implies that the trivial bundle, with all k( = 0, is stable under small deformations.
A simple geometrical example will illustrate the way in which the integers k( can suddenly jump. Consider the case n = 2, and assume that the topological invariant \ + k2 = 0. Instead of vector bundles with fibre G2 we can equally well work with projective bundles with fibre PX(G). The trivial bundle is P1xP1, while an example of a non-trivial bundle corresponding to integers (1, — 1) is provided by the family of generators of a quadric cone in P3 (we regard the vertex of the cone as a different point on each generator). If we now take a general quadric in Ps it can be identified with P1xP1 (cf. a real hyperboloid). By continuously altering the coefficients of the quadratic form we can end up with a singular quadric i.e. a cone.
On Pj((7) therefore all holomorphic vector bundles are known. In particular they are all algebraic, i.e. in suitable gauges the gauge transformations are rational functions of z. Moreover this algebraic structure is essentially unique. This is a special case of a general theorem of Serre  which applies to all complex algebraic varieties in projective space of any dimension, and in particular to projective spaces themselves. However the simple and complete classification available over Py(G) does not extend to other algebraic varieties.
One way to study holomorphic (or algebraic) vector bundles on a projective space Pm(G) (e.g. m = 3 which is our case) is to consider the restriction of the bundle to all the projective lines in Pm(C). The semi-continuity property mentioned above tells us that over the «general line» we get a set of integers (fcj, ..., 7c „) but there will in general be some exceptional or jumping lines where the differences \ht—kf\ jump upwards. If for some line all kt are 0 (i.e. the restricted bundle is trivial) then this must be true for the general line. According to Theorem (2.9) we see that the holomorphic vector bundles on PS(G) arising from anti-instantons on have this property, since every real line has all = 0. The jumping lines are not real and correspond to points in the complexification of Sl. Serre's theorem assures us that, in a suitable gauge, our anti-self-dual gauge potential on S1 will be given by rational functions in the coordinates. The complex poles of these rational functions account for the jumping lines.
If E is a holomorphic vector bundle on P3(G) trivial on general lines then, up to a constant matrix, E has a unique global holomorphic gauge along any general line (since liolomorpliic functions on Pj are constant). However if we take a triangle formed by three coplanar general lines there is no reason why the three holomorphic gauges on these lines should be consistent. One can think of a trivial holomorphic bundle on Px as having a distinguished connection or parallel transport. Then, going round the triangle formed by three coplanar lines, parallel transport may not return to the identity. Thus we get a version of curvature for E purely out of the holomorphic structure and the compactness of the projective lines. This « curvature » is a global notion, associated with each general triangle, but we can obviously infinitesimalize it to derive something more like the differential-geometric curvature.
From this point of view it is now natural to introduce the space which parametrizes all lines in P3(0). As explained in Chapter III this space is the Klein quadric Qi in P6(0). The «general» lines of P3, i.e. lines for which E is trivial, correspond to an open set U of Ql. We can now construct a holomorphic vector bundle 8 over U by defining the fibre ZL to be the space of holomorphic sections of E\l, where Le U represents the line I cP3. All the lines I through a point A correspond to points L of a plane a in Qi and the vector spaces SL for I e a n U can all be naturally identified with EA. This gives 8 a fiat connection along all a-planes through L. These a-planes generate the tangent cone to Q4 at L and so we get components for a connection along all these directions in the tangent cone. But any holomorphic function on this cone which is homogeneous of degree one on each generating line automatically extends (uniquely) to a linear function on the whole tangent space. This is an elementary geometric property of quadric cones (in any number of dimensions); cf. Chapter VI, §3. Hence 8 has a holomorphic connection, flat along the a-planes. On the other hand on the other system of planes on Q4, corresponding to planes of P3, the connection need not be flat. This corresponds to the other notion of curvature on P3 described above. A connection for a bundle on Qi which is flat along a-planes precisely corresponds to anti-self-duality for the curvature. This is essentially Ward's approach, passing from holomorphic bundles on P3 to holomorphic bundles with holomorphic anti-self-dual connection on an open set of Q4.
If we now want to consider unitary structures and work on 84 c we would impose appropriate conjugations throughout. The only drawback of this approach is that, in order to pass from 84 to a neighbourhood in Q4 and apply the above argument, we have to assume that our bundle on 84 is real-analytic. Our treatment in Section 2, based on the Kewlander- Nirenberg integrability theorem, had the advantage of requiring only differentiability. Analyticity, and eventually rationality of the solutions, is then an automatic consequence.
As mentioned in Section 2 (and will be proved in Chapter VI) a liolo- morpliic bundle E on PS(G) wliicli is trivial on a given line I is also trivial on the first neighbourhood of I. Thus a basis of E along I can be extended formally up to first derivatives normal to I. This extension essentially is just the connection on 8 at the point L defined by Ward's approach. In fact the property of quadric cones used in Ward's construction is closely related to the extension property of E: both can be formulated in terms of the vanishing of a certain sheaf cohomology group (cf. Chapter VI, §3).