1. — Cohomology of the Horrocks construction.

In Chapter Y we gave the construction of Horrocks for producing sym- plectic vector bundles on P3(G). In this chapter we shall prove a theorem due to Barth [8] which shows how the Horrocks bundles may be character­ized in cohomological terms. Combined with the results of the preceding chapter on the vanishing of appropriate sheaf cohomology groups for in­stanton bundles this will finally prove that the Horrocks construction gives all instanton bundles.

We begin in this section by examining the Horrocks construction in greater detail and deducing its cohomological properties. At this stage, and also in the next section, we work purely over the complex numbers. Beality questions do not enter until later.

Throughout this chapter we shall be making extensive use of the ma­chinery and standard results of sheaf cohomology which have only been lightly touched on in previous chapters. Inevitably therefore this chapter will be more technical, but we shall try to recall basic facts as and when they are needed. In any case, it is the arguments of this chapter which exhibit the full power of cohomology theory and the reader may find it instructive to try, where possible, to reinterpret the methods in real 4-space terms. We shall at the appropriate stage make some comments in this direction.

We recall that the Horrocks construction for 8p(n, <7)-bundles, as ex­plained in Chapter Y, Section 2, starts from a linear map

image Uz = A(z)W c V is assumed fc-dimensional and isotropic, i.e. Vz is contained in its polar space J7°. We then put Ez = EaJUz which gives a vector space of dimension 2n with an inherited non-degenerate skew-form. This is the fibre over (z)eP3(C) of the required bundle E.

Since A(z) is linear in 2 we can view (1.1) as a homomorphism of vector bundles over P3(C):

 

 

where W(—1) = ~WC?)L. In other words the bundle U c V is isomorphic to W(— 1). By duality VjU0 is then isomorphic to W*(l). We can exhibit all bundles relevant to the construction in the following display of exact sequences:

 

 

where Q = V/U, Q*= U°. This diagram is self-dual, and the duality is skew in an appropriate sense, being induced by that of V.

In dealing with sheaf cohomology groups on P3(C) we shall, where no confusion can arise, omit the symbol P3(G) and simply write Ha(S) for the cohomology of a sheaf S. The cohomology of the bundles E, Q in (1.3) is then given by

Proposition (1.4).

 

 

Peoof. The middle row of (1.3), together with the cohomology properties of W(— 1) and V give at once the results on Q Jwe use here the vanishing of H«(0(n)) for q = 1, 2 and all n, for q = 0 and n < oj. Using this informa­tion on Q in the last column we then deduce the results on E.

As noted in Chapter V the second Chern class c«(E) can be read off from (1.3) and we get c2(E) = 1c. More formally if x is the standard gen­erator of II"{I\, Z) the total Chern polynomials are given by:

 

 

showing that = Tc.

Returning now to (1.3) let us use the last column to investigate H1(E(n)) for w>— 1. We see that

 

 

 

is surjective. Using the identification of W* given by (1.4) this can be replaced by the surjectivity of

 

 

 

Thus, if we introduce the graded module

 

 

 

over the ring of polynomials in zlt..., zt, we see that M has the properties:

 

 

The properties (1.5) of the module M reflect the simple nature of the Hor­rocks construction. There are in fact vector bundles E given by more com­plicated constructions, in which A is not assumed linear in zx, ..., for which the associated module M is more complicated. This is part of the general theory of Horrocks but fortunately for our purposes the simple case leading to (1.5) is sufficient.theorems on algebraic bundles