3. — Reality constraints.

We come now to the question of imposing the necessary reality constraints on the Horrocks construction to produce instanton bundles. In Chapter V, § 2 we imposed a Galois action a on the triple (A, V, W), with a2 = — 1 on V and a2 = 1 on W, which enabled us to obtain an $p(w)-bundle over Sl with anti-self-dual connection. We want now to show the converse, i.e. that every such connection arises in this way.

According to Chapter IY, §2 every such connection corresponds to a holomorphic vector bundle E over P3{C) with

a holomorphic symplectic structure

an anti-linear a, compatible with (i), with a on P3(0), satisfying (T2 = — 1, and inducing a positive hermitian form on sections on E along all real lines of P3(0).

Since E is trivial on all real lines of P3(0) and satisfies the vanishing con­dition H1(E(— 2)) = 0 (Chapter VI, § 3) we can apply Barth's theorem of the previous section and deduce that E is constructed canonically from a triple (A, V, W). The anti-linear map a then induces an antilinear map on V. This follows from the uniqueness part of Barth's theorem: we compare E and the complex conjugate of a*(E). The uniqueness also shows that, on F, cr2 = — 1 and is compatible with the symplectic form.

To derive all the conditions imposed in Chapter V, § 2 we need finally to show that the hermitian form induced by a on V is positive. Now recall

that, for any (z)eP3(C) we have an orthogonal decomposition:

 

 

where UgZ= U°z = UZ@EZ. The hermitian form restricted to Uz is there­fore definite. Applying a shows that the sign of this definite form is the same in TJoz as in Uz. The form is positive on Ez because of its original definition. Hence the form is either positive on all F or else it has signature (2n, 21c). But in the latter case the fibres Ez would lie in the positive cone and the bundle would be topologically trivial (deformable to a fixed positive subspace), contradicting the fact that

Since F is a trivial bundle H°(V)^V, E\V) = 0. From the top row we have

This completes the proof that the construction of Chapter Y, made explicit in Chapter II, gives all anti-instantons for 8p(n). Exactly analogous arguments work for the orthogonal group. Finally for the unitary group U(n) we can embed this in 80(2n) and consider an additional anti-linear J. Again using the uniqueness part of Barth's theorem we deduce that we get an operator J on F with the appropriate properties.