4. — The Drinfeld-Manin description.

In the proof of Barth's theorem in § 2 we reconstructed the display (1.3) from the symplectic vector bundle E over P3. The vector space W* was identified with the cohomology group H1(E(— 1)). The vector space F can also be identified as a cohomology group in such a way that the linear map A (or its dual .A*) acquires a simple cohomological interpretation. This enables one to reformulate Barth's theorem rather more elegantly by directly exhibit­ing the triple {A, F, W) in terms of the cohomology of E. This reformulation has the advantage of making the additional reality conditions more transparent.

We shall explain how to interpret F cohomologically. For fuller details and a more systematic account see [18].

From the middle column of (1.3) we get the cohomology sequence:

 

 

 

 

 

 

 

Substituting these into (4.1) we get

 

 

and finally we recall that

where u can be identified with the natural module multiplication of the module M.

where T is the tangent bundle. Dualizing and tensoring with E{— 1) gives

 

The cohomology sequence of this is

 

On the other hand we have a natural exact sequence on P3:

 

 

where u is the natural multiplication (recall that 2T°(0(1)) is the space of linear forms (04)*).

Comparing (4.2) with (4.3) suggests that there should be a natural isomorphism

 

 

This is clear if H"(E) = 0 which is the case for instanton bundles not con­taining a trivial summand (e.g. for $p(l)-bundles and k =^0). In the gen­eral case V splits 8)8 cl direct sum

 

 

using the skew-form on V. Now H1(E(x) D1) has a natural skew-form given by the cup-product (and the skew-form on E):

(4.6)    H1{E®Q1)®H1(E®Q1)-+H*(Q*)gÉC

and so it decomposes analogously to (4.5). This gives us the isomorphism (4.4) and one must then check directly that this isomorphism is compatible with the skew-forms.

Thus the linear map A* associated to E can be identified with the map

 

 

occurring in (4.3).

Any additional structure on E such as a or J then passes naturally to these cohomology groups. Note that in the orthogonal case the multipli­cation (4.6) is symmetric.

Since H2(P3, Q2) maps isomorphically onto 7T2(P2, Q1) it follows that (4.6) factors through the restriction to P2 where it coincides with Serre duality. The non-degeneracy of (4.6) therefore implies the injectivity of

 

 

Concerning restriction from P3 to P2 we draw attention to the following. Let P2 and <r(P2) be considered as a degenerate quadric with real structure. In general for any real quadric Q c P3 two real bundles E, F on P3, cor­responding to anti-instanton $p(w)-bundles on are isomorphic if and only if their restrictions to Q are isomorphic. To see this we have to show that any isomorphism <p: E -> F over Q extends as a homomorphism over P3 (it is then necessarily an isomorphism by an argument used earlier). Now G = Horn (E, F) again corresponds to an anti-instanton bundle and so satisfies the vanishing condition H1(P3, G(— 2)) = 0. Applied to the exact sequence

 

 

this shows precisely that every section of G over Q extends to a section over P3. Taking Q = P2 U <r(P2) we deduce that E ^F over P3 (as real bundles i.e. commuting with a) if and only if E ^ F over P2 and the iso­morphism respects a over the real line Pj = P2 n <r(P2). Since E, F are trivial over real lines the real structure over Px corresponds to a real struc­ture or hermitian metric on 02n. Thus we deduce that the map from the space of moduli of real 8p(n) anti-instantons to the space of moduli of complex symplectic bundles over P2(C) has fibre the symmetric space 8p(n, C)/8p(n). For example when n = 1 this fibre is the hyperbolic 3-space. The (8fc— 3)-manifold of 8U(2) — fc-instantons therefore fibres over an 8fc—6 = 2(47c — 3)-manifold of bundles on P2(C). But a dimension count (cf. [3]) shows that (47c — 3) is the complex dimension of the relevant space of (stable) bundles on P2(0). Hence, after factoring out by the hyperbolic 3-space, we get a complex structure on the real moduli space.

This result can be interpreted roughly as follows. Starting in Til we pick coordinates to identify it with (J2. Then an anti-self-dual connection gives a holomorphic bundle on (J2. If we require the bundle to extend to $4 our holomorphic bundle extends to Pa(C) and acquires a real structure at oo. The surprising result is that this data uniquely determines the original solution: in other words it is enough to fix one set of complex coordinates if we work globally, whereas the main local result of Chapter IY was that we needed all complex coordinates to interpret the anti-self-duality equations. It would be very interesting to have a direct differential-geometric proof of this result. For this it would be necessary to prove the existence of a unique hermitian metric for the holomorphic bundle which satisfied ap­propriate conditions.