2. — Theorem of Barth.

We shall now give Barth's theorem [8] which gives sufficient criteria on a symplectic bundle E in order that it should arise from the Horrocks con­struction. We consider first the following assumption on E:

(2.1) For some line I in P3, the restriction of E to I is trivial.

This assumption is related to the notion of semi-stability: for example when E is 2-dimensional (2.1) is equivalent to semi-stability and to the vanishing of H°(E(— 1)) [7]. Note that (2.1) implies automatically that E is trivial on the «general» line of P3 (see Chapter IV). For bundles coming from .S'4, via the basic construction of Chapter IV, (2.1) is certainly satisfied since E is trivial on all real lines, i.e. the fibres of P3(C) £4.

Our second assumption is that E satisfies the vanishing theorem:

 

 

which is satisfied by bundles coming from the Horrocks construction (see (1.4)), and also for bundles corresponding to anti-self-dual Sp(n)- connections on $4 by the results of Chapter VI.

Barth's theorem asserts that (2.1) and (2.2) are sufficient, namely

(2.3) Theorem. Let E be a symplectic vector bundle on P3{C) satisfying (2.1) and (2.2). Then E arises by the Horrocks construction from a linear map

 

unique up to isomorphism.

 

The uniqueness in (2.3) means that, if (A, W, V) and (A', W', V') give isomorphic symplectic bundles, there are isomorphisms W->W, F-> V (preserving skew-forms) taking A into A'.

The idea of the proof of Theorem (2.3) is to show that the display (1.3) can be canonically reconstructed from E alone. As a preliminary we shall first show that the module M=® H1(E(n)) satisfies the conditions (1.5),

a necessary consequence of (1.3).

Take any plane P2 containing the line I, then E is trivial on the general line in this P2 and so, for n< 0, E(n) can have no non-zero section over this P2. Now consider the standard exact sequence

 

 

Tensoring with 0(n) and taking cohomology gives the long exact sequence

 

 

 

Since E is trivial on I, II1 (I, E(n)) = 0 for n > — 1, and by Serre duality H2{E{n— 2)) is dual to H1(E(— n— 2)) and so vanishes for n>0 (as proved above): note that we have used E ^.E* here. Hence (2.5) asserts that the map

 

Taking n = — 2, — 3, ... in turn and using the vanishing of H"(P2, E(nj) we deduce inductively that H1(P3, E(n)) = 0 for all «<-- 2, which is the first part of (1.5). To prove the last part we take coordinates in P3 so that I is given by zx = % = 0. Then we have the exact sequence, resolving 0,

 

 

where x = (z2, — zx) and f) — (z1, z2). The image of [i is the ideal sheaf J of the line I. Tensoring (2.4) with E(n), and taking cohomology we deduce the exact sequences

is surjective for all m> 0. This implies that M x generates M as a module (and is a somewhat stronger statement since only two of the variables are required). Finally to check the dimension of M_x = IIl(E(— 1)) we note that all the other cohomology groups H"(E(— 1)), q # 1, vanish. For q = 2 this follows from Serre duality and the vanishing of Jf_3, for q = 0 it follows from the triviality of E on general lines and for q = 3 we apply Serre duality and reduce to the same argument. Thus dim Jf-j can be computed from the Biemann-Roch theorem which evaluates

 

 

in terms of the Chern classes. Since ct(E) = 0, c2(E) = 1c we get dim = = a + where a, b are constants independent of E. These can either be obtained from the detailed Riemann-Roch formula or more simply we take

explicit examples arising from — ft-instantons. Either way we find dim Jf_!= ft completing the verification of (1.5).

 

while for m>0 we get

 

We turn now to the construction of the display (1.3) and we begin with the last column. Defining W* = E(— 1)) and using the interpretation of E1 in terms of extensions (cf. Chapter VI) we construct the extension

 

 

 

corresponding to the identity element of

 

 

The effect of this is that, in the cohomology sequence of (2.6) tensored with 0(— 1)

 

 

the coboundary d is the identity. Since H^W*) = 0 it follows that if1(Q(— 1)) = 0. If we tensor (2.6) in general with 0{n) then for w<— 2, we deduce

In (2.7) 8 can be identified with the module multiplication from M^ to Mn, and the surjectivity just proved (i.e. (1.5)) tells us that 2?1(Q(w)) = 0. Thus we have proved

 

 

We have now constructed the last column of (1.3) and established the key property (2.8) of the bundle Q. Dualizing we get the first row of (1.3). The cohomology sequence of this first row tells us that

 

 

Interpreted in terms of extensions (with n = — 1) this shows that there is an extension of Q* by W*(l) compatible with each such extension of E. In particular this gives us the middle column of (1.3), for a suitable vector bundle V.From the middle row of (1.3), using (2.8), we deduce

 

 

 

By Serre duality (2.8) implies that H2(Q*(n)) = 0 for all n, and hence from the middle column of (1.3) we get

 

 

From (2.9) and (2.10), we can now deduce that F is a trivial bundle. We require the following special case of the general theory of Horrocks [28].

Proposition (2.11). Let V be a vector bundle over P3 such that Hx(V(n)) = = H2(V(n)) = 0 for all n. Then V is isomorphic to a direct sum of line- bundles.

Proof. Fix P1cl\cP3 and let

 

 

where K ^ Z">©... © L"m is a sum of line-bundles. It will be enough to shows that the isomorphism <p of (2.12) extends to P3 as a homomorphism, because the points where <p is not an isomorphism form an algebraic surface (local equation det <p = 0) not meeting I\ and so necessarily vacuous. We show that <p extends first to P2 then to P3 by using the exact sequences

 

 

The second sequence and the vanishing of H1(V(n)), H2(V(n)) shows that H1(Pi, F(ra)) = 0 and that every section of V(n) over P2 extends to P3. The vanishing of H1(P2, F(«)) = 0 applied to the first sequence shows similarly that every section of V{n) over I\ extends to P2. Since <p (or rather <pis a direct sum of sections V{%i), the proposition follows.

We now apply (2.11) to the bundle F constructed above. To prove finally that the integers n( are all zero, i.e. that F is trivial, it is enough to restrict F to one line I. But when (1.3) is restricted to I, since E\l is trivial, all the extensions split and it follows that V\l is trivial.

We have now reconstructed the display (1.3) from E. It remains only to show that it is skew-symmetric and in particular that F has a canonical skew-form. But dualizing (1.3) and using the skew duality E ^ E* we get an isomorphic diagram, showing that F^F*. With a little more care

(cf. [8]) one checks that this isomorphism is skew, completing the proof of Barth's theorem. The uniqueness of the display (1.3) follows from the canonical nature of our reconstruction, in which no arbitrary choices were involved.

It is clear from our proof that exactly the same argument works for an orthogonal bundle, i.e. a vector bundle E with a non-degenerate quadratic form. The duality of the corresponding diagram to (1.3) is now symmetric.

In conclusion we should point out that a more general theorem than (2.3) is given in [17] (and has been further generalized in [9]) which gives neces­sary and sufficient conditions for a bundle to arise from the Horrocks con­struction. Condition (2.2) is still kept but (2.1) is relaxed. In terms of the module M, conditions (1.5) are still fulfilled but it is not assumed that 2 of the 4 variables (zx, ..., zt) are enough to generate M from M^. For in- stanton bundles condition (2.1) is satisfied and therefore the extra generality of [17] is not needed.