Gauge theory for embedded surfaces, II

64 P. B. Kronheimer and T. S. Mrowka
dim Θ = g − 1, and the extension class gives a point in P(H 1 (L 2 )), which has
dimension 2g−3, by an application of Riemann Roch. This gives a total of 3g−4,
so the bundles E for which (b) fails are a set of complex codimension at least
one in the moduli space.
Finally consider (c). We may assume that (a) and(b) both hold. In particular,
under assumption (b), the inclusion of L in E generates H 0 (L −1 ⊗E), and
(c) is then equivalent to the surjectivity of the map
H 1 (O) → H 1 (L −1 ⊗E)
appearing in the cohomology sequence of the extension sequence O→L −1 ⊗E →
L −2 . This surjectivity is in turn equivalent to the vanishing of H 1 (L −2 ), which
again says that L −2 does not lie in the theta divisor. Thus (c) follows from the
same parameter count as before. ⊓⊔
Proof of Proposition 9.15. Fix an E satisfying the conditions of the lemma; we
will show that S(E) has2 g elements. As in the proof of Proposition 9.1, consider
the family of bundles L −1 ⊗E as L runs over the Jacobian torus J−l, and let σ be
the index of the family of operators ¯ ∂ L −1 ⊗E, asanelementoftheKtheory of the
base. When L is not in S(E), we have h 0 (L −1 ⊗E)=0andh 1 (L −1 ⊗E)=g−1;
by condition (b) of the lemma, both h 0 and h 1 jump by one at each point of
S(E), so at these points h 1 = g. In the proof of (9.1) we used the fact that, if S(E)
is empty, then cg(−σ) = 0. Each point of S(E) makes a positive contribution to
cg, and this contribution is equal to 1 provided that the family of ¯ ∂ operators
is transverse to the set, Sg, of operators of corank g in the space of Fredholm
operators of index −(g − 1). Generally, if ϕ ∈ Sg, then the normal bundle to Sg
at ϕ is Hom(ker ϕ, cokerϕ), so in the case at hand, the transversality condition
is exactly what is expressed in condition (c) of the lemma: the space H 1 (O)
is the tangent space to the Jacobian. Thus under the conditions of the lemma,
cg(−σ) is equal to the number of points in S(E). On the other hand, in the proof
of (9.1), we calculated that cg(−σ)=2 g . ⊓⊔
Appendix 1: Topology of the configuration space and
orientability of the moduli space
In this appendix we give a more detailed discussion of the topology of the configuration
space Bk,l. There are two issues of interest to us here. The first is the
orientability of the moduli space. In outline the argument is the same as in the
untwisted case; see [D3]. We determine the fundamental group of the configuration
space and show directly that the determinant line bundle of the deformation
complex is trivial on an explicit set of generators.
For a proper discussion of orientation of the moduli space we need to work
with SO(3) bundles, or more precisely with U(2) bundles with fixed connection
on their determinant line. For this section let E → X be a C 2 bundle with
determinant line bundle det(E) =c.OverΣwe fix a decomposition of E as