Gauge theory for embedded surfaces, II

Gauge theory for embedded surfaces, II 7
omit mention of the restriction map, and write, for example, M α k,l ∩ Vi in place
of { [A] ∈ M α k,l | ri([A]) ∈ Vi }.
The section si can be chosen so that it is transverse to ri on each M α k ′ ,l ′,
or in less careful language, the intersections M α k ′ ,l ′ ∩ Vi are all transverse. Each
such intersection is therefore a smooth codimension-2 submanifold of the corresponding
moduli space. We can further arrange transversality for all multiple
intersections
M α k ′ ,l ′ ∩ Vi1 ∩ ...∩Vic, (2.2)
so that each is a smooth manifold ([DK], Proposition 5.2.12). The dimension
of (2.2) is 2e(k ′ ,l ′ )−2c. In Appendix 1 we will show the moduli spaces are
orientable and that the choice of homology orientation fixes an orientation of
each one. Since the normal bundle to each Vi is oriented, the intersection (2.2)
is also naturally oriented. In particular, the intersection
M α k,l ∩ V1 ∩ ...∩Ve, (2.3)
whose dimension is zero, is a collection of signed points.
At this point we would like to be able to assert that, under the assumptions
already in place, the intersection (2.3) must be compact. However, due to the
limitations of the main compactness theorem proved in [KrM], we need first to
put an additional restriction on ν: wemustmakeνlarge, depending on α; this
is explained in the next subsection.
Taking the compactness as given for the moment, we can define qk,l by the
familiar process of counting the points in (2.3), making the count algebraically,
taking notice of the signs:
(ii) Compactness.
qk,l(u1,...,ue)=#(M α k,l ∩ V1 ∩ ...∩Ve). (2.4)
Let [An] be a sequence without a cluster point in the moduli space M α k,l .By
Proposition 7.1 of [KrM], after passing to a subsequence, we can assume that
[An] converges off a finite set to some limit [A ′ ] ∈ M α k ′ ,l ′. The difference between
(k,l) and(k ′ ,l ′ ) is accounted for by what ‘bubbles off’ at the points where
convergence fails (the points where curvature concentrates). Thus, with points
of concentration xj in X\Σ, there is an associated positive integer kj, andfor
points of concentration xi in Σ there is an associated pair (ki,li)sothat
k ′ =k− ki− kj
l ′ =l− li.
(2.5)
In general, we do not have results which give control on the possible values
of the pairs (ki,li), but for large ν we are in a better situation: according to
Proposition 8.3 of [KrM], for each ε>0 there is a ν such that, provided α is in
− ε ], the pairs satisfy the inequalities
the interval Iε =[ε, 1
2
ki +2εli ≥ 0 and ki+(1−2ε)li≥0. (2.6)