Gauge theory for embedded surfaces, II

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16 P. B. Kronheimer and T. S. Mrowka Corollary 3.9. Suppose that κ ′ ≤ κ and let R⊂R(Y)be any component. If the inequality 2g − 2 > 1 2n +1 holds then the formal dimension of the moduli space Mκ ′(X o ; Rs ) is not greater than the dimension of the ordinary moduli space Mκ(X) for the compact manifold: dim Mκ ′(X o ; R s ) ≤ dim Mκ(X). Equality holds if and only if κ ′ = κ and R is either R+ or R−. ⊓⊔ Remark. In the case that R = R−, equality between the actions κ ′ and κ is not possible unless n is divisible by 4, because of the Chern-Simons formula (3.4)(b) and the fact that κ needs to be an integer. The corollary above will be useful because it provides a condition under which we will be able to exclude the moduli spaces M(X o ; Rm) from our considerations, on the grounds of dimension. Note that the corollary, like our main theorem (1.1), involves an inequality between the genus and self-intersection number; but the inequality here is weaker, roughly by a factor of two for large n. (iii) Transversality and compactness. The familiar transversality results have their analogues for the moduli spaces in the cylindrical-end setting (see [MMR]). (We now drop the condition that n be positive and consider positive and negative self-intersection together.) Proposition 3.10. For a generic compactly-supported perturbation of the metric g o , the moduli space M(X o ; R s ) is regular, in that the map e2 in the deformation complex (3.5) is surjective, at all irreducible connections. Furthermore, if b + (X) > 1, orifb + (X)>0and n is negative, then generically there are no reducible connections in the moduli spaces M(X o ; R), except perhaps for flat ones. ⊓⊔ Proposition 3.11. Let Si be a countable collection of manifolds with smooth maps si : Si →R s . Then for a generic compactly supported perturbation of the metric g o as in (3.10), the map rX : M(X o ; R s ) →R s will be transverse to all si. ⊓⊔ So far, we have said little about the inverse image of the singular strata of R(Y ) in the moduli space M(X o ). The story here is rather more complicated than the smooth case, but fortunately, all we need from the theory is a condition which allows us to conclude that these singular strata need not enter our calculations later. At first glance, in line with Proposition 3.11 above, one might suppose that M(X o ; R+), for example, would contain no connections asymptotic to the singular strata provided only that the dimension of the moduli space was strictly less than the codimension of the strata. (This codimension is 4g−6, since
Gauge theory for embedded surfaces, II 17 R+ has dimension 6g −6 while the stratum of connections reducible to U(1) has the same dimension as the Jacobian.) This first guess is not correct, since the local model of the part of the moduli space which lies over the singular strata is more subtle and (depending on the sign of n) the formal dimension of the fibres of rX may not be the same from one stratum to the next. The picture is correct, however, when n is positive: the following result is proved in [MMR] and [T]. Proposition 3.12. Let R be either R+ or, in the case that n is even, R−, and suppose n is positive. Then, for a generic compactly-supported perturbation of the metric g o , the moduli space Mκ(X o ; R s ) is the whole of Mκ(X o ; R) ∩ {irreducibles} when the formal dimension of Mκ(X o ; R s ) is less than 4g −6. ⊓⊔ Corollary 3.13. If the formal dimension of Mκ(X o ; R s ) is zero or less, the genus of Σ is at least 2 and n is positive, then for a generic metric the moduli space Mκ(X o ; R) contains no connections asymptotic to the singular strata, except perhaps for flat or reducible connections. ⊓⊔ If {ui} is a collection of compact surfaces in X\Σ then the cylindrical-end moduli spaces can, as usual, be cut down by the corresponding 2-dimensional constraints via the restriction maps. That is, we can form a cut-down moduli space M(X o ) ∩ V1 ∩ ...∩Ve. The usual transversality arguments can be applied to the construction of the Vi, and the above results extend. In particular, Corollary 3.13 is valid as stated if one replaces the moduli space Mκ(X o ; R) with a cut-down moduli space. Partnering the transversality results are compactness results for the moduli spaces. The proof of the following proposition follows the usual argument without change: Proposition 3.14. Let Am be a sequence of connections in the moduli space Mκ(X o ). Then there exists a subsequence Am ′, a finite collection of points xi in X o ,andaconnectionA∈Mκ ′(Xo )such that [Am ′] converges to [A] on compact subsets of X o \{xi}. The action densities |F (Am ′)|2 converge as measures on compact subsets of Xo to |F (A)| 2 + kiδ(xi), where δ(xi) is the delta function at xi and ki are positive integers. ⊓⊔ This proposition is not the end of the story, since some action may be lost on the end of the manifold in the limit; that is, the non-negative quantity γ defined by κ(Am)=κ(A)+ ki +γ i
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