Gauge theory for embedded surfaces, II

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$Math 126 Solution Set 2$

$Math S?101. Worksheet 1. Axioms of Integer Arithmetic$

40 P. B. Kronheimer and T. S. Mrowka A technical device of a slightly different sort is our use of cone-like orbifold metrics throughout. The exposition could have been streamlined somewhat if Conjecture 8.2 from [KrM] had been proved in complete generality, rather than the untidy approximation (2.12). Recall that in our arguments above for the proof of the key Proposition 5.1, we first needed to choose a suitably small value of α, smaller than a quantity α0 determined by n and g alone, and then ν had to be chosen so that the conclusion of (2.12) held for some interval I containing α. (iv) Other remarks. We gather together here a few remarks and questions related to Theorem 1.1. First of all, for an oriented 4-manifold X with non-zero invariants qk, the main theorem gives a lower bound for the genus of an embedded surface in terms of the self-intersection number; but this lower bound is trivial if the self-intersection number is negative. This contrasts, for example, with the lower bound obtained in [HS], which is independent of the sign. The point here is that the hypothesis on qk is sensitive to orientation; at present, no 4-manifold is known for which the polynomial invariants are non-zero for both orientations. To put some flesh on this remark, consider for X a K3 surface. As was mentioned in [KrM], the lower bound provided by Theorem 1.1 is sharp in this case, for all classes of positive square: Proposition 6.6. If ξ is a class of positive square in a K3 surface, then the smallest possible genus of an embedded surface representing ξ is given by 2g−2= ξ·ξ. If the class ξ has negative square however, it may well be represented by a sphere. The usual description of the Kummer surface exhibits 16 disjoint spheres of square −2 inK3; by joining any number of these together by piping, we obtain embedded spheres of square −4, −6,andsoondownto−32. With a little more ingenuity (as was pointed out to the authors by Ruberman), it is possible to go quite a bit further down, but it is not clear whether there is eventually a lower bound. Up to the action of the diffeomorphism group, there is in K3 a unique primitive class of square −2n, for every n, and we do not know the smallest genus surface which can represent this class. So we ask: Question 6.7. Can every primitive class in K3 with negative self-intersection be represented by an embedded sphere? The phrasing of this question is not meant to imply that the authors conjecture that the answer is ‘yes’. Some results about immersed (rather than embedded) surfaces are easily deduced from Theorem 1.1. For example, let Σ now be an immersed sphere in a 4-manifold X, having d+ double points of positive sign and d− double points of
Gauge theory for embedded surfaces, II 41 negative sign. Let n = Σ·Σ be the homological self-intersection number of the sphere. Suppose that X satisfies the hypotheses of Theorem 1.1 and that, as in that theorem, Σ is essential and n is not −1. Then we have: Proposition 6.8. The number of positive double points in the sphere Σ is bounded below by the inequality 2d+ − 2 ≥ n. Proof. As a first attempt, we might smooth out each intersection point, increasing the genus by one each time. Applying the result of Theorem 1.1 to the smoothedout surface would then give 2(d+ + d−) − 2 ≥ n. However, we can obtain the stronger inequality in the proposition by the following device (pointed out to the authors by Gompf): we blow up the manifold X at each one of the negative double points, and take the proper transform ˜ Σ in the new manifold ˜ X.When the sign of an intersection point is negative, this procedure does not affect the self-intersection number n (see [G] for example); and since ˜ X continues to satisfy the conditions of Theorem 1.1, the inequality in the proposition then follows by smoothing the positive double points. ⊓⊔ It has been pointed out to the authors by Gompf that, in the case of K3, this result is sharp whenever n is positive, in the same spirit as Proposition 6.6. 7. Closing up the ruled surface (i) Introduction. In this section we begin the calculation of the invariant p0,l. Recall that this integer can be interpreted as the degree of the natural map rl : M α 0,l (W o ; Rs +) → Rs + , or rather as the degree as the extension of rl to the natural compactification, for l = 1 4 (2g − 2) and small α. This quantity depends only on the genus g and self-intersection number n (and possibly – since we did not rule this out – on ν and the geometry of the end). A natural strategy is to exploit the excision principle encapsulated in Theorem 5.10: if we can calculate non-trivial values of qk and qk,l in just one special case of an acceptable pair (X,Σ), then we can calculate p0,l, for the corresponding values of g and n, by looking at the ratio of the two invariants. Of course, this approach as it stands is not going to move us forward. We are interested in p0,l when (2g − 2)
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Gauge theory for embedded surfaces, II

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