Gauge theory for embedded surfaces, II

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$Math 126 Solution Set 2$

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

70 P. B. Kronheimer and T. S. Mrowka Now suppose we have bundles E and E ′ with corresponding reductions along Σ, sayE|Σ=(C⊕L0)⊗L1 and E ′ |Σ =(C⊕L ′ 0 )⊗L′ 1 .Ifc2(E)−1/4c 2 1 (E)= c2(E ′ )−1/4c 2 1(E ′ ), c1(E)=c1(E ′ ) (mod 2), and L0 =L ′ 0 then the corresponding moduli space are identified. In this case an excision argument and comparison with the complex orientation on a Kähler surface, shows that the orientations compare just as in the closed case by the sign (−1) (c1(E)−c1(E′ ))/2) 2 . In the case of SU(2) bundles, the line bundle c is trivial, and we take the procedure above as defining the standard orientations of the moduli spaces M α k,l (X,Σ), to be used in the definition of the invariants qk,l. There are two other comparisons which one can make for the orientations in the twisted case. First, an orientation of Σ was used in the above constructions, which is not otherwise needed in the definition of α, k and l. One can show that changing the orientation of Σ changes the canonical orientation of the SU(2) moduli spaces by the sign (−1) (g−1) . The second comparison is more subtle. Recall from [KrM], section 2(iv), that the the SO(3) moduli space M α k,l,w is identified with M α′ k ′ ,l ′ ,w ′, where the “flipped” parameters are given by α′ = 1 2 − α, k′ = k + l − 1 4Σ·Σ, l′ = 1 2Σ·Σ − l and w′ = w +[Σ] mod 2. In the special case that [Σ] = 0 mod 2, these two moduli spaces are related by instanton and monopole additions and deformation of the parameter α. In this sense then, one can ask whether the flip identification is orientation-preserving. Based on the above considerations, one may show that the flip preserves or reverses orientation according to the parity of the quantity 1 4Σ·Σ − (g − 1). (A1.5) (An elementary calculation shows that this quantity is always even in the case of a complex pair. This is as it should be, since the flip preserves the natural complex orientation.) In the proof of Theorem 1.1, it is important to be able to keep track of orientations through the course of the excision argument in section 5; this involves defining orientations for moduli spaces associated with manifolds with cylindrical ends, in such a way that the gluing construction respects these orientations. The presence of twisting on the surface Σ makes no difference to these constructions, and we therefore refer to [MMR] or [GM] for examples of how this is done. (iii) Factoring the µ map. We have seen that the moduli spaces of twisted connections define invariants qk,l of the pair (X,Σ). From their definition, the qk,l are multilinear functions on H2(X o ), but it is asserted in Theorem [B4] that these factor through H2(X). The kernel of the map from H2(X o )toH 2 (X) comes from H1(Σ) in the long exact sequence of the pair (X,X o ). The following proposition is therefore what is needed to establish the assertion about the invariants. Proposition A1.6. Let (X,Σ) be an admissible pair. Fix s1,...,sd ∈ H2(X o ) such that s1 ∈ Im(δ : H3(X,X o ) → H2(X o )). Then qk,l(s1,...,sd)=0.
Gauge theory for embedded surfaces, II 71 Proof. Webeginwithalemma. Lemma A1.7. For all s ∈ Im(δ : H3(X,X o ) → H2(X o )) we have µ(s) =0in H 2 (Bk,l). Proof. Fix a compact surface S ⊂ X o representing s. Consider the universal SO(3) bundle U = P ×A∗ k,l /Gk,l → X ×Bk,l. Just as in the untwisted case we have µ(s) =−1 4p1(U)/s ∈ H2 (B∗ k,l ). But s maps to 0 in H2(X) and hence µ(s)=0. ⊓⊔ Fix compact embedded submanifolds S1,...,Sd representing the classes s1,...,sd.WechooseV1,V2,...,Vd to be generic divisors on Mk,l,α representing the restriction of µ(s1),...,µ(s2)toMk,l,α, so that compactness theorem implies that the intersection I = V2 ∩···∩Vd is a compact smooth oriented two manifold. Hence the signed number of points in V1 ∩ V2 ∩···∩Vd is equal to the evaluation µ(s1), [I] , as an honest pairing, and the result follows from the lemma. ⊓⊔ Appendix 2: Cone-like singularities in Kähler metrics Let X be a smooth, compact Kähler surface containing a smooth curve Σ. Let ω0 be any smooth Kähler metric on X. We wish to modify the metric ω0, by adding a small term supported near Σ, so that the new metric ω has a cone-like singularity along Σ, with cone-angle 2π/ν; we would like ω to be of orbifold type, so that it is smooth and non-degenerate when pulled back to the local ν-fold complex branched covering spaces. Denote the line-bundle O[Σ] byL, and let e be a section of L vanishing along Σ. Choose a hermitian metric h on L, and let ρ = h(e, e) 1 2 be the lenght of e. Fix ν ∈ N, and let ϕ = r 2/ν . The function ϕ is smooth on X\Σ, and continuous on X. Lemma A2.1. The (1, 1)-form (1/2i) ¯ ∂∂ϕ on X\Σ is bounded below; that is, comparing them as hermitian forms, we have 1 ¯∂∂ϕ ≥−Kω0 2i for some K. Proof. We calculate 1 2i ¯ ∂∂ϕ = 1 2i = r 2/ν 2 ν 2 ν r 2/ν ¯ ∂(r −1 ∂r)+ 1 1 1 F + i 2i 2 ν 2i 2 2 ν ¯α∧α , r 2/ν r −2 ¯ ∂r∂r
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Gauge theory for embedded surfaces, II

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