Gauge theory for embedded surfaces, II

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$Math 126 Solution Set 2$

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

50 P. B. Kronheimer and T. S. Mrowka sheaf O (a) , this means looking at the weight of the action on the local generating section ˜z −a ; this weight is a, because the sections of M are carried forward, not pulled back, by the group action. We see from this that O (a) is locally isomorphic to O (a′ ) when a = a ′ mod ν, though they are globally isomorphic only when a = a ′ . (iii) From orbifold bundles to parabolic bundles. Let F be an orbifold line bundle on X, that is, a locally free sheaf of rank 1. From F wecanobtainasheafon ¯ Xby the construction µ. We modify this construction slightly by first tensoring F by the orbifold line bundle O ([ν/2]) , where [ν/2] is the integer part: we define ¯F = µ F⊗O ([ν/2]) . (8.5) (The reason for this slightly awkward shift is related to our decision that the weight of a parabolic line bundle should lie in [ − 1 1 2 , 2 ).) Locally, the orbifold bundle F is isomorphic to O (b) for some b. We fix the convention that b should lie in the interval −[ν/2] ≤ b
Gauge theory for embedded surfaces, II 51 the natural inclusion O⊂O (c) gives an equality µ(O)=µ O (c) . This situation is a local model for the natural inclusion µ ϕ ∗ ( ¯ F) ⊂ µ ϕ ∗ ( ¯ F) ⊗O (c) , which must therefore be an equality also. So (8.9) is indeed isomorphic to ¯ F. Now consider the other direction: let F be an orbifold line bundle of weight b, define ¯ F by (8.5) and then apply the construction (8.8) to ¯ F to form the line bundle G = ϕ ∗ µ F⊗O ([ν/2]) ⊗O (b) . In general, ϕ∗ (µ(F)) can be identified with the OX-module which is generated, on each Ũ, by the invariant sections of F. There is therefore a natural inclusion ϕ∗ (µ(F)) →F, and hence in our case a map ɛ : G→F⊗O (b+[ν/2]) . Since b +[ν/2] is positive, F is a subsheaf of the orbifold sheaf on the righthand side. This subsheaf is precisely the image of ɛ, as one can verify locally by considering the case F = O (b) ,whereϕ ∗ µ F⊗O ([ν/2]) is equal to ϕ ∗ (O ¯ X), which is OX. This completes the verification that the two constructions are mutually inverse. The correspondence defined by (8.7) is functorial, as is the inverse construction. This is manifest when the line-bundles involved all have the same weight. When the weights are different, suppose (to take just one of two directions) that θ :( ¯ F1,β1)→( ¯ F2,β2) is a map between parabolic line bundles, with βi = bi/ν. If b1 ≤ b2 then O (b1) ⊂O (b2) ,andsoθinduces a natural map ϕ ∗ ( ¯ F1) ⊗O (b1) θ ′ −→ ϕ ∗ ( ¯ F 2) ⊗O (b1) ↩→ϕ ∗ ( ¯ F2)⊗O (b2) , as required. If b1 >b2 then the definition of map in the parabolic setting requires that θ should vanish along Σ in ¯ X. The first map θ ′ above is still there in this case, but its image lies in the subsheaf ϕ ∗ ( ¯ F2) ⊗O (b1−ν) ; and this is contained in ϕ ∗ ( ¯ F2)⊗O (b2) now, because b2 >b1−ν.Soineithercase we obtain a map between the corresponding orbifold bundles as defined by (8.8). To summarize, then: Proposition 8.10. The correspondence (8.7) is invertible and functorial, from the category of orbifold line bundles to the category of parabolic line bundles whose weights are rational with denominator ν. ⊓⊔ We can now repeat these constructions with vector bundles in place of line bundles. Let E be an orbifold SL(2, C) bundle: a locally free orbifold OX module of rank 2, with Λ 2 E ∼ = OX. Locally, such bundle is isomorphic to O (a) ⊕O (−a)
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Gauge theory for embedded surfaces, II

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