Gauge theory for embedded surfaces, II

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48 P. B. Kronheimer and T. S. Mrowka an orbifold version of the theorems of Donaldson et al. on the hermitian Yang- Mills equations [D1], we obtain a correspondence between M α and a moduli space of stable bundles on an orbifold; there is really no new work to be done here (see also [W]). The second step is to show that orbifold holomorphic bundles can be identified with parabolic bundles in a way which preserves stability (see also [SW]). This second step is basically elementary (the correspondence is local and quite straightforward), but some care is required to set things up properly, for otherwise the local distinction between the orbifold and parabolic viewpoints is liable to melt away. To begin with, therefore, we should clarify what is meant by a holomorphic bundle on a Kähler orbifold. Up until this point, we have been content to think of an orbifold rather loosely as a manifold with a singular geometry; now we must be more pedantic. One more remark is appropriate here. The identification of two moduli spaces of twisted anti-self-dual connections, noted in [KrM] as equation (2.16), is wellknown in the literature on parabolic bundles, where it results from an ‘elementary modification’ along L and is part of the so-called ‘Hecke correspondence’. In the context of the twisted connections, we refer to this identification as a ‘flip’. (ii) Orbifolds. For our limited purposes, a complex orbifold (X,Σ) with a finite quotient singularity of order ν along Σ can be regarded as comprising the following data. We are given a compact hausdorff space X and a closed subset Σ. Foreachpointof Σthere is given an open neighbourhood U in X and an orbifold chart, which is a homeomorphism ϕ : Ũ/Zν →U, where Ũ ⊂ C2 is an open subset on which the cyclic group Zν acts by (˜w, ˜z) ↦→ (˜w,e 2πi/ν ˜z) in standard coordinates; the homeomorphism ϕ carries the locus ˜z =0onto U∩Σ.WhenV =U1∩U2 is the intersection of the ranges of two such charts ϕ1 and ϕ2, the identity map on V has given a lifting to a holomorphic Zν- equivariant map ϕ −1 1 (V ) → ϕ−1 2 (V ). However, these liftings need not satisfy a cocycle condition on triple overlaps, though such a condition is automatically satisfied modulo Zν. These orbifold charts give a complex manifold structure to N\Σ for some neighbourhood N; we require that X\Σ has a complex structure which extends this, given as usual by an atlas of ordinary charts. Because there is no cocycle condition on the lifts, the definition above does not mean that any neighbourhood N of Σ is globally a quotient of a space Ñ by Zν. The obstruction to finding such covering is the self-intersection number Σ·Σ mod ν. With any such orbifold X as above, we can associate a smooth complex surface ¯ X as follows. The underlying topological space of ¯ X is the same as that
Gauge theory for embedded surfaces, II 49 underlying X, and as local complex coordinates on a patch U as above we take (w, z), where z =˜z ν w=˜w. A more invariant way to say this is to define a sheaf O ¯ X by declaring that the local sections on U are to be the Zν-invariant sections of the structure sheaf of Ũ. Generalizing the construction, we first define an orbifold sheaf E on X to be an assignment of a Zν-equivariant sheaf ˜ EŨ to each local covering space Ũ, together with equivariant transition isomorphisms which satisfy the cocycle condition up to the action of Zν. An example of such an orbifold sheaf is the structure sheaf OX, which assigns to each local covering space Ũ the sheaf OŨ . All our orbifold sheaves on X will be sheaves of OX modules, in the obvious sense. We say that E is locally free if each OŨ is. Now to any orbifold sheaf E on X, we can associate an ordinary sheaf of µ(E) on ¯ X by saying that, for each open set U with local covering Ũ, µ(E)U =( ˜ E Ũ )Zν – the Zν-invariant part. Our previous definition of the structure sheaf O ¯ X amounts to saying that O ¯ X is µ(OX), and if E is an orbifold sheaf of OX-modules then µ(E) isasheafofO¯ X modules in the ordinary sense. An important family of orbifold sheaves is the following. Let M be the orbifold sheaf on X which, on each local covering space Ũ, consists of the sections which are meromorphic with a pole only along Σ (if anywhere). For a>0an integer, define O (a) to be the subsheaf of M consisting of sections which, on Ũ, have a pole of order not more than a. Similarly, define O (−a) to be those sections having a zero of order a or more. These orbifold sheaves are locally free of rank 1; thus O (a) is locally generated on Ũ by the section ˜z− a ∈MŨ . They are all powers of O (1) , and their definition provides inclusions O (a) ⊂O (a+1) . (In the case of complex dimension 1, the corresponding objects were called the orbifold point bundles in [FuS].) Applying the construction µ to these orbifold sheaves, we find µ O (a) = OX for 0 ≤ a
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Gauge theory for embedded surfaces, II

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