A Quantum Kirwan Map: Bubbling and Fredholm Theory for ... - KIAS

2.2. CONVERGENCE TO A STABLE MAP 23(ii) If α,β ∈ T are such that αEβ then (ϕ ν α) −1 ◦ϕ ν β → z αβ, uniformly on compactsubsets of S 2 \ {z βα }.(iii) • Let α ∈ T 1 and Ω ⊆ R 2 be an open connected subset with compactclosure and smooth boundary. Then the restriction (ϕ ν α) ∗ W ν | Ωconvergesto W α | Ω, with respect to the topology τ Ω(as in Definition 19).• Fix α ∈ T ∞ . Let Q be a compact subset of S 2 \ (Z α ∪ {z α,0 }). For νlarge enough, we haveū Wν ◦ ϕ ν α(Q) ⊆ M ∗ /G,and ū Wν ◦ϕ ν α converges to ū α in C 1 on Q. (Here Z α and ū Wν are definedas in (2.9,2.2).)(iv) We have (ϕ ν α i) −1 (z ν i ) → z i for every i = 1,...,k.The meaning of this definition is illustrated by Figure 3. It is based on thenotion of convergence of a sequence of pseudo-holomorphic spheres to a genus 0pseudo-holomorphic stable map. 10 An example in which it can be understood moreexplicitly, is discussed in the next section.Remark. The condition in the first part of (iii), that (ϕ ν α) ∗ W ν | Ω→ W α | Ωwith respect to τ Ω, is equivalent to the requirement that there exist representativesw ν of (ϕ ν α) ∗ W ν | Ω(for ν ∈ N) and w of W α | Ωsuch that w ν converges to w in theC k -topology, for every k ∈ N. This follows from a straight-forward argument, usingLemma 120 (Appendix A.7). ✷Remark. The last part of condition (i) and the second part of condition (iii)capture the idea of catching a pseudo-holomorphic sphere in M by “zooming out”:Fix α ∈ T ∞ , and consider the case z α,0 = ∞. Then there exist λ ν α ∈ C \ {0}and z ν α ∈ C such that ϕ ν α(z) = λ ν αz + z ν α. It follows from a direct calculation that(ϕ ν α) ∗ W ν is a vortex class with respect to the area form ω Σ = |λ ν α| 2 ω 0 , where ω 0denotes the standard area form on C. The last part of condition (i) means thatλ ν α → ∞, for ν → ∞. Hence in the limit ν → ∞ we obtain the equations¯∂ J,A (u) = 0, µ ◦ u = 0.These correspond to the ¯J-Cauchy-Riemann equations for a map from C to M. (SeeProposition 116 in Appendix A.7.) The second part of (iii) imposes that in fact thesequence of rescaled vortex classes converges (in a suitable sense) to a ¯J-holomorpicsphere and that this sphere equals ū α .It is unclear whether the bubbling result, Theorem 3, remains true if we replacethe C 1 -convergence in this part of condition (iii) by C ∞ -convergence. (Compare toRemark 39 in Section 2.5.) ✷Remark. The “energy-conservation” condition (2.15) has the important consequencethat the stable map (W,z) represents the same equivariant homologyclass as the vortex class W ν , for ν large enough. (See [Zi1, Proposition 5.4] and[Zi4].) ✷21. Remark. The purpose of the additional marked point (α 0 ,z 0 ) is to be ableto formulate the second part of condition (iii). For α ∈ T ∞ and ν ∈ N the mapGu ν ◦ ϕ ν α is only defined on the subset (ϕ ν α) −1 (C) ⊆ S 2 . Since by condition (i) we10 For that notion see for example [MS2].