By Fabian Ziltener
Think of a Hamiltonian motion of a compact attached Lie team on a symplectic manifold M ,w. Conjecturally, below compatible assumptions there exists a morphism of cohomological box theories from the equivariant Gromov-Witten thought of M , w to the Gromov-Witten concept of the symplectic quotient. The morphism will be a deformation of the Kirwan map. the belief, because of D. A. Salamon, is to outline one of these deformation through counting gauge equivalence sessions of symplectic vortices over the advanced aircraft C. the current memoir is a part of a undertaking whose objective is to make this definition rigorous. Its major effects take care of the symplectically aspherical case
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Extra info for A quantum Kirwan map: bubbling and Fredholm theory for symplectic vortices over the plane
35) is satisﬁed by hypothesis. 51) |dAν uν | + Rν |μ ◦ uν | = c−1 ν fν ◦ ϕν ≤ 2, on Bεν cν (0). 50) and the fact fν (zν ) → ∞, that εν cν → ∞. 36) follows, for every compact subset Q ⊆ C. Therefore, applying Proposition 38, there exists an r0 -vortex w0 = (A0 , u0 ) ∈ WC and, passing to some subsequence, there exist gauge transformations gν ∈ W 2,p (C, G), with the following properties. For every compact subset Q ⊆ C, gν∗ Aν converges to A0 in C 0 on Q, and gν−1 uν converges to u0 in C 1 on Q. 5.
27. Proposition (Convergence in the Ginzburg-Landau setting). Let k ∈ N0 , for ν ∈ N let Wν ∈ M<∞ be a vortex class and z1ν , . . ,k be a stable map. Then the following conditions are equivalent. (i) The sequence Wν , z0ν := ∞, z1ν , . . , zkν converges to (W, z). 25) deg(Wν ) = deg(Wα ) =: d. α∈T1 Furthermore, there exist M¨obius transformations ϕνα : S 2 → S 2 for α ∈ T and ν ∈ N such that conditions (i,ii,iv) of Deﬁnition 20 are satisﬁed, and for every α ∈ T1 the point in the symmetric product degWν ◦ϕνα ∈ Symd (C) ⊆ Symd (S 2 ) converges to ιd (degWα ) ∈ Symd (S 2 ), as ν → ∞.
Assume that R0 = ∞. 1. Claim. 39) sup ν F Aν Lp (Q) ν ∈ N : Q ⊆ Ων < ∞. Proof of Claim 1. Let Ω ⊆ C be an open subset containing Q such that Ω is compact and contained in C \ Z. 40) sup dAν uν ν L∞ (Ω) < ∞. It follows from our standing hypothesis (H) that there exists δ > 0 such that G acts freely on K := x ∈ M |μ(x)| ≤ δ . Since μ is proper the set K is compact. Recall that Lx : g → Tx M denotes the inﬁnitesimal action at x. It follows that |ξ| x ∈ K, 0 = ξ ∈ g < ∞. 41) sup |Lx ξ| Using the second Rν -vortex equation, we have |μ ◦ uν | ≤ ν eR wν .