Adiabatic limits and Kazdan-Warner equations
Aleksander Doan
 






We study the limiting behaviour of solutions to abelian vortex equations when the volume of the underlying Riemann surface grows to infinity. We prove that the solutions converge smoothly away from finitely many points. The proof relies on a priori estimates for functions satisfying generalised Kazdan–Warner equations. We relate our results to the work of Hong, Jost, and Struwe on classical vortices, and that of Haydys and Walpuski on the Seiberg–Witten equations with multiple spinors.

Adiabatic limits and Kazdan–Warner equations
    Aleksander Doan
    December 30, 2023
============================================





§ INTRODUCTION

The vortex equations on Riemann surfaces originate from the Ginzburg–Landau model of superconductivity. They were first brought to the attention of mathematicians by Jaffe and Taubes <cit.>. Since then there has been a considerable body of work aimed at understanding these equations and their many generalisations. The purpose of this article is to study one such generalisation in the context of adiabatic limits and compactifications in two- and three-dimensional gauge theories. 

Let Σ be a closed Riemann surface and L →Σ a Hermitian line bundle. A (classical) vortex is a pair of a unitary connection A on L and a section φ of L satisfying

    {[                   _Aφ = 0,; i Λ F_A    =  1 - | φ |^2, ].

where  Λ F_A is the Hodge dual of the curvature form. The space of vortices modulo gauge equivalence is simply the symmetric product Sym^d Σ with d =  L. A point in Sym^d Σ corresponds to a degree d effective divisor D and there exists a unique, up to gauge equivalence, solution (A,φ) having  D as the zero divisor of φ <cit.>.

Scaling the metric on Σ by a factor ϵ^-1 results in the modified equations

    {[                     _Aφ = 0,; ϵ^2 iΛ F_A    = 1 - | φ |^2. ].

A question arises about the behaviour of solutions in the limit ϵ→ 0, as the volume of Σ grows to infinity. This is the idea of the adiabatic limit which has been used in many contexts in gauge theory and Riemannian geometry <cit.>. The question for the vortex equations was answered by Hong, Jost, and Struwe <cit.> whose result can be stated as follows. Fix a degree d effective divisor D = ∑_k m_k x_k, where m_k ∈_≥ 0 and x_k ∈Σ. If (A_i, φ_i, ϵ_i) is a sequence of solutions to (<ref>) having D as the zero divisor and satisfying ϵ_i → 0, then after passing to a subsequence and changing (A_i, φ_i) by gauge transformations,


  * i/2πΛ F_A_i converges as measures to the sum of Dirac deltas δ_D := ∑_k m_k δ_x_k, and

  * ∇_A_iα_i → 0, | α_i | → 1, and  F_A_i→ 0 in C^0_ on Σ∖ D [By abuse of notation, we denote by the same symbol a divisor and the underlying set of points.].


In this paper we consider the following generalisation of (<ref>). Fix auxiliary unitary bundles E_1, …, E_N over Σ together with respective connections B_1, …, B_N and non-zero integer weights k_1, …, k_N. Let ϵ > 0 and τ∈. The generalised equations for a connection A on L and a section φ = (φ^1, …, φ^N) ∈Γ( ⊕_j=1^N E_j ⊗ L^⊗ k_j) are
 
    {[       _A ⊗ B_jφ^j = 0     for  j = 1, …,  N,; ϵ^2 i Λ F_A + ∑_j=1^N k_j | φ^j |^2 + τ = 0. ].

Equations (<ref>) fit into the more general setting of framed vortex equations discussed in <cit.>. As before, the moduli space of solutions has a holomorphic description <cit.>. 

Our main result concerns sequences of solutions to (<ref>) with ϵ→ 0. Suppose that either the weights k_i are of mixed signs or k_i > 0 for all i and τ < 0 as otherwise there are no solutions.


Let (A_i, φ_i, ϵ_i) be a sequence of solutions to (<ref>) such that  ϵ_i → 0 and the sequence of norms φ_i _L^2 is bounded. Then there is a finite set of points D ⊂Σ such that after passing to a subsequence and applying gauge transformations (A_i, φ_i  )  converges in C^∞_ on Σ∖ D. The limit (A,φ) satisfies (<ref>) with ϵ = 0 on Σ∖ D.



More generally, we can assume τ = τ_i to depend on i as long as τ_i converges. If the sequence λ_i := φ_i _L^2 is unbounded, we obtain the convergence of (A, λ_i^-1φ_i, λ_i^-1ϵ_i) with τ_i = λ_i^-1τ. Thus, Theorem <ref> describes also the limiting behaviour of solutions to (<ref>) with ϵ=1 and the L^2 norms diverging to infinity. In other words, we provide a description of the ends of the non-compact moduli space of solutions to (<ref>). This should be compared with recent results on sequences of solutions to the Hitchin equations <cit.>.


When N=1, k_1 = 1, and τ = 1, we recover the classical vortex equations (<ref>). As a result, we reprove and strengthen the result of <cit.>. We should point out that our method of proof is different from that of <cit.> and will be outlined at the end of this introduction. 



Let (A_i, φ_i, ϵ_i) be a sequence of solutions to (<ref>) such that  ϵ_i → 0. Then there is a degree d effective divisor D on Σ such that after passing to a subsequence and applying gauge transformations (A_i, φ_i) converges in C^∞_ on Σ∖ D and i/2πΛ F_A_i→δ_D as measures. The limit (A, φ) satisfies F_A = 0, |α| = 1 and ∇_A α = 0 on Σ∖ D.
 


In the general setting of Theorem <ref>, the limiting connection A is no longer necessarily flat; we will see an example of this in point (4) of Theorem <ref> below.




 §.§ Seiberg–Witten theory
 
The main application of Theorem <ref> concerns generalised Seiberg–Witten equations in dimension three. To set the stage, let Y be a closed Riemannian spin three-manifold. Let  S be the spinor bundle and E, L vector bundles over Y with structure groups  (n) and (1) respectively; we equip E with a connection B. The Seiberg–Witten equations with multiple spinors for a connection A on L → Y and Ψ∈Γ((E, S ⊗ L)) are

    {[             D_A ⊗ BΨ = 0,; F_A = ΨΨ^* - 1/2 | Ψ |^2. ].
 
Here D_A ⊗ B is the Dirac operator twisted by A and B and in the second equation we use the identification i Λ^2 T^*Y ≅ i 𝔰𝔲(S) given by the Clifford multiplication. 

An analogous set of equations on four-manifolds was introduced in <cit.>. 
The three-dimensional version was studied by Haydys and Walpuski in relation to enumerative theories for associative submanifolds and G_2–instantons on G_2–manifolds <cit.>. The principal result of <cit.> concerns the limiting behaviour of sequences  of solutions (A_i, Ψ_i) such that Ψ_i _L^2→∞. Haydys and Walpuski showed that there is a closed nowhere dense subset Z ⊂ Y such that after passing to a subsequence and applying gauge transformations 

    A_i → A  weakly in  W^1,2_  and  Ψ_i / Ψ_i _L^2→Ψ weakly in  W^2,2_

on Y ∖ Z, and the limiting configuration (A, Ψ) defined on Y ∖ Z satisfies

    {[            D_A⊗ BΨ = 0,; ΨΨ^* - 1/2 | Ψ |^2 = 0. ].

Moreover, Z is the zero locus of Ψ in the sense of <cit.> and,  if E = 2, A is flat with holonomy contained in _2. If E > 2, then A induces a flat _2–connection on a rank two subbundle of E twisted by a line bundle; see <cit.>.

A number of problems in this theory remain open despite their importance for the possible applications of generalised Seiberg–Witten equations to G_2–gauge theory:


  
  * Taubes has made significant progress in the study of the local structure of Z, proving in particular that Z has Hausdorff dimension at most two <cit.>; yet the question whether Z is a smooth curve is still unanswered.
  
  * For the applications in enumerative theories it is crucial to improve the convergence statement for (A_i,Ψ_i/Ψ_i_L^2), as exemplified by <cit.> where,  as part of the main proof,  C^∞ convergence is established under the assumption that Z is empty.
  
  * There are two ways of associating weights to the connected components of Z: one based on Taubes' frequency function <cit.> and one developed by Haydys using topological methods  <cit.>. It is currently unknown whether these constructions are related.
  
  * Haydys conjectured that, equipped with appropriate weights, Z has the structure of an integral rectifiable current and that i/2π F_A_i converges to Z as currents <cit.>.


Using Theorem <ref> and the methods involved in its proof, we refine the compactness theorem of <cit.> and solve all of the above problems in the case Y = S^1 ×Σ. 


Suppose that  Y = S^1 ×Σ equipped with a product metric, B is pulled back from Σ, and (A_i, Ψ_i), (A,Ψ), and Z are as above, see also <cit.>. Then

  
  * The singular set Z is of the form S^1 × D for a degree 2d divisor D = ∑_k m_k x_k.
  
  * After passing to a subsequence and applying gauge transformations 
  
    A_i → A   and  Ψ_i / Ψ_i _L^2→Ψ

   in C^∞_ on Y ∖ Z.
   
  * | Ψ |^4 extends to a smooth function on Y whose zero set is Z and for all k
   
    | Ψ (x) | = O( dist(x,S^1 ×{ x_k })^|m_k|/2).

  In particular, the weight of the connected component S^1 ×{ x_k } of Z in the sense of <cit.> is smaller than or equal to its weight in the sense of <cit.>.
  
  * If E = 2, then i/2π F_A_i→1/2 Z as currents. If E > 2, then there is a rank two subbundle F ⊂EY ∖ Z such that Ψ∈Γ(Y ∖ Z, (F, S ⊗ L)) and the previous statement holds if we replace A and A_i by the tensor product connections on L ⊗ (  F)^1/2. Here F and F are  equipped with the unitary connections induced from B.  



The relationship between Seiberg–Witten monopoles with multiple spinors and generalised vortices (<ref>) is the subject of the author's paper <cit.> where further consequences of the results presented here are explored. In particular, Theorem <ref> is used to construct a compactification of the moduli space of Seiberg–Witten monopoles with multiple spinors on S^1 ×Σ and to compare it with a corresponding algebro-geometric moduli space <cit.>. This, in turn, leads to the first known examples of the non-compactness phenomenon predicted by the Haydys and Walpuski's theorem <cit.>. 



 §.§ Symplectic vortex equations
 
Coming back to dimension two, equations (<ref>) fit into the general framework of symplectic vortex equations or gauged σ-models. One associates vortex-type equations on Σ to any pair (G,M), where G is a compact Lie group acting in a Hamiltonian way on a symplectic manifold M. In the spirit of Gromov–Witten theory one wishes to extract numerical invariants of (G,M) from the moduli space of solutions. The parameter ϵ can be incorporated into the equations in the same way as before. In the adiabatic limit ϵ→ 0, we obtain the equation for pseudoholomorphic curves in the symplectic quotient M  G. Thus, we expect a relation between the invariants of (G,M) and the Gromov–Witten invariants of M  G. This programme has been proposed and successfully carried out by Cieliebak, Gaio, Mundet i Riera, Salamon  <cit.>, <cit.>, and others. In order to establish Gromov compactness for symplectic vortices, more constraints are imposed on the pair (G,M), a crucial condition being the properness of the Hamiltonian moment map.  Already for linear actions this is a rather restrictive assumption. The simplest example is the one discussed here with the corresponding equations (<ref>).  In this case M = ^n and G = (1) acts diagonally with weights (k_1, …, k_N). The moment map is not proper unless all the weights have the same sign. Theorem <ref> shows that if the properness condition is dropped we have to take under account formation of singularities in considerations regarding compactness and adiabatic limits. 



 §.§ Outline of the proof
 
One consequence of the improperness of the moment map is that, unlike classical vortices, solutions to (<ref>) do not obey an a priori L^∞ bound. This causes a major difficulty in establishing the convergence. While the proof in <cit.> is based on local ϵ–regularity estimates, we employ here a complex-geometric description of the moduli space of solutions to (<ref>). To be more specific, we use the action of 𝒢^c = C^∞(Σ, ^×), the group of complex automorphisms of L, on the space of pairs (A, φ). The moduli space of solutions to (<ref>) is homeomorphic to the quotient of the set solutions to the Cauchy–Riemann equation by 𝒢^c; this is a Hitchin–Kobayashi type correspondence proved in  <cit.>. Using elliptic estimates for Dolbeault operators, we show that this quotient is compact modulo the rescaling action of ^×. Thus, there are complex gauge transformations g_i = e^f_i u_i for f_i ∈ C^∞(Σ, ) and u_i ∈ C^∞(Σ, (1)) such that  after rescaling and applying g_i the original sequence (A_i, φ_i) converges. In order to obtain the convergence in the real rather than the complex moduli space, we need to control the functions f_i. The original equations (<ref>) translate in this setting to a partial differential equation for f_i of the form

    ϵ^2 Δ f + ∑_j=1^n A_j e^α_j f - ∑_j=1^m B_j e^-β_j f + w = 0

for some functions A_j ≥ 0, B_j ≥ 0, w, and positive constants α_j, β_j. This is a generalisation of the Kazdan–Warner equation <cit.>, <cit.>. In section <ref> we establish a priori bounds for solutions of this equation. Importantly, they are independent of ϵ∈ (0, 1] and uniform on compact subsets of Σ∖ D, where  D is the set of common zeroes of  A_j and B_j. Consequently, the Arzelà–Ascoli theorem guarantees the existence of a subsequence of f_i converging smoothly on compact subsets of Σ∖ D.

To the best of our knowledge, the strategy of passing to the holomorphic moduli space by means of a Hitchin–Kobayashi correspondence, obtaining good control there using –methods, and deducing from it compactness for the original sequence in the real moduli space, has not been used before. We believe that this idea—and some of the related analytical results such as Lemma <ref>—might be useful in studying other gauge-theoretic equations on Kähler manifolds.



 §.§ Acknowledgements
 The work presented in this article is part of my doctoral thesis at Stony Brook University. I am grateful to my advisor Simon Donaldson for his guidance and support. Thanks to Andriy Haydys and Thomas Walpuski for their encouragement and many helpful discussions, and to Gonçalo Oliveira, Oscar Garcia–Prada, Song Sun, Alex Waldron, and the anonymous referee for valuable comments on the previous versions of this paper. 
I am supported by the https://sites.duke.edu/scshgap/Simons Collaboration on Special Holonomy in Geometry, Analysis, and Physics.



§ BACKGROUND AND NOTATION

The set of solutions to (<ref>) is invariant under the action of the gauge group 𝒢 of unitary automorphisms of L, identified with C^∞(Σ,(1)). The action of a map u Σ→(1)  on (A, φ^1, …, φ^N) is given by

    u (A, φ^1, …, φ^N) = (A - u^-1 du, u^k_1φ^1, …, u^k_Nφ^N).

The Dolbeault equation in (<ref>), as well as the algebraic condition φ^1 φ^2 = 0 in the equation (<ref>) below, are also invariant under the action of the complex gauge group 𝒢^c. It consists of complex automorphisms of L and is identified with C^∞(Σ, ^×), where ^× = ∖{ 0 }. The action of g Σ→^× is given by

    g(A, φ^1, …, φ^N) = (A + g^-1∂g - g^-1 g, g^k_1φ^1, … , g^k_Nφ^N ).
 
In terms of the associated Dolbeault operators, for s ∈Γ(Σ, E_j ⊗ L^⊗ k_j) we have

    _B_j, g(A) s = g^k_j_B_j A( g^-k_j s ).

The action of 𝒢^c does not preserve the last equation in (<ref>) involving the curvature. Indeed, if we write g = e^f u for functions f Σ→ and u Σ→(1), then

    F_g(A) = F_A + 2 ∂ f,

or equivalently

    i Λ F_g(A) = i Λ F_A + Δ f,

where Δ is the Hodge Laplacian acting on functions. 

We will need also the following lemma whose elementary proof we omit.


Let L →Σ be a Hermitian line bundle, D ⊂Σ a finite set of points, A a unitary connection on LΣ∖ D and α∈Γ(Σ∖ D, L). If _A α = 0, and | α | = 1 everywhere on Σ∖ D, then 

    ∇_A α = 0     and     F_A = 0.

Moreover, let B be a small ball around a point p ∈ D such that in a local unitary trivialisation A = d + a for a one-form a ∈Ω^1(B ∖{ p }, i), and α is identified with a smooth function α B ∖{ p }→ S^1. Then 

    i/2π∫_∂ B  a = deg( α∂ B).
















We end this section by restating Theorem <ref> in terms of generalised vortex equations. In <cit.> we show that all irreducible solutions to the Seiberg–Witten equations with multiple spinors are pulled back from solutions to (<ref>) of the following form. Using the notation of the introduction, set N=2, k_1 = 1, k_2 = -1. Fix a spin structure on Σ thought of as a square root K^1/2 of the canonical bundle, and let E be an (n)-bundle; then set E_1 = E ⊗ K^1/2 and E_2 = E^* ⊗ K^1/2.

The equations we consider next involve a connection A on L and sections φ^1 ∈Γ(E ⊗ K^1/2⊗ L) and φ^2 ∈Γ(E^* ⊗ K^1/2⊗ L^*):

    {[                        _A⊗ Bφ^1 = 0,;                       _A ⊗ Bφ^2 = 0,;                         φ^1 φ^2 = 0,; ϵ^2 i Λ F_A + |φ^1|^2 - |φ^2|^2 = 0, ].

The third equation is an additional algebraic condition for the section φ^1 φ^2 ∈Γ(K) defined as the image of (φ^1,φ^2) under the pairing

    Γ(E ⊗ K^1/2⊗ L) ×Γ(E^* ⊗ K^1/2⊗ L^*) ⟶Γ(K).


We will deduce Theorem <ref> from the following result.

 
Let (A_i, φ_i, ϵ_i) be a sequence of solutions to (<ref>) with φ_i _L^2 = 1 and ϵ_i → 0. Then

 
  * There exist a degree 2d divisor D = ∑_k m_k x_k and a configuration (A,φ) defined on Σ∖ D and satisfying (<ref>) with ϵ = 0,
  
  * (A_i, φ_i) → (A,φ) in C^∞_ on Σ∖ D,

  * The function | φ |^4 extends to a smooth function on all of Σ whose zero set consists of the points in D and for all k
   
    | φ (x) | = O( dist(x,x_k)^|m_k|/2).


  * If E = 2, then the limiting connection A is flat, has holonomy contained in _2, and i/2πΛ F_A_i→1/2δ_D as measures. If E > 2, then there exists a rank two subbundle F ⊂EΣ∖ D  such that 

    φ^1 ∈Γ(Σ∖ D, F ⊗ L ⊗ K^1/2),     φ^2 ∈Γ(Σ∖ D , F^* ⊗ L^* ⊗ K^1/2),

 and the previous statement holds if we replace A and A_i by the tensor product connections on L ⊗ ( F)^1/2. Here, F and F are  equipped with the unitary connections induced from B.




In contrast to Theorem <ref> here the divisor D need not be effective. In a way, replacing classical vortices by solutions to (<ref>) is analogous to replacing holomorphic sections by meromorphic sections. This idea will play a role in the proof of Theorem <ref>.




§ A PRIORI ESTIMATES


The main analytical input are a priori estimates for solutions to (<ref>). 


 
Let X be a compact Riemannian manifold with (possibly empty) boundary ∂ X, and Ω⊂ X an open subset such that Ω⊂ X ∖∂ X. Let ϵ_0, α_1, … , α_n, β_1, … , β_m be positive numbers, and let A_1, …, A_n, B_1, …, B_m, and w be smooth functions on X such that A_j ≥ 0 and B_j ≥ 0 for all j and

    A_1 + … + A_n > 0,       B_1 + … + B_m > 0.

Then there exist constants M_0, M_1, M_2, …, depending only on the data listed above, such that for any ϵ∈ [0, ϵ_0] and f ∈ C^∞(X) satisfying the equation

    ϵΔ f + ∑_j=1^n A_j e^α_j f - ∑_j=1^m B_j e^-β_j f + w = 0,

the following inequalities hold:

    f _C^k(Ω)≤ M_k     for k=0, 1, 2, …



  
M_k depends on A_j, B_j, and w and their derivatives. Later we will consider sequences ϵ_i → 0 and f_i, A_1^i, …, A_n^i, B_1^i, …, B_m^i, w_i satisfying for all i

    ϵ_i Δ f_i + ∑_j=1^n A_j^i e^α_j f_i - ∑_j=1^m B_j^i e^-β_j f_i + w_i = 0.

It will be clear from the proof that in this case the C^k estimate still holds for large i (depending on k) provided that A_j^i, B_j^i, and w_i converge smoothly to A_j, B_j, and w respectively, satisfying  

    A_1 + … + A_n > 0,       B_1 + … + B_m > 0.







The proof of Proposition <ref> is preceded by three lemmas.

 
Let X and Ω be as in Proposition <ref>, and let V_0 ⊂ X be an open subset such that Ω⊂ V_0. Then there exist an open subset V and ϕ∈ C^∞(X) such that:


  * Ω⊂ V ⊂ V_0. 

  * 0 < ϕ≤ 1 on V.

  * ϕ = 1 on Ω. 

  * ϕ = 0 on X ∖ V.

  * There is a constant K such that for any α∈ [0,2), 

    sup_V'| ∇ϕ |^2/ϕ^α≤K/(2-α)^4.





Let V' ⊂ X be any open subset such that Ω⊂ V' ⊂ V and V' has smooth boundary. We can construct such a subset for example by taking any smooth function h  X → with h < 0 on Ω and h> 1 on X ∖ V, and setting V' = h^-1( (- ∞, c)) where c ∈ (0,1) is a regular value of h. Let N = h^-1(c) be the boundary of V'. Assume for simplicity that X is orientable. By the tubular neighbourhood theorem, there is an embedding

    (- ϵ, ϵ) × N ↪ X

such that { 0 }× N is mapped diffeomorphically onto N ⊂ X and the image of (0, ϵ) × N is contained in V'.  We may also assume that the image of this embedding is disjoint from Ω.  

Using a partition of unity (and passing to a slightly smaller ϵ), we construct a function ϕ with properties (1)–(4), which for (t,x) ∈ (-ϵ, ϵ) × N agrees with

    ϕ(t, x) = {[     M exp( - 1/t)   for  t ∈ (0, ϵ);                 0 for  t ∈ (- ϵ, 0);                   ].

for some constant M required for the normalisation ϕ_C^0(X) = 1. Now let α < 2. Away from N we have ϕ > 0 and | ∇ϕ |^2 / ϕ^α is bounded. In a neighbourhood of N, 

    | ∇ϕ |^2/ϕ^α≤C/t^4exp( -(2-α)/t)

for some constant C depending on the Riemannian metric on X and the embedding of the tubular neighbourhood. Define

    g(t) = 1/t^4exp( -(2-α)/t).

Then g is smooth and bounded on [0, ∞) and its global maximum is

    g( 2-α/2) =  4e^-2/(2-α)^4,

which shows that | ∇ϕ |^2 / ϕ^α≤ K (2-α)^-4 as desired.



 
Let X and Ω be as in Proposition <ref>. Fix positive numbers ϵ_0, p, and γ > 1, and consider functions A ∈ C^∞(X) and Q ∈ C^∞(X × (0, ∞)) such that  for all (x,y) ∈ X × (0, ∞),

    A(x) ≥η > 0     and     | Q(x,y) | ≤∑_j=1^k a_j(x) y^γ_j,

where a_j ∈ C^∞(X) and γ_j < γ. Under these assumptions there exists a constant M such that for any ϵ∈ [0, ϵ_0] and  u ∈ C^∞(X) satisfying u ≥ 0 and 

    ϵΔ u + A u^γ + Q(x,u) ≤ 0,

the following inequality holds:

    u _L^p(Ω)≤ M.

Moreover, M depends only on Ω, X, ϵ_0, p, γ, γ_j, η, and the norms a_j _L^q(X) for a certain q < ∞  depending on p.  












We adopt here the convention that C always denotes a constant depending only on the fixed data and not on u or ϵ. Its value might change from line to line. 

It is enough to prove the statement for some ϵ_0 > 0, which we will later assume to be sufficiently small. Indeed, the corresponding statement for any other ϵ_0' > ϵ_0 can be reduced to the one for ϵ_0 by multiplying both sides of (<ref>) by ϵ_0 / ϵ_0' at the cost of appropriately scaling A and Q.  
Furthermore, it suffices to prove the statement for p=1 since for p >1 we have

    Δ( u^p )    = - p(p-1) u^p-2 | ∇ u |^2 + p u^p-1Δ u 
       ≤ -pA u^p - 1 + γ - pu^p-1 Q(x,u) 
       ≤ - A' (u^p)^γ' - Q'(x,u^p),

where 

    A' = pA,     γ' = 1 + γ-1/p,      Q'(x,y) = py^p-1/pQ(x, y^1/p).

We easily check that the new data (A', γ', Q') satisfies the hypotheses of the lemma. The only non-trivial condition is the estimate for Q' which follows from

    | Q'(x,y) | ≤  p ∑_i^k a_j(x) y^1 + γ_j - 1/p = p ∑_i^k a_j(x) y^γ_j'.

Note that γ_j' < γ'. Therefore, the statement for p > 1 reduces to that for p=1 after replacing (u, A, γ, Q) by (u^p, A', γ', Q'). 

In fact, we will bound u _L^1+γ(Ω) . Let V_0 ⊂ X be an open subset containing Ω such that the volume of V_0 ∖Ω is sufficiently small. We will specify later what we mean by that, but for the moment let us stress that the choice of  V_0 will depend only on the fixed data and not the function u. Once V_0 is fixed, choose a subset V ⊂ V_0 and a bump function ϕ∈ C^∞(X) as in Lemma <ref>.
Note that vol(V ∖Ω) ≤vol(V_0 ∖Ω).


Multiply inequality (<ref>) by u ϕ^2 and integrate it over X: 

    ∫_X   ϵ (Δ u) u ϕ^2 + A u^1+γϕ^2 + Q u ϕ^2 ≤ 0 .

Since ϕ has compact support, integration by parts yields

    ∫_X ( Δ u) u ϕ^2    =   ∫_X ⟨∇ u, ∇ (u ϕ^2) ⟩
        =  ∫_X ⟨∇ u , 2uϕ∇ϕ + ϕ^2 ∇ u ⟩
        = ∫_X 2 u ϕ⟨∇ u, ∇ϕ⟩ + ϕ^2 | ∇ u |^2 
       =  ∫_X | u ∇ϕ + ϕ∇ u |^2 - u^2 | ∇ϕ |^2 
       ≥ - ∫_X u^2 | ∇ϕ |^2.

Together with (<ref>), this implies the inequality

    ∫_X - ϵ u^2 | ∇ϕ |^2 + A u^1+γϕ^2  + Q u ϕ^2 ≤  0.

Recall that ϕ is supported in V and ϕ = 1 on Ω. Let P = V ∖Ω so that V = Ω∪ P. Splitting the integral on the left-hand side according to this decomposition and rearranging the inequality, we obtain

    ∫_Ω A u^1+γ + I ≤∫_Ω | Q | u,

where we have collected all integrals over P into a single term,

    I = I_0 + I_1 + I_2,


    I_0 = ∫_P A u^1 + γϕ^2,    I_1 = ∫_P Q u ϕ^2,    I_2 = -  ϵ∫_P u^2 | ∇ϕ |^2.

The next goal is to estimate I. We will show that for a suitable choice of V and ϵ_0, depending only on the initial data and not on u, we may assume that I ≥ 0, provided that ϵ≤ϵ_0. Strictly speaking, this will not always be true, but in the case when our estimate fails, we will obtain an upper bound for | I | so that we can move it to the right-hand side of (<ref>).

Before proving this, let us show that the inequality I ≥ 0 gives us a bound for u _L^1+γ(Ω). If I ≥ 0, then by (<ref>) and Hölder's inequality,

    η∫_Ω u^1+γ   ≤∫_Ω A u^1+γ≤∫_Ω | Q | u  ≤∫_Ω∑_j=1^k a_j u^1+γ_j
       ≤∑_j=1^k  a_j _L^p_j(X)( ∫_Ω u^1+γ)^1/q_j,

where the Hölder exponents p_j and q_j are given by

    q_j = 1+γ/1+γ_j,     1/p_j + 1/q_j = 1.

Note that q_j > 1 for each i, because γ_j < γ.  An equivalent way of writing (<ref>) is

    u _L^1+γ(Ω)^1+γ≤η^-1∑_j=1^k  a_j _L^p_j(X) u _L^1+γ(Ω)^1+γ/q_j,

which, in view of q_j > 1, results in an upper bound for u _L^1+γ(Ω). 

The dependance of the bound on the initial data is clear. 

In order to finish the proof, it remains to estimate the integral I = I_0 + I_1 + I_2.  We will deal separately with each of the three terms. The first one contributes positively to I and is bounded below by

    I_0 ≥η∫_P u^1 + γϕ^2.

The terms that can contribute negatively are I_1 and I_2. We estimate the former using our assumption on Q and Hölder's inequality:

    | I_1|    ≤∑_j=1^k ∫_P | a_j | u^1+γ_jϕ^2 
       ≤∑_j=1^k ( ∫_P |a_j|^p_j)^1/p_j( ∫_P u^1+γϕ^2q_j)^1/q_j
       ≤∑_j=1^k vol(P)^1/2p_j a_j _L^2p_j(X)( ∫_P u^1+γϕ^2q_j)^1/q_j,

where the Hölder exponents p_j and q_j are as before. Let S = ∫_P u^1+γϕ^2q_j. If S ≤ 1, then |I_1| is bounded by a constant independent of u, say C, and we can move it on the right-hand side of (<ref>). Next we replace I by the sum of the remaining two terms I' = I_0 + I_2 and if we can show that I' ≥ 0, then repeating the previous discussion we arrive at a bound for u _L^1+γ(Ω) with an extra term involving C. Thus, let us assume that S ≥ 1. In this case, we have S^1/q_j≤ S and
 
    | I_1|    ≤∑_j=1^k vol(P)^1/2p_j a_j _L^2p_j(X)∫_P u^1+γϕ^2q_j
       ≤∑_j=1^k vol(P)^1/2p_j a_j _L^2p_j(X)∫_P u^1+γϕ^2,

 where we have also used that ϕ≤ 1 and so ϕ^2q_j≤ϕ^2. Comparing the right-hand side of the inequality with the previously obtained upper bound (<ref>) for I_0 we see that if P has sufficiently small volume (which can be guaranteed by the choice of the initial open set V_0), then
  
    |I_1| ≤I_0/2.

Furthermore, how small P has to be depends only on η and a_j _L^p_max(X), where p_max = max{ p_1, …, p_k }. 
Note that at this point the sets V and P are chosen and will not be changed.

The second potentially negative term I_2 is dealt with in a similar manner. For every α∈, Hölder's inequality implies that

    |I_2|    = ϵ∫_P u^2 | ∇ϕ|^2 = ϵ∫_P   | ∇ϕ |^2/ϕ^α  u^2 ϕ^α
       ≤ϵ | ∇ϕ |^2 /ϕ^α_L^q(P)( ∫_P u^1+γϕ^α (1+γ)/2)^2/1+γ,

where q is given by 1/q + 2/(1+γ) = 1.  Observe that 1+ γ > 2, so we can choose α so that

    4/1+γ < α < 2.

Then, | ∇ϕ |^2 / ϕ^α is bounded on P, and the first factor on the right-hand side of (<ref>) is finite. As regards the integral in the second factor, assuming as before that it is greater than or equal to one (as otherwise we can rearrange and get a bounded factor on the right-hand side of (<ref>)), we arrive at

    | I_2| ≤ C ϵ∫_P u^1+γϕ^α (1+γ)/2≤ C ϵ∫_P u^1+γϕ^2,

where we have used that  ϕ <1 and α(1+γ)/2 > 2. 
Note that here the constant C depends on the choice of P and can be potentially large.
However, we still have the freedom to choose ϵ small enough—as remarked at the beginning of the proof, it suffices to establish an estimate for ϵ sufficiently small.
Thus, comparing the right-hand side of the above inequality with the lower bound (<ref>) for I_0, we conclude that if ϵ_0 is sufficiently small, then for all ϵ≤ϵ_0,

    | I_2| ≤I_0/2.

Together with the estimate for |I_1|, this implies that I = I_0 + I_1 + I_2 is non-negative (or else we can rearrange (<ref>)), which finishes the proof of the lemma. 


Our proof does not work in the case γ = 1. What fails  is the last estimate for I_2, because we cannot set α = 2. Indeed, there is no cut-off function ϕ such that  | ∇ϕ | / ϕ is bounded. However, we can still prove a slightly weaker statement.


  
  If γ = 1, then the statement of Lemma <ref> still holds provided that u satisfies an estimate u _L^2p(X)≤ K ϵ^-1 for some constant K. Apart from the rest of the data, the final bound for u _L^p(Ω) depends also on K. 


Suppose for simplicity that p=1, so that 

    u _L^4(X)≤ Kϵ^-1.

Following the proof of Lemma <ref>, we can obtain a bound for u _L^2(Ω)  The only modification that we have to make is the estimate (<ref>) which now should be

    | I_2 |    = ϵ∫_P u^2 | ∇ϕ |^2 = ϵ∫_P (u^1/2) ( u^3/2ϕ^α)  | ∇ϕ |^2/ϕ^α
       ≤ϵ  u^1/2_L^8(X) | ∇ϕ |^2/ϕ^α_L^8(P)( ∫_p u^2 ϕ^4α/3)^3/4
       ≤ϵ^1/2 K^1/2 | ∇ϕ |^2/ϕ^α_L^8(P)( ∫_P u^2 ϕ^4α/3)^3/4,

where we have used Hölder's inequality with weights (8,8,4/3). Now for α satisfying 2 > α > 3/2, the function | ∇ϕ |^2 / ϕ^α is bounded 4α / 3 > 2, so that (again, assuming that the integral on the right-hand side is greater than one) we obtain

    | I_2 | ≤ C ϵ^1/2∫_P u^2 ϕ^2.

Recall that in the case γ = 1, the positive integral I_0 is bounded below by

    I_0 ≥η∫_P u^2 ϕ^2,

so that for ϵ small enough we have I_2 ≤ I_0 / 2. This leads to a bound for u _L^2(Ω) as before. In the same way we obtain a bound for u _L^p (Ω) 

    u _L^2p(Ω)≤ Kϵ^-1.




We will establish bounds of the form f _W^k,p(Ω)≤ M_k,p  for all k and p by induction over k. Let us start with k=0. Let η > 0 be such that 

    A_1 + ⋯ + A_n ≥ n η,     and     B_1 + ⋯ + B_m ≥ m η,

and set Ω_j = Ω∩{ A_j ≥η}. The subsets { A_j ≥η}_j=1,…, n cover X and therefore Ω_1, …, Ω_n cover Ω. Let u = e^f. For any given j we have

    ϵΔ u    = - ϵ e^f | ∇ f|^2 + ϵ e^f Δ f  
       ≤ u ( - ∑_j=1^n A_j u^α_j + ∑_j=1^n B_j u^- β_j - w ) 
       ≤ -A_j u^1 + α_j + ∑_j=1^n B_j u^1-β_j - w u,

or equivalently,

    ϵΔ u + A_j u^1+ α_j + Q(x,u) ≤ 0,

where Q(x,u) = - ∑_j B_j u^1-β_j + wu. It follows from Lemma <ref> that u _L^p(Ω_j) is bounded by a constant depending only on the fixed data. Since Ω_1, …, Ω_n cover Ω, we obtain a bound for u _L^p(Ω). Similarly, considering the subsets Ω∩{ B_j ≥η} and the function e^-f we find bounds for e^-f_L^p(Ω). Combining them with the inequality

    | f| ≤ e^f + e^-f,

we obtain bounds for f _L^p(Ω).

Suppose that W^k-1,p bounds have been established for some k ≥ 1 and all p. We may assume that they hold on a slightly larger domain containing Ω, which we assume to be all of X to keep the notation simple. First consider the case when k = 2l is even. Consider the function 

    v = Δ^l f =  Δ⋯Δ_l  times f.

Applying Δ^l to both sides of (<ref>) and using the formula

    Δ( gh ) = g Δ h - 2 ⟨∇ g, ∇ h ⟩ + hΔ g,

we inductively show that v satisfies a differential equation of the form

    ϵΔ v + A v + P(e^α_j f, e^- β_j f, ∇ f, … , ∇^2l-1 f ) = 0,

where

    A = ∑_j=1^n α_j A_j e^α_j f + ∑_i=1^m β_j B_j e^- β_j f,

and P is a polynomial function of the functions e^α_j f, e^-β_j f, and the first 2l-1 covariant derivatives of f. Its coefficients depend only on A_j, B_j, w, and their derivatives. In particular, P is a finite sum P = ∑_γ P_γ say, where each term P_γ satisfies an inequality of the form

    | P_γ | ≤ C e^a f e^- b f | ∇ f |^c_1… | ∇^2l-1 f |^c_2l-1

with some exponents a,b,  c_1, …, c_2l-1 and a coefficient C depending only on A_j, B_j, w, and their derivatives. Therefore, by the induction hypothesis and Hölder's inequality, we can bound the L^p norm of P for any p, by a constant depending only on the fixed data and not on f. 

At every point A is bounded below by either ∑_j α_j A_j or ∑_j β_j B_j,  depending on the sign of f. In any case, there is a positive  constant η̃, depending only on the fixed data, such that A ≥η̃. Note that v is raised to the first power in (<ref>). Thus, we are in place to apply Lemma <ref> to obtain a bound for v _L^p(Ω). In order to do so, we need to make sure that v obeys an estimate of the form 

    v _L^2p(Ω)≤ K_p ϵ^-1
 
for some constant K_p. Such an estimate follows from the induction hypothesis and the fact that v' =  Δ^l-1 f satisfies an equation analogous to (<ref>):

    ϵ v = ϵΔ v' = - Av' + P' (e^α_j f, e^-β_j f, ∇ f, …, ∇^2l-3 f ),

where P' is a polynomial function as before. Since the right-hand side depends only on the derivatives of f up to the order 2l-2, we obtain an estimate for ϵ v as required. Thus, Lemma <ref> yields a bound for v _L^p(Ω). Of course, we can as well assume that it holds on a slightly larger domain containing Ω. Then, in view of v = Δ^l f, the elliptic estimate for the Laplacian implies a bound for f _W^2l, p(Ω). This finishes the proof of the induction step in the case k = 2l.

The odd case k = 2l+1 is similar. Assume that the assertion is true for k-1 = 2l. Let v = Δ^l f as before and ψ = | v |^2. By the Bochner formula,

    1/2Δψ   = - | ∇^2 ψ |^2 - Ric( ∇ψ, ∇ψ ) + ⟨∇ψ, ∇ ( Δψ ) ⟩
       ≤Ric_C^0(X) | ∇ψ |^2 +  ⟨∇ψ, ∇ ( Δψ ) ⟩,

where Ric is the Ricci curvature of X. After taking the gradient of (<ref>) and plugging it to the inequality above, we arrive at

    ϵ/2Δψ + (A - ϵRic_C^0(X)) ψ + Q(e^α_j f, e^-β_j f, ∇ f, …, ∇^2l f ) ψ^1/2≤ 0 ,

where Q is a polynomial function of e^α_j, e^- β_j f, and the first 2l derivatives of f. Provided that ϵ is sufficiently small, the function A - ϵRic is bounded below by a positive constant and we can apply Lemma <ref> as before to obtain a bound for ψ_L^p(Ω). Again, by the elliptic estimate for the Laplacian, this yields  W^k,p bounds for f. The statement for general ϵ≤ϵ_0 follows from a scaling argument as described in the proof of Lemma <ref>. 




§ PROOFS OF THE THEOREMS

We prove the theorems in the order of increasing generality.



Since ϵ_i → 0, we may assume that none of the sections φ_i is identically zero. Let 𝒜 be the space of unitary connections on L and  𝒢^c be the complex gauge group of L, that is the space of smooth maps from Σ to ^×. 

[Convergence modulo 𝒢^c] We claim that there are sequences of complex gauge transformations g_i ∈𝒢^c  such that, after passing to a subsequence, g_i(A_i, φ_i) converges in C^∞(Σ) to a pair (A', φ'). The limiting section φ' is not identically zero and satisfies _A'φ' = 0.  

Let us prove this claim. The quotient 𝒜 / 𝒢^c  with the C^∞ topology is homeomorphic to the Jacobian torus H^1(Σ, ) / H^1(Σ, ). In particular, it is compact and there is a sequence of g_i ∈𝒢^c such that, after passing to a subsequence, A_i' = g_i A_i converges in C^∞ to a connection A', say.  After replacing g_i by μ_i g_i, where  μ_i =  g_i φ_i _L^2^-1 , we may assume that g_i φ_i _L^2 = 1 for all i.  Note that the constant gauge transformations μ_i act trivially on the space of connections so that we still have A_i' → A'. The final remark about our choice of g_i is that we will assume them to be purely "imaginary" gauge transformations. Any complex gauge transformation is of the form g = u e^f for a (1) gauge transformation u and real function f Σ→. By incorporating the (1) part into the original sequence (A_i, φ_i, β_i) we may assume that g_i = e^f_i/2 for a smooth function f_i Σ→.  

Set φ_i' = g_i φ_i. The action of 𝒢^c preserves the Cauchy-Riemann equation:

    _A_i'φ_i' = 0.

As a consequence,

    _A'φ_i' _L^2 =  (_A' - _A_i') φ_i' _L^2≤ A' - A_i' _L^∞φ_i' _L^2

Since A_i → A', the sequence of norms _A'φ_i' _L^2 is bounded by a number independent of i. From the elliptic estimate for _A' we conclude that the sequence φ_i' is bounded in W^1,2. Bootstrapping gives us C^k for any k and we can choose a subsequence (denoted for simplicity by the same symbols) that converge in C^∞ to a section φ', say, satisfying

    _A'φ' = 0     and    φ' _L^2 = 1,

which finishes the proof of the claim.


[C^0 estimates]
Let D be the set of zeroes of φ'. Counted with multiplicities, there are exactly d = (L) of them. The next step is to show that the sequence f_i is uniformly bounded on compact subsets of Σ∖ D. First, we compute

    ϵ_i^2 ( 2 i Λ F_A_i')    = ϵ_i^2 ( 2i Λ F_A_i + Δ f_i ) 
       = 1 - | φ_i |^2 + ϵ_i^2 Δ f_i 
       = 1 - e^-f_i | φ_i' |^2 + ϵ_i^2 Δ f_i,

so after rearranging, we obtain the following partial differential equation for f_i:

    ϵ_i^2 Δ f_i    = e^-f_i | φ_i' |^2 - 1 + ϵ_i^2 ( 2 i Λ F_A_i') 
       =   q_i e^-f_i - w_i,

where q_i = | φ_i' |^2 and w_i = 1 - ϵ_i^2 ( 2 i Λ F_A_i'). Now set u_i = e^f_i. Then

    ϵ_i^2 Δ u_i    =  ϵ_i^2 ( - e^f_i | ∇ f_i |^2 +  e^f_iΔ f_i ) 
       ≤ q_i - w_i u_i.

Since w_i → 1 uniformly, for i large enough we have w_i ≥ 1/2. The functions q_i are bounded because φ_i' converges. Thus, the maximum principle yields an upper bound for u_i, and consequently for f_i. On the other hand, we easily compute that 

    ϵ_i^2  Δ |φ_i|^2 + 2 ϵ_i^2 | ∂_A_iφ_i |^2 = |φ_i|^2 ( 1 -|φ_i|^2 ),

which again by the maximum principle shows that |φ_i|^2 ≤ 1 for all i. Since |φ_i|^2  = e^-f_i | φ_i '|^2 and | φ_i' |^2 converges uniformly to | φ' |^2, it follows that f_i is bounded below uniformly on compact subsets of Σ∖ D. 


[Convergence outside D] 
Once the C^0 estimate is established, it follows from equation (<ref>),  Proposition <ref>, and Remark <ref> that the sequence f_i is bounded uniformly with all derivatives on compact subsets of Σ∖ D. Thus, we can choose a subsequence of f_i which converges uniformly with all derivatives on compact subsets of Σ∖ D to a smooth function f Σ∖ D →. Let g = e^f/2 be the corresponding complex gauge transformation. Set (A, φ) = ( g^-1 A', g^-1φ'). The pair is well-defined on Σ∖ D and (A_i, φ_i) → (A, φ) in C^∞_(Σ∖ D. Indeed, we have

    φ_i - φ   = g_i^-1φ_i' - g^-1φ' 
       = g_i^-1φ_i' - g_i^-1φ' + g_i^-1φ' - g^-1φ' 
       = g_i^-1( φ_i' - φ' ) + ( g_i^-1 - g^-1) φ', 
    ,

so the convergence of φ_i' and g_i' guarantees that for any compact subset K ⊂Σ∖ D there are constants M_l,K for l = 0,1, … such that

    φ_i - φ_C^l(K)≤ M_l,K( φ_i' - φ' _C^l(K) +  g_i^-1 - g^-1_C^l(K)).

As the right-hand side converges to zero, we see that φ_i converges to φ in C^l for any l on K. A similar argument shows the convergence of connections.


[The limiting configuration] 
Passing to the limit in equation (<ref>), we see that f Σ∖ D → is given by

    f = log | φ' |^2,

which, by φ = e^-f/2φ', is equivalent to | φ | = 1.  Since we also have _A φ = 0, Lemma <ref> implies that ∇_A φ = 0 and F_A = 0 on Σ∖ D.
 

[Convergence of measures] 
It remains to show that 

    i/2πΛ F_A_i→∑_j = 1^d δ (x_j)

in the sense of measures, or, equivalently, that for any small disc B around x_j, 

    lim_i →∞∫_Bi/2π F_A_i = k,

where k is the  multiplicity of the section φ' at x_j. Choose local coordinates on B together with a unitary trivialisation of L. Then A_i is of the form A_i = d + a_i for a_i ∈Ω^1(B, i) and the curvature is F_A_i = da_i. By Stokes' theorem,

    lim_i →∞∫_Bi/2π F_A_i = lim_i →∞∫_Bi/2π da_i = lim_i →∞∫_∂ Bi/2π a_i = ∫_∂ Bi/2π a,

where a ∈Ω^1(B ∖{ x_j }, i) is the one-form corresponding to the singular connection A = d + a. By Lemma <ref>, the integral on the right-hand side is the degree of the limiting section φ around x_j. Since φ' and φ differ by a non-zero real function on B ∖{ x_j }, their degrees around x_j are the same and equal to the multiplicity of φ' at x_j. 



We now turn to Theorem <ref>. 

 For simplicity, we write _A instead of _BA.




To avoid double upper indices, denote α_i = φ_i^1 and β_i = φ_i^2. Integrating the last equation of (<ref>), we obtain

    α_i _L^2^2 - β_i _L^2^2 = - 2 πϵ_i   L,

which together with

    α_i _L^2^2 + β_i _L^2^2 =  φ_i _L^2^2 = 1

implies that for sufficiently large i neither α_i nor β_i is identically zero. 



[Convergence modulo 𝒢^c] We claim that there are sequences g_i ∈𝒢^c and λ_i > 0 such that after passing to a subsequence (g_i A_i, λ_i g_i α_i, λ_i g_i^-1β_i) converges in C^∞ on Σ to a triple (A', α', β'). The limiting sections α' and β' are not identically zero and satisfy

    _A'α' = 0     and    _A'β' = 0.
 

By the same argument as in the proof of Theorem <ref>, we can find g_i ∈𝒢^c such that, after passing to a subsequence, A_i' = g_i A_i converges in C^∞ to a connection A'. We want to rescale the corresponding sequences of sections g_i α_i and g_i^-1β_i so that they also converge, and that the limiting sections are non-zero. First, by replacing the sequence g_i by μ_i g_i, where μ_i are constant complex gauge transformations given by

    μ_i = √( g_i^-1β_i _L^2/ g_i α_i _L^2),
 
we may assume that g_i α_i _L^2 =  g_i^-1β_i _L^2. After changing the original sequence by real gauge transformations we can assume that g_i = e^f_i/2 for smooth functions f_i Σ→.  

Set  λ_i =  g_i α_i _L^2^-1 and consider the rescaled sequences

    α_i' = λ_i g_i α_i     and    β_i' = λ_i g_i^-1β_i.

We have α_i' _L^2 = β_i' _L^2 = 1 so as in the proof of Theorem <ref> we can choose subsequences (denoted for simplicity by the same symbols) that converge in C^∞ to sections α' and β' say. They are holomorphic with respect to A' and satisfy

    α' _L^2 = lim_i →∞α_i' _L^2 = 1,     β' _L^2 = lim_i →∞β_i' _L^2 = 1.



[An upper bound for λ_i] Next, we show that the sequence λ_i is bounded above. Assume by contradiction that after passing to a subsequence λ_i →∞. Then

    | α_i | | β_i | = λ_i^-2 | α_i' | | β_i' | ≤λ_i^-2sup_iα_i ' _L^∞β_i' _L^∞ .

Since α_i' →α' and β_i' →β' uniformly, we conclude that

    lim_i →∞ | α_i | | β_i | _L^∞ = 0.


We will argue that this cannot happen. Set ψ_i = | β_i |^2 and Q_i = | β_i' |^2. Differentiating  λ_i^2ψ_i = e^f_i Q_i twice, we arrive at

    λ_i^2 ∇ψ_i = e^f_i Q_i ∇ f_i + e^f_i∇ Q_i,

and

    λ_i^2 Δψ_i    = Δ( e^f_i ) Q_i - ⟨∇ ( e^f_i ), ∇ Q_i ⟩ + e^f_iΔ Q_i, 
       = e^f_i Q_i Δ f_i - e^f_i Q_i | ∇ f_i |^2 - e^f_i⟨∇ f_i , ∇ Q_i ⟩+ e^f_iΔ Q_i 
       = e^f_i Q_i Δ f_i - ⟨∇ f_i , e^f_i Q_i ∇ f_i + e^f_i∇ Q_i ⟩ + e^f_iΔ Q_i  
       = e^f_i Q_i Δ f_i - ⟨∇ f_i, λ_i^2 ∇ψ_i ⟩ + e^f_iΔ Q_i.

Let x_i ∈Σ be a global maximum of ψ_i, so that ∇ψ_i(x_i) = 0 and Δψ_i (x_i) ≥ 0. The calculation above implies that

    Q_i (x_i) Δ f_i(x_i) ≥ - Δ Q_i(x_i).

We would like to conclude that Δ f_i is bounded below for all sufficiently large i. Consider i large enough so that Q_i is sufficiently close to Q = | β' |^2 in the C^2 norm. Since β' is holomorphic and not identically zero, it vanishes at finitely many points. Therefore, for any δ > 0 there is a small open neighbourhood, V_δ say, of the zero set of β' such that Q_i(x) ≥δ whenever x ∈Σ∖ V_δ and i is large. It follows that if the sequence x_i is isolated from the zero set of β', then there exists δ > 0 such that 

    Δ f_i(x_i) ≥ - 1/δsup_i Δ Q_i _L^∞.

On the other hand, assume that after passing to a subsequence, x_i → x, where β'(x) = 0. If x is a simple root of β', then in local holomorphic coordinates centred around x we have Q(z) = c |z|^2 + O(|z|^3) for some c > 0 and as a result Δ Q (x) < 0. It follows that Δ Q_i (x_i) < 0 for i sufficiently large and by (<ref>), we must have that Q_i(x_i) ≠ 0 since

    0 ≤λ_i^2 Δψ_i (x_i)  = e^f_i(x_i) Q_i(x_i) Δ f_i(x_i) + e^f_i(x_i)Δ Q_i(x_i).

Thus, dividing both sides by e^f_i(x_i) Q_i(x_i) results in the inequality Δ f_i(x_i) ≥ 0. In the general situation, the section β' vanishes to the order k, say. In this case, locally Q^1/k(z) = c |z|^2 + O(|z|^3) for some c > 0. We can therefore, at least locally, replace ψ_i by ψ_i^1/k (note that ψ_i(x_i) →∞, so ψ_i^1/k is a smooth function around x_i), f_i by f_i / k, and Q_i by Q_i^1/k and repeat the previous argument to obtain Δ f_i(x_i) ≥ 0. 

From the lower bound for Δ f_i(x_i) we easily arrive at a contradiction. Indeed, since A_i' = g_i( A_i), the third equation of (<ref>) can be written in the form

    ϵ_i^2  Λ i F_A_i' = ϵ_i^2 Λ i F_A_i + ϵ_i^2 Δ f_i = - |α_i|^2 + |β_i|^2 + ϵ_i^2 Δ f_i,

or after rearranging,

    | β_i |^2 =  ϵ_i^2 Λ i F_A_i' - ϵ_i^2 Δ f_i + |α_i |^2.

Since Δ f_i(x_i) is bounded below and A_i' converges in C^∞, at x_i we have

    | β_i(x_i) |^2 ≤ C ϵ_i^2 + | α_i(x_i)|^2.

Equation (<ref>) implies that for large i 

    | β_i(x_i) | = β_i (x_i) _L^∞≥ C β_i _L^2 = C(1/2+ πϵ_i  L) ≥C/4 > 0,

so | β_i(x_i) | is separated from zero and (<ref>) forces | α_i(x_i) | to converge to zero. Then (<ref>) yields | β_i(x_i) | → 0, which is a contradiction. This finishes the proof that the sequence λ_i is bounded above.


[Convergence of λ_i] The next step is to show that the sequence λ_i is separated from zero. This follows from the inequality

    λ_i^2 ( | α_i|^2 + |β_i|^2 ) ≥ 2 λ_i^2 | α_i | |β_i| = 2  | λ_i e^f_i/2α_i | | λ_i e^-f_i/2β_i | = 2 | α_i' | | β_i' |.

Integrating over Σ and taking the square root, we obtain

    λ_i ≥√( 2 ∫_Σ |α_i' | | β_i' | )

Since α_i' →α' and β_i' →β' uniformly on Σ and the limiting sections are holomorphic and non-zero, the integral on the right-hand side is bounded below by a positive number. Since λ_i is bounded above and separated from zero, after passing to a subsequence we can assume that λ_i converges to a positive number, λ say. Therefore, 

    e^f_i/2α_i = λ_i^-1α_i' →λ^-1α',


    e^-f_i/2β_i = λ_i^-1β_i' →λ^-1β'

 and after  rescaling α', β' we may assume that λ_i = λ = 1.


[Convergence outside the singular set]
 Let D be the union of the zero sets of α' and β'. Since the sections are holomorphic and not identically zero, D is empty or consists of finitely many points.  We claim that after passing to a subsequence, f_i converges in C^∞_(Σ∖ D). To see this, we translate the third equation of (<ref>) for the triple (A_i, α_i, β_i) into a partial differential equation for f_i:

    ϵ_i^2 Λ i F_A_i'   = ϵ_i^2 Λ i F_A_i + ϵ_i^2 Δ f_i = - |α_i|^2 + |β_i|^2 + ϵ_i^2 Δ f_i 
       = -  e^-f_i | α_i' |^2 +  e^f_i | β_i' |^2 + ϵ_i^2 Δ f_i,

or after rearranging

    ϵ_i^2 Δ f_i - P_i e^-f_i + Q_i e ^f_i - w_i = 0,

where P_i = | α_i' |^2, Q_i = | β_i' |^2, and w_i = ϵ_i^2 Λ i F_A_i'. Note that ϵ_i → 0 and the functions P_i, P_i, and w_i converge in C^∞ to, respectively, P = |α'|^2, Q = | β' |^2, and w = 0. Furthermore, P > 0 and Q > 0 on Σ∖ D.

It follows from Proposition <ref> and Remark <ref> that the sequence f_i and its derivatives are uniformly bounded on every compact subset of Σ∖ D. Therefore, we can choose a subsequence of f_i which converges in C^∞_(Σ∖ D). The limit is a smooth function f Σ∖ D →. Let g = e^f/2 be the corresponding complex gauge transformation. Now set (A, α, β) = ( g^-1( A' ), g^-1α', gβ'). The triple is well-defined on Σ∖ D and (A_i,  α_i, β_i) → (A, α, β) in C^∞_ on Σ∖ D.


[The limiting configuration]
After passing to the limit i →∞ in (<ref>), we get

    - e^-f |α'|^2 + e^f | β'|^2 = 0,

or equivalently e^f = | α'| / |β'|. Therefore, the limiting configuration φ = (α, β) satisfies

    - |α|^2 + |β|^2 = - e^-f | α'|^2 + e^f | β'|^2 = 0.

The remaining equations _A α = 0, _A β = 0, and αβ = 0 are obtained from passing to the limit in the corresponding equations for A_i, α_i, and β_i.  Moreover, 

    | φ |^4 = ( |α|^2 + |β|^2 )^2 = ( e^-f | α' |^2 + e^f | β'|^2 )^2 = 4 |α'|^2 | β'|^2

extends to a smooth function on Σ whose zeroes are the points in D. At a point x ∈ D, the norm | φ | = 2 √( | α' | | β' |) vanishes to the order 

    1/2( ord_x (α') + ord_x (β') )

where ord_x (α') and ord_x (β') are the orders of vanishing of α' and β' at x.

Assume now that E = 2. Let A^2 = A ⊗ A denote the tensor product connection on L^2 = L ⊗ L. We will show that the pair (L^2, A^2) is trivial as a bundle with a connection. It will follow then that A is flat and has holonomy contained in _2. Since E = 2 and αβ = 0,  we have the following exact sequence over Σ∖ D:

    0 r    L^* ⊗ K^-1/2rα    E rβ    L^* ⊗ K^1/2r    0,

which yields a bundle isomorphism over Σ∖ D

    ψ_αβ  L^-2⊗ K^-1/2⊗ K^1/2⟶ E.

Both line bundles E and K^-1/2⊗ K^1/2 are trivial as bundles with connections. Thus, ψ_αβ is a section of L^2 over Σ∖ D which satisfies

    _A^2ψ_αβ = 0,      | ψ_αβ | =  | α | /|β| = 1

By Lemma <ref>, ∇_A^2ψ_αβ = 0 and (L^2, A^2) is trivial as a bundle with a connection.

If E > 2, let F be the complex subbundle of EΣ∖ D spanned by the subspaces α and (β)^⊥, interpreting α and β as in the short exact sequence (<ref>). 
Equip F with the unitary connection B̃ obtained from B using the orthogonal projection π_F  E → F, and let _B̃ = (∇_B̃)^0,1 = π_F _B be the induced holomorphic structure on F.
Note that F ⊂ E is not, in general, a holomorphic subbundle. 
Nevertheless, α and β descend to smooth sections

    α∈Γ(Σ∖ D, F  ⊗  L  ⊗ K^1/2),     β∈Γ(Σ∖ D, F^* ⊗ L^* ⊗ K^1/2) 
    which are nowhere zero and satisfy αβ = 0; we claim that these sections are in fact holomorphic with respect to _B̃. This is clear for α since _B̃Ãα = π_F _BAα = 0. To prove that β is also holomorphic, it is enough to show that for every locally defined section s of F satisfying _B̃ s = 0 we have _A ( β(s)) = 0 (thinking of β as a homomorphism from F to L^* ⊗ K^1/2).
    The condition 0 = _B̃ s = π_F _B s implies that _B s takes values in β and since β E → L^* ⊗ K^1/2 is holomorphic, we have _A(β(s)) = β(_B s) = 0. 
    
    Thus, α and β in (<ref>) are holomorphich with respect to _B̃ and so they fit into the short exact sequence (<ref>) with F in place of E.
    The previous argument shows that there is a nowhere vanishing section ψ_αβ∈Γ(Σ, ∖ D,  F ⊗ L^2) satisfying _BAψ_αβ = 0 and | ψ_αβ| =1. Here F is equipped with a connection obtained from B by the orthogonal projection E → F, and F has the induced connection. Lemma <ref> implies that ∇_BAψ_αβ = 0 and F ⊗ L^2 is trivial over Σ∖ D. In particular, there is a well-defined square root ( F)^1/2 and the connection on ( F)^1/2⊗ L induced from B and A is flat with holonomy contained in _2. 
    [Convergence of measures]
    We claim that if x is a point in D at which the sections α' and β' vanish to the order k = ord_x (α') and l = ord_x (β') respectively, and B is a small disc around x, then
    
        lim_i →∞∫_B   i/2πΛ F_A_i  = k-l/2.
    
    This is proved in the same way as the corresponding statement in Theorem <ref>, except that now Lemma <ref> is applied to the connection A^2 on L^2 and section ψ_αβ (under the assumption that E = 2; in the higher rank case we need to twist L^2 by F). The degree of the restriction of ψ_αβ to ∂ B is
    
        deg( ψ_αβ) = deg( α/ |α|) - deg( β/ |β|) = k - l,
    
    and the factor 1/2 enters because we consider A instead of A^2. This shows that D is the set underlying the divisor
    
        D = ∑_x ( ord_x (α') - ord_x (β') )x
    
    and i Λ F_A = 1/2δ_D as measures. In particular, D has degree 2d.
    
    Our assumptions guarantee that all irreducible solutions to the Seiberg–Witten equations with multiple spinors are gauge-equivalent to configurations pulled back from Σ satisfying (<ref>) with ϵ = 1<cit.>. The bundle L is necessarily pulled back from a bundle over Σ. Let ϵ_i = Ψ_i _L^2^-1 and Ψ_i' = ϵ_i Ψ_i. Then the rescaled sequence (A_i, Ψ_i') satisfies Ψ_i' _L^2 = 1 and equations (<ref>) with ϵ = ϵ_i. The statements about the convergence of (A_i, Ψ_i') and the structure of Z follow now from Theorem <ref>. 
    
    It remains to show the convergence of currents. Every η∈Ω^1(Y) is of the form
    
        η = f (t,x) dt + ξ (t),
    
    where f ∈ C^∞(Y), (t,x) denote the product co-ordinates on S^1 ×Σ, and ξ is an S^1-family of one-forms ξ(t) ∈Ω^1(Σ). The forms F_A_i are pulled back from Σ, so
    
        ∫_Y i/2π F_A_i∧η = ∫_S^1∫_Σi/2π F_A_i f(t,x) dt = ∫_Σi/2π g(x) F_A_i,
    
    where g(x) = ∫_S^1 f(t,x) dt. Passing to the limit i →∞ and using the convergence of measures (i/2π) Λ F_A_i→δ_D / 2 proved in Theorem <ref>, we arrive at
    
        lim_i →∞∫_Y i/2π F_A_i∧η = 1/2∑_k m_k g(x_k) = 1/2∑_k m_k ∫_S^1 f(t,x) dt = 1/2∑_k m_k ∫_S^1 ×{ x_k }η,
    
    which is the equality that we wanted to prove.
    
    It remains to address the statement made in point (3) of the theorem about the weights associated with the connected components of Z.
    If the singular set Z is a disjoint union ⋃_k Z_k of connected one-dimensional submanifolds, the weight prescribed by Taubes to Z_k is the highest number N_k such that |Ψ| vanishes to the order N_k / 2 along Z_k<cit.>. In our case, Z_k = S^1 ×{ x_k }, N_k is an integer, and by point (3) of Theorem <ref>, we have |m_k| ≤ N_k. On the other hand, m_k is the coefficient of the current 1/2 Z and equals the weight prescribed to Z_k by Haydys: the degree of the section of L^2 induced from Ψ restricted to a small loop linking Z_k<cit.>; see step 5 in the proof of Theorem <ref>.
    
    
    
    Finally, we prove Theorem <ref>. The following inequality will be useful.
    
    
    Let a, b, x, and y be positive numbers. Then for all ξ∈ (0, ∞)
        ξ^-a x + ξ^b y ≥ K x^b/a+b y^a/a+b
    
    where K is positive and depends only on a and b. 
    
    Consider the function f  (0, ∞) → (0, ∞) given by 
    
        f(ξ) = ξ^-a x + ξ^b y.
    
    It attains a global minimum at
    
        ξ_0 = ( ax/by)^1/a+b.
    
    and the value of f at ξ_0 is
    
        f(ξ_0) = K x^b/a+b y^a/a+b
    
    with  
    
        K =  (a/b)^b/a+b + (b/a)^a/a+b
      First, order the weights according to their signs:
    
        (k_1, …, k_N) = (a_1, …, a_n, -b_1, …, -b_m),
    
    where n+m = N and a_j  > 0 and b_j > 0. Denote 
    
        α_i = ( α^1_i, …, α^n_i) = (φ^1_i, …, φ^n_i),
    
        β_i = ( β^1_i, …, β^m_i) = (φ^n+1_i, …, φ^N_i).
    
    When all the weights are positive and τ < 0, or when one of the sequences α_i _L^∞ or β_i _L^∞ converges to zero, the proof is the same as for Theorem <ref>. Let us focus on the more difficult case when the weights are of mixed signs and α_i _L^∞, β_i _L^∞ are bounded below by a positive number. 
    
    As in the proof of Theorem <ref>, we find sequences of λ_i > 0 and complex gauge transformations g_i ∈𝒢^c such that the sequence 
    
        (A_i', α_i', β_i') = g_i(A_i, λ_i α_i, λ_i β_i)
    
     converges uniformly with all derivatives on Σ to a limit (A', α', β'). In order to do this, first choose g_i so that g_i (A_i) converges. After changing g_i by constant complex gauge transformations we may assume that g_i (α_i) _L^2 =  g_i ( β_i) _L^2. Indeed, for each i such a constant gauge transformation is given by a number μ > 0 satisfying
    
        ∑_j=1^n μ^a_jα_i^j _L^2^2 - ∑_j=1^mμ^-b_jβ_i^j _L^2^2 = 0.
    
    Since the left-hand side diverges to ∞ when μ→∞, and to - ∞ when μ→ 0, such μ exists. Then we choose the scaling constants λ_i so that
    
        λ_i g_i ( α_i ) _L^2 = λ_i g_i ( β_i) _L^2 = 1,
    
    which, after passing to a subsequence, guarantees the existence of non-zero limiting sections α' = ((α')^1, …, (α')^n) and β' = ((β')^1, …, (β')^m). After changing the original sequence by a sequence of unitary gauge transformations, we may assume that g_i = e^f_i/2 for smooth functions f_i Σ→. 
    
    The next step is to show that λ_i is bounded above and separated from zero. To prove the latter observe that at least one of the sections (α')^i is non-zero. Without loss of generality assume that (α')^1≠ 0. Likewise, we may assume that (β')^1 ≠ 0. Then 
    
        λ_i^2 ( | α_i |^2 + | β_i |^2 ) ≥ | λ_i α_i^1 |^2 + | λ_i β_i^1 |^2 =  e^-a_1 f_i  | (α_i')^1 |^2 + e^ b_1 f_i  | (β_i')^1 |^2.
    
    Applying Lemma <ref> to the right-hand side, integrating over Σ, and using the L^2-bound for φ_i = ( α_i, β_i), we obtain  
    
        C λ_i^2 ≥ K ∫_Σ | (α_i')^1 |^κ_1 | (β_i')^1 |^κ_2
    
    for some positive exponents κ_1 and κ_2 depending on a_1 and b_1. Now the right-hand side of the inequality is bounded below by a positive constant, because of the convergence (α_i')^1 → (α')^1 and (β_i')^1 → (β')^1 and the assumption that the limiting sections are non-zero. This shows that the sequence λ_i is separated from zero. 
    
    An upper bound is found in the same way as in the proof of Theorem <ref>. Let
    
        u_i = ∑_j=1^n a_j | α_i^j |^2     and     v_i = ∑_j=1^m b_j | β_i^j|^2.
    
    Suppose without loss of generality that τ≤ 0. We have, as in (<ref>),
    
        v_i = ϵ_i^2 Λ i F_A_i' - ϵ_i^2 Δ f_i + u_i + τ.
    
    Using the maximum principle, one shows that at the point x_i where v_i attains a global maximum, there is a lower bound for Δ f_i(x_i). As in the situation of Theorem <ref>, if λ_i is unbounded, then  u_i v_i → 0 uniformly. It follows that u_i(x_i) → 0 and passing to the limit i →∞ we arrive at a contradiction v_i(x_i) → 0. This shows that λ_i must be bounded above. Since it is also separated from zero, after passing to a subsequence it may be assumed to be convergent and we may as well assume that λ_i = 1. 
    
    The function f_i satisfy the partial differential equation
    
        ϵ_i ^2 Δ f_i - ∑_j=1^n P_i^j e^- a_j f_i + ∑_j=1^m Q_i^j e^b_j f_i - w_i = 0,
    
    where P_i^j = a_j | g_i (α_i^j) |^2, Q_i^j = b_j | g_i ( β_i^j) |^2, and w_i = ϵ_i^2 i Λ F_g_i(A_i). We have
    
        P_i^j ⟶ P^j,    Q_i^j ⟶ Q^j  ,    w_i ⟶ 0   in  C^∞,
    
    where P^j = a_j | (α')^j|^2 and Q^j = b_j | (β')^j |^2. Let D be the union of the zero sets of the limiting sections α' and β'. Since they are not identically zero, D is a (possibly empty) finite subset of Σ.  Over Σ∖ D we have 
    
        P^1 + … + P^n > 0,     and     Q^1 + … + Q^m > 0.
    
    Therefore, by Proposition <ref> and Remark <ref> we can bound  the the functions f_j and their derivatives on any compact subset of Σ∖ D, and consequently, extract a subsequence converging smoothly to a function f on Σ∖ D. As before, this leads to the smooth convergence of (A_i, α_i, β_i) to e^-f/2(A', α', β') on Σ∖ D.
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