Moduli spaces of PU(2)-monopoles

This paper has been withdrawn by the author. A substantially improved
  version, arXiv:math/9906163, entitled "Moduli spaces of PU(2)-monopoles
  (revised version)" which gives simpler, clearer proofs of the transversality
  results, has been published on arxiv in June 99 (see also the article "Moduli
  spaces of PU(2)-Monopoles", Asian J. Math. Vol. 4, No. 2 (2000), 391-436.)

Abstract:
  The goal of this article was the S^1-equivariant transversality-problem and
the compactification-problem for the moduli spaces of (perturbed)
PU(2)-monopoles. A substantially improved version entitled "Moduli spaces of
PU(2)-monopoles (revised version)" which gives simpler, clearer proofs of the
transversality results, has been published on arxiv in June 99 and appeared in
Asian J. Math, see Moduli spaces of PU(2)-Monopoles, Asian J. Math. Vol. 4, No.
2 (2000), 391-436.

\section{Introduction}

It is known ([OT4], [OT5], [PT2]) that the most natural way to prove the
equivalence between
Donaldson theory and Seiberg-Witten theory is to consider a suitable moduli
space of
"non-abelian monopoles". In [OT5]  it was shown that an $S^1$-quotient of a
moduli space of
quaternionic monopoles should give an homological equivalence between a
fibration over  a
union of Seiberg-Witten moduli spaces  and  a fibration over   certain
$Spin^c$-moduli spaces
  [PT1].

By the same method, but using moduli spaces of $PU(2)$-monopoles instead of
quaternionic
monopoles,  one should be able to express any Donaldson polynomial invariant
  in terms of Seiberg-Witten invariants associated with the
\underbar{twisted} abelian
monopole equations of [OT6]. In  [T1], [T2], we have shown that this idea
can be generalized
  to express   Donaldson-type polynomial invariants associated with higher
symmetry
groups in terms of new Seiberg-Witten-type invariants.

This paper is dedicated to   two important problems  (see  [T1], [T2],
[OT5]) concerning
moduli spaces of $PU(2)$-monopoles, which are crucial for all
further developments.

First we prove an $S^1$-equivariant transversality theorem: we define
perturbations of the
equations which lead to $S^1$-spaces which, for generic choices of the
perturbing parameters,
are smooth, at least outside the "Donaldson locus"  (the vanishing locus of
the projection on the
spinor component) and of the abelian locus.  The abelian subspaces become
smooth, and the
complement  of the zero sections in the fibrations of "normal deformations"
and of "normal
obstructions" of an abelian subspace becomes smooth of the corresponding
expected
dimension.   We show that the  proofs of  the corresponding transversality
theorems for
other versions of moduli spaces of non-abelian connections coupled with
harmonic spinors
([PT1], [PT2]) are incomplete, by indicating counterexamples to one of the
statements on
which these proofs are based. It is interesting to notice that in fact {\it
any}  solution of
the equations in the K\"ahler case gives a counterexample to this statement.

The perturbed moduli spaces we construct
become   smooth excepting in the abelian points, and in the
Donaldson-points. These points are
exceptions to transversality. In general transversality {\it cannot} be
achieved in the
these points by using
$S^1$-equivariant perturbations.

Second, we prove the existence of an "Uhlenbeck compactification" for the
perturbed moduli
space. Note that in   the problem is much more difficult   in the perturbed
case as in the
non-perturbed one. The    terms containing first order  derivatives are no
longer linear,
and the symbol of the linearized  equations has not diagonal form.  It is
also no longer
possible to get an apriori ${\cal C}^0$-bound for the spinor component, but
we are able
to prove that such an apriori bound does exist in a sufficiently small
neighbourhood of
the non-perturbed moduli space in the space of ideal pairs, as soon as the
perturbations
are sufficiently small. We construct  an Uhlenbeck compactification for the
moduli
space of those solutions of the perturbed equations which  belong to this
neighbourhood.

The problem is much simpler for the non-perturbed equations. For instance
on closed manifolds
one gets easily an apriori ${\cal C}^0$-bound of the spinor component, as
in the abelian case.
This shows in particular that the self-dual component of the curvatures of
a sequence of
solutions cannot bubble, which is essential in constructing the Uhlenbeck
compactification.
The corresponding result in the case of perturbed equations is a more
delicate point. On the
other hand, one can prove stronger local regularity properties for
solutions of the
non-perturbed equations. For example   local estimates,  convergence properties of
sequences of solutions with bounded curvature, Removable
Singularities-theorem also hold
without assuming the
$L^{\infty}$-boundedness of the spinor component. This
is why  in section 2   we give  a complete proof in the non-perturbed case,
which can be easily
generalized to other non-abelian Seiberg-Witten equations with larger
symmetry groups
([T1], [T2]).  We follow the same strategy as in the instanton case [DK],
which can be
summarized as follows:
\\
\\
Local estimates --  Regularity --  Removable Singularities --  Compactification.
\\

In section 3.3, dedicated to the Uhlenbeck compactification of the
perturbed moduli space, we
prove only the results which require new arguments.
\\

Note that in order to prove the equivalence between the Donaldson and the
Seiberg-Witten
theories, it remains only to give explicit descriptions of the ends of the
moduli space
towards the Donaldson locus and the the ideal locus. The first problem  is
treated in [T1].
Further results on the   description of the ends, as well as
generalizations to larger symmetry
groups will be the  subject of a future paper.

I thank Professor Ch. Okonek for  valuable suggestions and  discussions, as
well as for   the
careful reading of the manuscript.

\section{$PU(2)$-monopoles}
\subsection{The $Spin^{U(2)}$ group and $Spin^{U(2)}$-structures}

The group $Spin^{U(2)}$ is defined by
$$Spin^{U(2)}:=Spin\times_{{\Bbb Z}_2} U(2)\ .
$$
Using the natural isomorphism $\qmod{U(2)}{{\Bbb Z}_2}\simeq PU(2)\times S^1$, we get
the exact
sequences
$$1\longrightarrow Spin\longrightarrow Spin^{U(2)}\textmap{(\bar\delta,\det)}PU(2)\times S^1\longrightarrow 1
$$
$$1\longrightarrow U(2)\longrightarrow Spin^{U(2)}\stackrel{\pi}{\longrightarrow}  SO\longrightarrow 1
\eqno{(1)}$$
$$1\longrightarrow {\Bbb Z}_2\longrightarrow Spin^{U(2)}\textmap{(\pi,\bar\delta,\det)}
SO\times PU(2)\times S^1
 \longrightarrow 1 \ .
$$

Let $X$ be a compact manifold and $P^u$  a   $Spin^{U(2)}$-bundle  over
$X$. We consider the
following associated bundles
$$\pi(P^u):=P^u\times_\pi SO, \ \bar\delta(P^u):=P^u\times_{\bar\delta} PU(2),\
\det(P^u):=P^u\times_{\det} S^1,
$$
$${\Bbb G}_0:=P^u\times_{Ad\circ\bar\delta} SU(2);\
{\scriptscriptstyle|}\hskip -4pt{\g}_0:=P^u\times_{ad\circ\bar\delta} SU(2)
\ ,
$$
where
$Ad:PU(2)\longrightarrow Aut(SU(2))$,
$ad:PU(2)\longrightarrow so(su(2))$ are induced by the adjoint morphism $SU(2)\longrightarrow
Aut(SU(2))$,
$SU(2)\longrightarrow so(su(2))$.

The group of sections ${\cal G}_0:=\Gamma(X,{\Bbb G}_0)$ can be identified with
the group of
automorphisms of $P^u$ over $\pi(P^u)\times_X\det(P^u)$. After suitable Sobolev
completions it
becomes  a Lie group, whose Lie algebra is the corresponding completion of
$A^0({\scriptscriptstyle|}\hskip -4pt{\g}_0)$.\\

Let $P$ be a $SO$ bundle over $X$. A $Spin^{U(2)}$-structure in $P$ is a
morphism
 $P^u \longrightarrow P$
of type $\pi$, where $P^u$ is a $Spin^{U(2)}$-bundle. Two
$Spin^{U(2)}$-structures
 $P^u\longrightarrow P$,
$P'^u\longrightarrow P$ in $P$ are called equivalent if the bundles  $P^u$, $P'^u$ are
isomorphic
over $P$.
A $Spin^{U(2)}(n)$-structure in an oriented Riemannian 4-manifold $(X,g)$ is a
$Spin^{U(2)}(n)$-structure in the bundle $P_g$ of oriented coframes.
\\

We refer to [T1], [T2] for the following classification result:
\begin{pr} Let $P$ be a principal $SO$-bundle, $\bar P$ a $PU(2)$-bundle, and
$L$ a Hermitian line bundle  over $X$.\\
 i) $P$ admits a $Spin^{U(2)}$-structure $P^u
\rightarrow  P$  with
$$P^u\times_{\bar \delta}PU(2)\simeq\bar P\ ,\ \ P^u\times_{\det}{\Bbb C}\simeq L$$
if and only if $w_2(P)=w_2(\bar P)+\overline c_1(L)$, where $\overline
c_1(L)$ is
the mod 2 reduction of $c_1(L)$ .\\
 ii) If the base $X$ is a compact oriented 4-manifold, then the
map
$$P^u\longmapsto \left([P^u\times_{\bar\delta}
PU(2)],[P^u\times_{\det}{\Bbb C}]\right)$$
 defines a 1-1 correspondence between the set of  isomorphism classes of
 $Spin^{U(2)}$-struc\-tures in  $P$  and   the set of pairs of isomorphism
classes
$([\bar P],[L])$, where
$\bar P$ is a $PU(2)$-bundle and $L$ an $S^1$-bundle with $w_2(
P)=w_2(\bar P)+\overline c_1(L)$. The latter set can be identified with
$$\{(p,c)\in H^4(X,{\Bbb Z})\times H^2(X,{\Bbb Z}) |\ p\equiv (w_2(P)+ \bar c)^2\ {\rm
mod}\ 4\}
$$
\end{pr}
\hfill\vrule height6pt width6pt depth0pt \bigskip

The group $Spin^{U(2)}(4)$ can be written as
$$Spin^{U(2)}(4)=\qmod{SU(2)_+\times
SU(2)_-\times U(2)}{{\Bbb Z}_2}\ ,$$
hence it comes with natural orthogonal representations
$${\rm ad}_{\pm}: Spin^{U(2)}(4)\longrightarrow so(su(2)),
$$
defined by the adjoint representations of $SU(2)_{\pm}$, and with natural
unitary
representations
$$\sigma_{\pm}:Spin^{U(2)}(4)\longrightarrow
U({\Bbb H}_{\pm}\otimes_{\Bbb C} {\Bbb C}^2)
$$
obtained by coupling  the canonical representations of $SU(2)_{\pm}$  with the
canonical representation of $U(2)$.

We denote by ${\rm ad}_{\pm}(P^u)$, $\Sigma^{\pm}(P^u)$ the corresponding associated
vector bundles. The Hermitian 4-bundles $\Sigma^{\pm}(P^u)$ are called the
spinor
 bundles of
$P^u$, and the sections in these bundles are called spinors.\\

We refer to [T2] the following simple  result
\begin{pr} Let $P$ be an $SO(4)$-bundle whose second Stiefel-Whitney class
admits
integral lifts.

There is a 1-1 correspondence between isomorphism classes of
  $Spin^{U(2)}$-structures   in $P$ and   equivalence classes of pairs
consisting of a $Spin^c(4)$-structure $P^c\longrightarrow  P$ in
$P$ and a  $U(2)$-bundle $E$. Two pairs are considered equivalent if, after
tensoring
the first one with a line bundle, they become isomorphic over $P$.
\end{pr}

Suppose that $P^u$ is associated with the pair
$(P^c,E)$, and let $\Sigma^{\pm}$ be the spinor
bundles corresponding to $P^c$. Then  the  associated bundles
$\Sigma^{\pm}(P^u)$,
$\bar\delta(P^u)$,
$\det(P^u)$, ${{\Bbb G}(P^u)}$, ${\Bbb G}_0(P^u)$
  can be expressed in terms of the pair $(P^c,E)$ as follows:
$$\Sigma^{\pm}(P^u)=[\Sigma^{\pm}]^{\vee}\otimes
E=\Sigma^{\pm}\otimes E^{\vee}\otimes[\det(P^u)]    ,\
\bar\delta(P^u)\simeq
\qmod{P_E}{S^1},\ {\rm ad}_{\pm}(P^u)=su(\Sigma^{\pm})$$   $$  \det(P^u)\simeq \det
(P^c)^{-1}\otimes (\det E),\
 {\Bbb G}_0(P^u)=SU(E),\ {\scriptscriptstyle|}\hskip -4pt{\g}_0(P^u)=su(E)\ .
 $$

A $Spin^{U(2)}(4)$-structure $P^u\longrightarrow P$ in a $SO(4)$-bundle $P$ defines a
\underbar{Clifford}
\underbar{map}
$$\gamma:P\times_{SO(4)}{{\Bbb R}^4}\longrightarrow{\rm Hom}_{{\Bbb G}_0}(\Sigma^+(P^u),
\Sigma^-(P^u))\subset
{\rm Hom}_{{\Bbb C}}(\Sigma^+(P^u), \Sigma^-(P^u))
$$
and canonical isomorphisms
$$\Gamma_{\pm}:\Lambda^2_{\pm}(P\times_{SO(4)}{{\Bbb R}^4})\longrightarrow {\rm ad}_{\pm}(P^u)\subset
su(\Sigma^{\pm}(P^u))\ .
$$
\subsection{The $PU(2)$-monopole equations}

Let $\sigma:P^u\longrightarrow P_g$ be a $Spin^{U(2)}(4)$-structure in the oriented compact
 Riemannian
4-manifold $(X,g)$. Fix a connection $a\in{\cal A}(\det(P^u))$. Using the
third exact
 sequence in
(1), we see that the data of a connection $A\in {\cal A}(\bar\delta(P^u))$
is equivalent
to the
data of a connection $B_{A,a}$ in $P^u$ which lifts the Levi-Civita
connection in $P_g$
and the
fixed connection $a$ in $\det(P^u)$  (via the maps $P^u\longrightarrow P_g$ and
$P^u\longrightarrow \det(P^u)$
respectively).  The
Dirac operator ${\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_{A,a}$ associated with the pair $(A,a)$ is the first
order elliptic
operator
$${\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_{A,a}:A^0(\Sigma^{\pm}(P^u))\textmap{\nabla_{B_{A,a}}}
A^1(\Sigma^{\pm}(P^u))\stackrel{\gamma}{\longrightarrow}A^0(\Sigma^{\mp}(P^u))
$$
Regarded as operator
$\Sigma^+(P^u)\oplus\Sigma^-(P^u)\longrightarrow\Sigma^+(P^u)\oplus\Sigma^-(P^u)$, the
Dirac
operator
${\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_{A,a}$ is also selfadjoint.

We define the quadratic map $\mu_{00}: \Sigma^{\pm}(P^u)\longrightarrow
{\rm ad}_+(P^u)\otimes{\scriptscriptstyle|}\hskip -4pt{\g}_0$
by
$$\mu_{00}(\Psi):=pr_{{\rm ad}_+(P^u)\otimes{\scriptscriptstyle|}\hskip -4pt{\g}_0}(\Psi\otimes\bar\Psi) \ ,
$$
where $pr_{{\rm ad}_+(P^u)\otimes{\scriptscriptstyle|}\hskip -4pt{\g}_0}$ denotes the orthogonal  projection
$${\rm Herm}(\Sigma^+(P^u))\longrightarrow{{\rm ad}_+(P^u)\otimes{\scriptscriptstyle|}\hskip -4pt{\g}_0}\ .$$

We introduce now the $PU(2)$-Seiberg-Witten equations $SW^\sigma_a$ associated
to the
pair $(\sigma,a)$, which are equations for a pair $(A,\Psi)$ formed by  a
$PU(2)$-connection $A\in{\cal A}(\bar\delta(P^u))$ and a positive spinor
$\Psi\in
A^0(\Sigma^+(P^u))$:
$$\left\{\begin{array}{ccc}
{\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_{A,a}\Psi&=&0\\
\Gamma(F_A^+)&=&\mu_{00}(\Psi)
\end{array}\right. \eqno{(SW^\sigma_a)}
$$

The natural symmetry group of the equations is the gauge group ${\cal
G}_0:=\Gamma(X,{\Bbb G}_0)$. We denote by ${\cal M}^\sigma_a$ the moduli space
$${\cal M}^\sigma_a:=\qmod{\left[{\cal A}(\bar\delta(P^u))\times
A^0(\Sigma^+(P^u))\right]^{SW^\sigma_a}}{{\cal G}_0} \ ,
$$
where $\left[{\cal A}(\bar\delta(P^u))\times
A^0(\Sigma^+(P^u))\right]^{SW^\sigma_a}$ denote the space of solutions of
the equations
$(SW^\sigma_a)$. Using the well-known Kuranishi method one can endow ${\cal
M}^\sigma_a$ with the structure of a ringed space, which has locally the form
$\qmod{Z(\theta)}{G}$, where $G$ is a closed subgroup of $SU(2)$ acting on
finite
dimensional vector spaces $H^1$, $H^2$, and
$Z(\theta)$ is the real analytic space cut-out by a $G$-equivariant real
analytic map
$H^1\supset U \stackrel{\theta}{\longrightarrow}  H^2\  $
(see [OT5],  [T1], [T2] for details).

\section{The Uhlenbeck Compactification}
\subsection{ Local estimates}

Fix two copies ${\Bbb H}_{\pm}$ of the quaternionic skew-field ${\Bbb H}$ regarded as
right complex
and quaternionic vector spaces. Let
$U$ be a manifold diffeomorphic to the 4-ball and let
$S^{\pm}_0:=U\times{\Bbb H}_{\pm}$ be two trivial $SU(2)$-bundles on $U$. Every linear
isomorphism
$$\gamma:\Lambda^1_U\longrightarrow {\rm Hom}_{{\Bbb H}}(S^+_0,S^-_0)=U\times{\Bbb H}$$
gives a trivialization of $T_U$ and hence defines a metric $g_\gamma$ on
$U$ such that
$\gamma$ becomes the Clifford multiplication of a $Spin(4)$-structure
${\germ c}_\gamma$ in
$(U,g_\gamma)$.

On the other hand,  the Clifford maps
$\Lambda^1_{U}\longrightarrow{\rm Hom}_{{\Bbb H}}(S^+_0,S^-_0)$ corresponding to a  metric $g$  are
well defined only  up to a transformation in ${\cal C}^{\infty}(U,SO(4))$.
Therefore
it is more convenient to
consider variable (linear isomorphic) Clifford maps
$$\gamma:\Lambda^1_U\longrightarrow {\rm Hom}_{{\Bbb H}}(S^+_0,S^-_0)=U\times{\Bbb H}$$
 rather than variable
metrics on  $U$.   In this way we get a simple way of parameterizing the
pairs consisting of
a metric on $U$ and a $Spin(4)$-structure with respect to this metric.  Let
${\cal C}l_U\subset A^0({\rm Hom}(\Lambda^1_U,{\rm Hom}_{{\Bbb H}}(S^+_0,S^-_0)))$ be the
space of
Clifford maps on $U$.

The $Spin(4)$-structure
${\germ c}_\gamma$ will be called the trivial $Spin^c(4)$-structure associated
with $\gamma$.
Similarly, let $E_0=U\times{\Bbb C}^2$ be the trivial Hermitian rank 2-vector
bundle on $U$.
The
$Spin^{U(2)}(4)$-structure associated to $\gamma$ and $E_0$ via the morphism
$SU(2)\times SU(2)\times U(2)\longrightarrow  Spin^{U(2)}(4)$ will be called the trivial
$Spin^{U(2)}(4)$-structure associated to $\gamma$, and will be denoted by
${\germ u}_\gamma$.

If $\sigma:P^u\longrightarrow P_g$ is an arbitrary $Spin^{U(2)}(4)$-structure on a
Riemannian
4-manifold $X$, a (local) trivialisation of $\sigma$ on a contractible open
subset $U$
is by definition an isomorphism between $\sigma|_U$ and a trivial
$Spin^{U(2)}(4)$-structure
on $U$ associated to a Clifford map $\gamma$   with $g_\gamma=g|_U$.  A local
trivialisation  of $\sigma$ trivializes also the spinor bundles and the
determinant bundles
$\bar\delta(P^u)$,   $\det(P^u)$ of $\sigma$.

The essential difference between the anti-self-dual and the     monopole
equations
is that the
latter are  not conformal invariant.  Under a conformal rescaling  of a metric
$g\mapsto\tilde
g=\rho^2 g$ on a 4-manifold $X$, the associated objects change as follows
$$\begin{array}{c}
\tilde{g^*}=\rho^{-2}g^*\ {\rm on\ 1-forms} ;\ \ \
vol_{\tilde g}=\rho^2 vol_g \ ;\ \ s_{\tilde
g}=\rho^{-2}s_g+2\rho^{-2}\Delta \rho\\
\Sigma^{\pm}_{\tilde g}=\Sigma^{\pm}_{g} ({\rm as\ Hermitian\ bundles})\ ,\ \
\tilde\gamma=\rho^{-1}\gamma\ ;\
\
\tilde\Gamma=\rho^{-2}\Gamma\ ;
\
\

{\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_{\tilde g}=\rho^{-\frac{5}{2}}{\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_g\rho^{\frac{3}{2}} \ .

\end{array}
$$
The main point which makes the whole analysis developed for the instanton
case work is
the invariance of the monopole equation under \underbar{constant}
rescalings. This means
precisely that if $(A,\Psi)$,  $\Psi\in A^0(\Sigma^+(P^G))$ is a solution of the
$G$-monopole equations for the metric $g$ and $\rho$ is a constant, then
$(A,\rho^{-1}\Psi)$
is a solution of the monopole equations for $\tilde g=\rho^2 g$.

This simple remark will be applied in the following way:
 %

Let $(\bar B,g_0)$ be the standard closed 4-ball  with interior $B$, and let
$ h_r:\bar B\longrightarrow \bar B_r\subset\bar B$ be  the homothety of slope $r<1$.
 %
Let $(X,g)$ be a  Riemannian 4-manifold,   $x\in X$, and fix an
isometric identification $T_x(X)={\Bbb R}^4$. Consider  the   smooth family of parameterizations
$f_r:\bar B\textmap{\simeq}\bar  D(r)\subset X$,  $f_r=\exp\circ h_r$,
where $\bar D(r)$ is
the compact geodesic ball of radius $r$, $0<r\ll 1$.

Let $\sigma:P^u\longrightarrow P_{g}$ be $Spin^{U(2)}(4)$-structure on $X$ and choose
a trivialisation
of
$\sigma$ in a neighbourhood $U$ of $x$ with Clifford map $\gamma_U$. Fix a
connection
$a\in{\cal A}(\det(P^u))$ which is defined on $U$ by the 1-form $\alpha\in
A^1_U(u(1))$.

\begin{pr}\hfill{\break}
1.  The metrics $g_r'':=\frac{1}{r^2}
f_r^*(g|_{D(r)})$ converge in the ${\cal C}^{\infty}$-topology to the
standard flat metric
$g_0$ on $\bar B$, as $r\rightarrow 0$.  \\
2.  $\gamma_r'':=r f_r^*(\gamma_U)$ is a Clifford map for $(B,g_r'')$
converging
to a Clifford map $\gamma_0''$ for $(B,g_0)$. \\
3. If  $(A,\Psi)$ is a solution of  the $PU(2)$-monopole
 equation for   $(\sigma,a)$, then the pair $(f_r^*(A), rf_r^*(\Psi))$ is a
solution of the  monopole equations associated to the trivial
$Spin^{U(2)}(4)$-structure
${\germ u}_{\gamma_r''}$  and   the connection $\alpha''_r:=f_r^*(\alpha)$.
\end{pr}
{\bf Proof: } Indeed, the maps
$\frac{1}{r}\circ (f_r)_*:T(\bar B)\longrightarrow T(X)$ converge in the ${\cal
C}^{\infty}$-topology
to the projection of $T(\bar B)=\bar B\times T_x(X)$ on the second factor.
\hfill\vrule height6pt width6pt depth0pt \bigskip

\begin{re} \hfill{\break}
 Let $g$ be a  metric on $\bar B$ with
$g|_{T_0}=g_0|_{T_0}$.  The same  assertions hold  for the family of metrics
$g'_r=r^{-2}h_r^*(g|_{B_r})$ and the family  of Clifford maps
$\gamma'_r:=rh_r^*(\gamma|_{B_r})$.\\
\end{re}
\begin{thry} (Gauge-fixing) There are constants
$\varepsilon_1,\ M>0$ such that  the following holds:

Any connection $A$ on the trivial bundle $E_0$ over $\bar B$ with
$\parallel F_A\parallel_{L^2}<\varepsilon_1$ is gauge equivalent to a
connection
$\tilde A$
over $B$ with\\
(i) $d^*_0\tilde A=0$,  where $d^*_0$ is the normal adjoint of $d$ with
respect to the
standard flat metric $g_0$.\\
 (ii) $\lim_{r\rightarrow 1} A_r=0$ on $S^{3}$,\\
(iii) $\parallel\tilde A\parallel_{L^2_1}\leq M\parallel F_{\tilde
A}\parallel_{L^2}$. \\
 The corresponding  gauge transformation is unique up to a constant matrix.
\end{thry}
 {\bf Proof: }  Theorem 2.3.7 in [DK].
\hfill\vrule height6pt width6pt depth0pt \bigskip

In the theorems below   all the Sobolev norms on the ball are computed with
respect to the
standard  flat metric $g_0$.
Fix a Clifford map
$$\gamma_0:\Lambda^1_{\bar B}\longrightarrow {\rm Hom}_{\Bbb H}(\bar
B\times{\Bbb H}_+,\bar B\times{\Bbb H}_-)$$
 compatible with $g_0$.
\begin{thry} (Local estimates)   There is a
positive constant $\varepsilon_2>0$  such that for any Clifford map
$\gamma'$ on $\bar B$ which is sufficiently
${\cal C}^{3}$-close to $\gamma_0$ and every $a'\in A^1(u(1))$ sufficiently
small in the
${\cal C}^2$-topology, the following holds:

For any solution $(A,\Psi)$  of the $PU(2)$-monopole equation for
the pair $({\germ u}_{\gamma'},a')$ over the open ball $B$  satisfying $d^*_0  A=0$,
$\parallel A\parallel_{L^4}\leq\varepsilon_2$, $\parallel \Psi\parallel_{L^4}\leq
\varepsilon_2$, and any interior domain $D{\  {\subset}\hskip -2.2mm{\raisebox{.28ex B$, one has
estimates of the form :
$$\begin{array}{l}\parallel A\parallel _{L^2_l(D)}\leq
C_{D,l,\gamma',a'}(\parallel A\parallel_{L^4}+\parallel\Psi\parallel_{L^4})\ ,\\
\parallel\Psi\parallel_{L^2_l(D)}\leq C_{D,l,\gamma',a'}
(\parallel A\parallel_{L^4}+\parallel\Psi\parallel_{L^4}) \end{array}$$
with  positive  constants  $C_{D,l,\gamma',a'}$ for all $l\geq 1$.
\end{thry}
{\bf Proof: }  First of all we identify the ball with the upper semi-sphere of
$S:=S^4$ and we endow the sphere with a metric $g_s$ which extends the flat
metric $g_0$ on the ball and which has non-negative sectional curvature.

Then we fix  a   $Spin(4)$-structure on the sphere with spinor bundles
$S^{\pm}_s$ given by a Clifford map
$\gamma_s:\Lambda^1_S\longrightarrow {\rm Hom}_{\Bbb H}(S_s^+ , S^-_s)$, and we choose
$SU(2)$-trivializations
$S_0^{\pm}:=\bar B\times{\Bbb H}_{\pm}\textmap{\simeq} S^{\pm}_s|_{\bar B}$  such
that the
Clifford map induced by
$\gamma_s$  in $(\bar B,S^{\pm}_0)$ is
$\gamma_0$. Let $E_s$ be the trivial $U(2)$-bundle on $S^4$.

Let $\gamma:\Lambda^1_S\longrightarrow{\rm Hom}_{\Bbb H}(S_s^+,S^-_s)$, $a\in
\Lambda^1_S(u(1))$ be   extensions of $\gamma'$ and  $a'$ to the
sphere. We can choose $\gamma$ as close to $\gamma_s$ as we please if
$\gamma'$ is sufficiently close to $\gamma_0$ and $a$ as small as we please
if $a'$
is sufficiently small.

Now consider the following first order elliptic operators on the sphere
$$\begin{array}{ccccc}
{\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_{\gamma,a}&:&A^0(S^+_s\otimes E_s)&\longrightarrow&
A^0(S^-_s\otimes E_s)\\
\delta_\gamma:=d_s^*+d_\gamma^+&:&A^1(su(2))&\longrightarrow& A^0(su(2))
\oplus A^2_{+,g_\gamma}\  (su(2))\ ,
\end{array}$$
where the symbol $+$ on the second line means the self-dual component
w.r. t. the metric $g_\gamma$, $d^*_s$ means the adjoint of $d:A^0(su(2))\longrightarrow
A^1(su(2))$ with respect to the fixed   metric $g_s$, and
${\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_{\gamma,a}:={\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_\gamma+\gamma(\frac{a}{2})$. Both operators  have
index 0; they
are injective in the special case $(\gamma,a)=(\gamma_s,0)$, by the
corresponding
Weitzenb\"ock formulas. Since   the coefficients of both operators in local
coordinates are
algebraic expressions in the components of $\gamma$, the components of $a$,
and in the
first order  derivatives of the components of $\gamma$, it follows by elliptic
semicontinuity that the two operators remain injective if $\gamma$ is
sufficiently ${\cal
C}^1$-close to
$\gamma_s$ and $a$ is sufficiently ${\cal C}^0$-small. Therefore they
define maps
$$\begin{array}{l}
(\gamma,a)\longmapsto {\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_{\gamma,a}\in{\rm Iso}\left(L^2_{k+1}(S^+_s\otimes
E_s),L^2_k (S^-_s\otimes E_s)\right),\\
(\gamma,a)\longmapsto \delta_\gamma\in{\rm Iso}\left(L^2_{k+1}(\Lambda^1(su(2))),
L^2_k(\Lambda^0 (su(2))\oplus
\Lambda^0(su(S_s^-)\otimes su(2))\right)
\end{array}
$$
which are continuous with respect the ${\cal C}^{k+1}\times {\cal
C}^k$-topology on the
 space
of pairs formed by a  Clifford map and a $u(1)$-valued 1-form on the sphere.

Therefore we get elliptic estimates
$$\begin{array}{l}
\parallel u\parallel_{L^2_{k+1}}\leq const(k,\gamma,a)\parallel{\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_{\gamma,a}
u\parallel_{L^2_{k}}\\
\parallel v\parallel_{L^2_{k+1}}\leq const'(k,\gamma,a)\parallel
 \delta_\gamma v\parallel_{L^2_{k}}\ ,
\end{array}
\eqno{(el_k)}$$
where $const(k,\gamma,a)$, $const'(k,\gamma,a)$ depend continuously on
$(\gamma,a)$
w.r. t. the ${\cal C}^{k+1}\times {\cal C}^k$-topology. In a sufficiently
small ${\cal
C}^3\times {\cal C}^2$-neighbourhood of $(\gamma_s,0)$ one has
the following estimates with $(\gamma,a)$-independent constants
$$\begin{array}{l}
\parallel u\parallel_{L^2_{k+1}}\leq const\parallel{\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_{\gamma,a}
u\parallel_{L^2_{k}}\\
\parallel v\parallel_{L^2_{k+1}}\leq const\parallel \delta_\gamma v\parallel_{L^2_{k}}\ ,\ \
k\in\{0,1,2\} \ .
\end{array}
\eqno{(el)}$$

The first step is  an input-estimate for the ${L^2_1(D)}$-norms:

Let $\varphi$ be a cut-off function supported in the open ball $B$ which
is identically 1 in a neighbourhood of $\bar D$. Then $\psi:=\varphi\Psi$
extends as section in the spinor bundle $S^+_s\otimes E_s$ and
$\alpha:=\varphi A$ extends as a $su(2)$-valued 1 form on the sphere.
Since $\Psi$ is harmonic with respect to the Dirac operator
$${\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_{\gamma',a',A}={\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_{\gamma',a'}+\gamma'(A)\ ,$$
 we get
$${\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_{\gamma,a}(\varphi\Psi)=\gamma(d\varphi)\Psi+\varphi{\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_{\gamma,a}\Psi=
\gamma(d\varphi)\Psi-\varphi\gamma(A)\Psi \ .\eqno{(1)}$$
Then by $(el)$ we obtain an elliptic estimate of
the form
$$\begin{array}{l}
\parallel\psi\parallel_{L^2_1}\leq
c\parallel{\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_{\gamma,a}\psi_0\parallel_{L^2}
\leq c'(\parallel A\parallel_{L^4}\parallel\psi\parallel_{L^2_1}+\parallel
d\varphi\parallel_{L^4}\parallel\Psi\parallel_{L^4})
\end{array}$$
where we have used on the right the existence of a bounded Sobolev
inclusion $L^2_1\subset L^4$. The constants $c$, $c'$
can be chosen to depend continuously on $(\gamma,a)$. Therefore, for a small
enough  bound for $\parallel A\parallel_{L^4}$,  we get an
estimate of the type
$$\parallel\psi\parallel_{L^2_1}\leq c_D\parallel\Psi\parallel_{L^4} \ ,
$$
where the constant is independent of  $(\gamma,a)$ but depends on $D$ via $\parallel
d\varphi\parallel_{L^4}$.

Since, by hypothesis, the $su(2)$-valued
1-form $A$ on $\bar B$ is a solution in Coulomb gauge of the second
monopole equation, it
satisfies the following non-linear equation
$$\delta_\gamma A+( A\wedge
A)^+_\gamma=\Gamma_{\gamma}^{-1}(\Psi\bar\Psi)_0\ .$$
On the other hand
$d^*_s(\alpha)=\varphi d_s^*A-A(\nabla \varphi)=-A(\nabla\varphi)$, where
$\nabla$ means the gradient with respect to the fixed metric  $g_s$.
Therefore
$$\begin{array}{ccl}\delta_\gamma\alpha&=&- A(\nabla\varphi)+\varphi
d^+_\gamma A+(d\varphi\wedge A)^+_\gamma \\
 &=&-  A(\nabla\varphi)+(d\varphi\wedge A)^+_\gamma
+\varphi\left(\Gamma_\gamma^{-1}(\Psi\bar\Psi)_0-(A\wedge
A)^+_\gamma\right),
\end{array} \eqno{(2)}$$
and one has by $(el)$ an estimate of the form
$$\parallel\alpha\parallel_{L^2_1}\leq c''\left(\parallel A\parallel_{L^4}\parallel
d\varphi\parallel_{L^4}+\parallel\Psi\parallel_{L^4}^2+\parallel A\parallel_{L^4}\parallel\alpha\parallel_{L^2_1}
\right)\ , $$
which gives, for a sufficiently small bound for
$\parallel A\parallel_{L^4}$, an estimate of the type
$$\parallel\alpha\parallel_{L^2_1}\leq c'_D\left(\parallel A\parallel_{L^4}+\parallel\Psi\parallel_{L^4}^2
\right)\ .
$$

Note that so far we did not need  any bound on $\parallel\Psi\parallel_{L^4}$.

In a next step we estimate the $L^2_2$-norms:
Put $\alpha_1=\lambda  A$, $\psi_1:=\lambda\Psi$, where $\lambda$ is
identically 1 on $D$, but the support ${\rm supp}\lambda$ is contained in the
interior of $\varphi^{-1}(1)$. Then we can also
write $\alpha_1=\lambda\alpha$, $\psi_1=\lambda\psi$.

We estimate first the $L^2_1$-norm of the right hand side in the
equation obtained by replacing $\varphi$ with $\lambda$ in (2) . We find
$$\parallel\delta_\gamma\alpha_1\parallel_{L^2_1}\leq const \left(\parallel\lambda
\alpha\otimes \alpha\parallel_{L^2_1}+\parallel\lambda\psi\bar\psi\parallel_{L^2_1}\right)
+const_{D}\parallel\alpha\parallel_{L^2_1}\ ,\eqno{(3)}$$
and again we can assume that the constants do not depend on $\gamma$. Using
the identity
$\nabla(\lambda \alpha\otimes\alpha)=\nabla(\lambda\alpha)\otimes \alpha
+\alpha\otimes\nabla(\lambda\alpha)-\alpha\otimes\nabla\lambda\otimes\alpha$
, where
$\nabla$ denotes the covariant derivative w.r.t. $g_s$, the same
computation as in the proof in Prop. 2.3.8  p. 63 [DK] gives
$$\begin{array}{cl}
\parallel\lambda\alpha\otimes\alpha\parallel_{L^2_1}&\leq\parallel\nabla(\lambda\alpha\otimes
\alpha)\parallel_{L^2} +\parallel\lambda\alpha\otimes\alpha\parallel_{L^2}\leq
const\parallel\nabla(\lambda\alpha)\parallel_{L^4}
\parallel\alpha\parallel_{L^4} \\ &+ const_D\parallel\alpha\parallel _{L^4}^2  \leq
const'\parallel\alpha_1\parallel_{L^4_1}\parallel\alpha\parallel_{L^4}+const_D\parallel\alpha\parallel_{L^4}^2 \ .
\end{array}\eqno{(4)}$$

By a  similar computation one has
$$\parallel\lambda\psi\bar\psi\parallel_{L^2_1}\leq
const\parallel\lambda\psi\parallel_{L^4_1}\parallel\psi\parallel_{L^4}+const_D\parallel\psi\parallel^2_{L^4} \ .
\eqno{(5)}$$

Now we replace $\varphi$ by $\lambda$ in (1) and  estimate the $L^2_1$-norm
of the right
hand side of the obtained equation:
$$\begin{array}{ll}\parallel\gamma(d\lambda)\psi-\lambda\gamma(
\alpha)\psi\parallel_{L^2_1}\leq&  const(\parallel\lambda\psi\parallel_{L^4_1}\parallel\alpha\parallel_{L^4}+
\parallel\lambda\alpha\parallel_{L^4_1}\parallel\psi\parallel_{L^4})+\\
&+const_D(\parallel\psi\parallel_{L^2_1}+\parallel\alpha\parallel_{L^4}\parallel\psi\parallel_{L^4}) \ .
\end{array}\eqno{(6)}$$
By $(el)$ and (1) one has an estimate of the form
$$\parallel\psi_1\parallel_{L^2_2}\leq const\parallel{\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_{\gamma,a}\psi_1\parallel_{L^2_1}=const
\parallel\gamma(d\lambda)\psi-\lambda\gamma(
\alpha)\psi\parallel_{L^2_1} \ .
$$
  By the same method as above, and using  the bounded inclusion
$L^2_2\subset L^4_1$, one gets for a
sufficiently small bound of $\parallel A\parallel_{L^4}$

$$\parallel\psi_1\parallel_{L^2_2}\leq
const\parallel\alpha_1\parallel_{L^2_2}\parallel\Psi\parallel_{L^4}+const_{D}[\parallel\psi\parallel_{L^2_1}+\parallel
A\parallel_{L^4} \parallel\Psi\parallel_{L^4}]\ .\eqno{(7)}$$

The inequalities (5) and (7) yield the estimate
$$\begin{array}{ll}
\parallel\lambda\psi\bar\psi\parallel_{L^2_1}\leq&
const\parallel\alpha_1\parallel_{L^2_2}\parallel\Psi\parallel^2_{L^4}+\\
&+const_{D}[\parallel\psi\parallel_{L^2_1}\parallel\Psi\parallel_{L^4}+
 \parallel A\parallel_{L^4}\parallel\Psi\parallel^2_{L^4}+\parallel\Psi\parallel^2_{L^4}] \ .
\end{array}
\eqno{(8)}$$
In the inequality (3) use  (4) and (8) in order to estimate the first bracket.
The   elliptic estimates $(el)$   for $k=1$ gives

$$\begin{array}{ll}
\parallel\alpha_1\parallel_{L^2_2}\leq& const\parallel\alpha_1\parallel_{L^2_2}(\parallel A\parallel_{L^4}+
\parallel\Psi\parallel^2_{L^4}) + const_{D}
\left[\parallel\psi\parallel_{L^2_1}\parallel\Psi\parallel_{L^4}+\right.\\&
\left.+\parallel\Psi\parallel^2_{L^4}+ \parallel A\parallel_{L^4}\parallel\Psi\parallel^2_{L^4}+\parallel
\alpha\parallel_{L^2_1}+\parallel A\parallel_{L^4}^2\right ] \ .
\end{array}\eqno{(9)}$$
For sufficiently small  bounds for  $\parallel A\parallel_{L^4}$
and $\parallel\Psi\parallel^2_{L^4}$, which are independent of $D$,  one has  (using again the
boundedness of
$L^2_1\subset L^4$ and the obtained inequalities for the $L^2_1$-norms) an
estimate of the type
$$
 \parallel\alpha_1\parallel_{L_2^2}\leq p_D\left(\parallel A\parallel_{L^4},\parallel\Psi\parallel_{L^4}
\right)\ , \eqno{(10)}
$$
for a polynomial $p_D$ without constant term. Then (7) gives  the estimate
$$\parallel\psi_1\parallel_{L^2_2}\leq q_D\left(\parallel A\parallel_{L^4},\parallel\Psi\parallel_{L^4}
\right) \eqno{(10')}$$
for another polynomial $q_D$ with $q_D(0,0)=0$. Choosing
$\varepsilon_2\leq 1$ one can bound of course any such polynomial by a
linear form in $\parallel A\parallel_{L^4},\ \parallel\Psi\parallel_{L^4}$.

The estimates for the third step can be proved by  the same algorithm,
using the existence of
a bounded inclusion $L^2_3\subset L^4_2$.

Since $L^2_3$ is already a Banach algebra, the
estimates for the higher Sobolev norms follow by the usual bootstrapping
procedure using the
inequalities $(el_k)$. Note in particular that we do \underbar{not} have
to take  smaller
bounds for $\parallel A\parallel_{L^4}$ to get estimates of the higher Sobolev norms, so
that a positive number $\varepsilon_2$ (independent of
$l$  and $D$ !) with the required  property   does exist.
\hfill\vrule height6pt width6pt depth0pt \bigskip
\begin{co} (Estimates in terms of the curvature) There exists a constant
$\varepsilon>0$,
and a neighbourhood $N\subset{\cal C}l_B\times A^1_B(u(1))$ of
$(\gamma_0,0)$ with
respect to the
${\cal C}^3\times{\cal C}^2$-topology with the following property:

For  any interior ball
$D{\  {\subset}\hskip -2.2mm{\raisebox{.28ex B$ and any
$l\geq 1$ there exists a  positive constant $C_{D,l,\gamma,a}$ such that
every solution
$(A,\Psi)$ of the $PU(2)$-monopole equations for the pair  $({\germ u}_\gamma,a)$ with
$(\gamma,a)\in N$ and
$\parallel F_A\parallel_{L^2}\leq\varepsilon$, is gauge equivalent on $B$ to a pair $(\tilde
A,\tilde\Psi)$ satisfying the estimates
$$\parallel\tilde A\parallel_{L^2_l(D)}\leq C_{D,l,\gamma,a}\parallel F_A\parallel_{L^2}\ ,\ \ \parallel\tilde
\Psi\parallel_{L^2_l(D)}\leq C_{D,l,\gamma,a}\parallel F_A\parallel_{L^2}^{\frac{1}{2}} \ .
$$
\end{co}
{\bf Proof: } We choose the neighbourhood $N$ such that the conclusion of the above
theorem holds
for
$(\gamma,a)\in N$ and such that, by the "Gauge fixing"-theorem there is
uniform bound
$\parallel  A \parallel_{L^2_1}\leq M   \parallel   F_A \parallel_{L^2}$.  On the other hand, there
exists a
positive constant $M_1$ such that  $\parallel\Gamma_\gamma(F_A^+)\parallel_{L^2}^2 \leq
M_1\parallel
F_A\parallel^2$ uniformly on
$N$. With this choice, the
$L^2$-norm of the
$g_\gamma$-self-dual component of the curvature controls uniformly w.r. t.
$(\gamma,a)\in N$  (the  square of) the $L^4$-norm of the spinor component
of any
solution
of the $PU(2)$-monopole equation for $({\germ u}_\gamma,0)$. Choose now
$$\varepsilon:=\min\left(\varepsilon_1,\frac{\varepsilon_2}{CM},
\left(\frac{\varepsilon_2}{M_1}\right)^{\frac{1}{2}}\right)\ ,$$
where $C$ is a bound for the Sobolev inclusion $L^2_1\subset L^4$.  Then
apply the
"Local estimates" theorem    to get a solution
$(\tilde A,\tilde \Psi)$ on the open ball  satisfying the assumption in the
"Local
estimates"-theorem, and take into account that the $L^4$-norms of $\tilde
A$ and $\tilde
\Psi$ are estimated in terms of the
$L^2$-norm of the curvature.
\hfill\vrule height6pt width6pt depth0pt \bigskip
\begin{co} (Local compactness) There exists a neighbourhood $N$ of
$(\gamma_0,0)$ in
${\cal C}l_B\times A^1_B(u(1))$ with respect to the ${\cal C}^3\times {\cal
C}^2$-topology, and a constant
$\varepsilon>0$ such that the following holds:

For any pair $(\gamma,a)\in N$, and any sequence $(A_n,\Psi_n)$ of
solutions of the
$PU(2)$-monopole equations   for the pair $({\germ u}_\gamma,a)$ on $\bar B$ with $\parallel
F_{A_n}\parallel_{L^2}\leq
\varepsilon$, there is a subsequence
$m_n$ of
${\Bbb N}$ and gauge equivalent solutions $(\tilde A_{m_n},\tilde \Psi_{m_n})$
converging in the
${\cal C}^{\infty}$-topology on the open ball $B$.
\end{co}
\hfill\vrule height6pt width6pt depth0pt \bigskip

We can prove now  the following result, which is analogous to   Proposition
4.4.9 p. 161 [DK].
\begin{co} (Global compactness)  Let $(\Omega,g)$ be an oriented Riemannian
manifold
with a
$Spin^{U(2)}(4)$-structure $\sigma:P^u\longrightarrow  P_g$ and   fix a connection
$a\in{\cal A}(\det(P^u))$.    Let
$(A_n,\Psi_n)$ be a sequence of solutions of the $PU(2)$-monopole equations associated
to $(\sigma,a)$. Suppose that any point $x\in
\Omega$ has a geodesic ball neighbourhood
$D_x$ such that for all large enough $n$,
$$\int_{D_x}| F_{A_n}|^2< \varepsilon^2$$
where $\varepsilon$ is the constant in Corollary 2.1.5 . Then there is a
subsequence
$(m_n)\subset{\Bbb N}$ and gauge transformations $u_{m_n}\in{\cal G}_0$ such that
$(A_{m_n},\Psi_{m_n})^{u_{m_n}}$ converges in the ${\cal
C}^{\infty}$-topology on
$\Omega$.

\end{co}
{\bf Proof: } For any $x\in\Omega$,  denote by $r_x$ the radius of $D_x$, and
consider the
exponential maps $\exp_x:B_0(r_x)\longrightarrow D_x$. Fix (as in Proposition 2.1.1)  a
trivialisation
    of the   $Spin^{U(2)}(4)$-structure $\sigma|_{D_x}$ with Clifford map
$\gamma_x$, and for any    $r<r_x$ consider the parameterization  $f_{x,r} :
B\textmap{\simeq} D_x(r)\subset \Omega$ defined by
$f_{x,r}:=\exp_x\circ h_r$.

By  Proposition 2.1.1 it follows that
the Clifford maps $\gamma_{x,r}:=\frac{1}{r}f_{x,r}^*(\gamma_x)$ converge
to the standard
Clifford map
$\gamma_0$ on the unit ball as $r\rightarrow 0$, so that the corresponding
metrics
$g_{x,r}:=\frac{1}{r^2}f_{x,r}^*(g|_{D_x(r)})$ converge to the standard
flat metric
$g_0$.

Moreover,   the pair
$(f_{x,r}^*(A_n|_{D_x(r)}) ,rf_{x,r}^*\Psi_n)$ is a solution of the
$PU(2)$-monopole equation
on  the unit ball for the pair $({\germ u}_{\gamma_{x,r}},f_{x,r}^*(a))$.  For
every $x\in \Omega$,
let
$\rho_x<r_x$ be sufficiently small such that
$(\gamma_{x,\rho_x},f_{x,r}^*(a))$ belongs to
the  neighbourhood $N$ given by the Corollary 2.1.6

By the conformal invariance of the $L^2$-norm on 2-forms, the curvature of
$(f_{x,\rho_x}^*(A_n|_{D_x(\rho_x)})$ still satisfies the inequality
$$\int\limits_{\bar B}| F_{f_{x,\rho_x}^*(A_n)}|^2 <\varepsilon^2$$
if we use the metric $g_{x,\rho_x}$ to compute the norm.   Since it holds
$\lim\limits_{r\rightarrow 0}g_{x,r}=g_0$, we can choose $\rho_x$ small
enough such
that the inequality still holds if the norm is computed with the metric
$g_0$. Therefore
Corollary 2.1.6 applies to the sequence
$(f_{x,\rho_x}^*(A_n|_{D_x(\rho_x)}) ,\rho_xf_{x,\rho_x}^*\Psi_n)$ and the
claim follows from
standard "patching" arguments (see Corollary 4.4.8 p. 160 [DK]).
\hfill\vrule height6pt width6pt depth0pt \bigskip

\begin{pr} (Global ${\cal C}^0$-boundedness of the spinor) Let $(X,g)$ be a
compact
Riemannian
manifold, $\sigma:P^u\longrightarrow P_g$  a $Spin^{U(2)}(4)$-structure  in $(X,g)$
and $a\in{\cal A}(\det(P^u))$ an abelian connection.

For any solution $(A,\Psi)\in{\cal
A}(\bar\delta(P^u))\times A^0(\Sigma^+(P^u))$ of the $PU(2)$-monopole
equations associated to
$(\sigma,a)$ the following apriori estimate holds:
$$\sup\limits_X|\Psi|^2\leq \max\left(0,
C\sup\limits_X(-\frac{s}{2}+|F_a^+|)\right)
$$
Here $s$ stands for  the scalar curvature of $g$ and $C$ is a universal
positive constant.
\end{pr}
{\bf Proof: } The Weitzenb\"ock formula for coupled Dirac operators gives for any
$(A,a,\Psi)\in
{\cal
A}(\bar\delta(P^u))\times {\cal A}(\det(P^u))\times A^0(\Sigma^+(P^u))$.
$${\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_{A,a}^2\Psi=\nabla_{A,a}^*\nabla_{A,a}\Psi+
\Gamma[(F_A+\frac{1}{2}F_a)^+]\Psi+\frac{s}{4}\ \Psi \cdot
$$

If $(A,\Psi)$ solves the $PU(2)$-monopole equations for $(\sigma,a)$ we get
pointwise
$$(\nabla_{A,a}^*\nabla_{A,a}\Psi,\Psi)+
\left(\mu_{0,0}(\Psi)(\Psi),\Psi\right)+\frac{1}{2}(\Gamma(F_a^+)(\Psi),\Psi)+
\frac{s}{4}|\Psi|^2=0\cdot
$$
By  definition of the quadratic map $\mu_{0,0}$ there exists a positive
universal constant $C'$ with
$$\left(\mu_{0,0}(\Psi)(\Psi),\Psi\right)=|\mu_{0,0}(\Psi)|^2\geq C'|\Psi|^4\ .
$$
Therefore
$$\frac{1}{2}\Delta|\Psi|^2=(\Delta_{A,a}\Psi,\Psi)-|\nabla_{A,a}\Psi|^2\leq
-C'|\Psi|^4+\frac{1}{2}(|F_a^+|-\frac{s}{2})|\Psi|^2 \ ,
$$
and the assertion follows easily by the maximum principle.

\hfill\vrule height6pt width6pt depth0pt \bigskip
\begin{co} If $\Omega$ is compact, the condition "$\int\limits_{D_x}|
F_{A_n}|^2<\varepsilon^2$ for all sufficiently large n"  in Corollary 2.1.7
can be
replaced by the condition
 $$"\int_{D_x}| F_{A_n}^{-}|^2<\frac{\varepsilon^2}{2} \   for\ all\
sufficiently\ large\ n\ ".$$
\end{co}

{\bf Proof: } By the proposition above  the pointwise norm $|F_{A_n}^+|$ of the self-dual
component  of the curvature is  apriori bounded by a constant (depending on
$s$ and $a$),
hence
$\int\limits_{D_x}| F_{A_n}^{+}|^2$ can be made arbitrarily small, by
replacing eventually
$D_x$ with a smaller ball.
\hfill\vrule height6pt width6pt depth0pt \bigskip

Fix as above a   $Spin(4)$-structure
$\gamma_s:\Lambda^1_{S^4}\longrightarrow{\rm Hom}_{{\Bbb H}}(S^+_s,S^-_s)$ on the 4-sphere $S$, such
that the corresponding metric $g_s:=g_{\gamma_s}$ has nonnegative sectional
curvature.

For a connection matrix $C\in A^1 (su(2))$, put as in [DK] sect. 2.3.4
$$Q_l(C):=\parallel
F_C\parallel_{L^{\infty}}+\sum\limits_{i=1}^l\parallel\nabla_C^{(i)}F_C\parallel_{L^2}\ .$$
\begin{pr} (Gauge invariant estimates on the sphere)  There exists
 a positive constant $\eta>0$ such that for any $l\geq 1$, any connection
matrix $C\in
A^1_{S} (su(2))$ satisfying $d_{g_s}^*C=0$, $\parallel C\parallel_{L^4}<\eta$, and any
positive
spinor
$\Psi\in A^0(S^+_s\otimes E_s)$,  the following  estimates hold:\\
(i) $\parallel C\parallel_{L^2_{l+1}}\leq f_l(Q_l(C))$,\\
(ii)  $\parallel\Psi\parallel_{L^2_{j}}\leq
\sum\limits_{i=0}^{j} p_i^l(\parallel C\parallel_{L^2_{l}})\parallel\nabla^{(i)}_C\Psi\parallel_{L^2}$\\
for any $j\leq l+1$, $l\geq 3$. Here $f_l$, $p_i^l$ are
for polynomial functions with  $f_l(0)=0$,   $p_i^l(0)=0$.
In particular one has "gauge invariant estimates" for the Sobolev norms
$\parallel\Psi\parallel_{L^2_{l}}$ in terms of the covariant derivatives
$\nabla_C^{(i)} F_C$,
$\nabla_C^{(i)}\Psi$.
\end{pr}

{\bf Proof: } If $g_s$ is the standard metric on the sphere with positive constant
curvature,
assertion $(i)$  is Lemma 2.3.11 [DK].  But all that is needed in the proof
of this
Lemma is the injectivity of the elliptic first order operator
$\delta =d_{g_s}^*+d^+_{g_s}:A^1 (su(2))\longrightarrow A^0( su(2))\oplus
A^2_+(su(2))$. This holds for
any metric $g_s$ with nonnegative sectional curvature,  by the corresponding
Weitzenb\"ock formula ([FU], p. 111).

The second statement follows since, in the stable range  $l\geq 3$,  $L^2_i$
becomes an
$L^2_l$-Banach module for all
$i\leq k$, so that
$$\parallel \Psi\parallel_{L^2_{i}}\leq
\parallel \Psi\parallel_{L^2_{i-1}}+\parallel\nabla \Psi\parallel_{L^2_{i-1}}
\leq\parallel\nabla_C \Psi \parallel_{L^2_{i-1}}+
(\parallel C\parallel_{L^2_l}+1)\parallel \Psi\parallel_{L^2_{i-1}}\ .$$
From this  assertion $(ii)$ follows by induction.
\hfill\vrule height6pt width6pt depth0pt \bigskip

We can use the cutting-off method as in Theorem 2.1.4 to prove the following
\begin{co}(Gauge invariant estimates on the ball) Let $g$ be a metric on
$\bar B$
sufficiently close to $g_0$. Then there is a constant
$\eta'>0$ such that for  any $l\geq 1$,  $D{\  {\subset}\hskip -2.2mm{\raisebox{.28ex B$, any connection
matrix $C\in
A^1(su(2))$ with
$d_g^*C=0$ , $\parallel C\parallel_{L^4}<\eta'$, and any positive spinor $\Psi\in
A^0(S^+_0\otimes
E_0)$  the following estimates hold:\\
(i) $\parallel C\parallel_{L^2_{l+1}(D)}\leq f_l(Q_l(C))$\ ,\\
(ii) $\parallel\Psi\parallel_{L^2_{j}(D)}\leq \sum\limits_{i=0}^{j}
p_i^l(\parallel C\parallel_{L^2_{l}})\parallel\nabla^{(i)}_C\Psi\parallel_{L^2}\  $\\
for all $j\leq l+1$, $l\geq 3$. Here
$f_l=f_l(D)$, $p_i^l=p_i^{l}(D)$ are polynomial functions with
$f_l(0)=0$, $p_i^l(0)=0$. In particular one has "gauge invariant estimates"
for the norms
$\parallel\Psi\parallel_{L^2_{l}(D)}$ in terms of the covariant derivatives
$\nabla_C^{(i)} F_C$,
$\nabla_C^{(i)}\Psi$.
\end{co}

\subsubsection{Regularity}

The pairs consisting  of a metric on the sphere and a $Spin(4)$-structure
for that metric
are parameterized by linear isomorphic Clifford maps
$$\gamma:\Lambda^1_S\longrightarrow {\rm Hom}_{\Bbb H}(S^+_s,S^-_s)\ .$$

\begin{pr} (Regularity of $L^4$-small
$L^2_1$-almost solutions) Let $\gamma$ be a Clifford map on the sphere
which is sufficiently ${\cal C}^1$-close to
$\gamma_s$,    $a\in A^1(u(1))$   sufficiently ${\cal C}^0$-small,  and
endow the
sphere with the metric
$g:=g_\gamma$   associated with $\gamma$.

There are positive constants
$\alpha$, $\psi$, $\mu$,
$\nu$ such that any  pair
$(A,\Psi)\in L^2_1(\Lambda^1 (su(2)))\times L^2_1(S^+_S\otimes E_s )$
satisfying:\\
(i) $d^*_g(A)=0$,\\
(ii) $\parallel A\parallel_{L^4}\leq \alpha$, $\parallel\Psi\parallel_{L^4}\leq\psi$,\\
(iii) $\parallel \Gamma_\gamma(F_A^+)-\mu_{0,0}(\Psi)\parallel_{L^2}\leq\mu$, $\parallel
{\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_{\gamma,A,a}\Psi\parallel_{L^2}\leq\nu$,\\
(iv)   $ \Gamma_\gamma(F_A^+)-\mu_{0,0}(\Psi)$ and
${\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_{\gamma,A,a}\Psi$ are smooth,\\
is also smooth. Moreover, there are positive constants $c_1$, $c_2$
such that the following estimates hold
$$\begin{array}{l}
\parallel\Psi\parallel_{L^2_1}\leq c_1\parallel{\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_{\gamma,A,a}\Psi\parallel_{L^2}\\
\parallel A\parallel_{L^2_1}\leq
c_2\left(\parallel\Gamma_\gamma(F_A^+)-\mu_{0,0}(\Psi)\parallel_{L^2}+\parallel
{\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_{\gamma,A,a}\Psi\parallel_{L^2}^2\right) \ .
\end{array}$$
\end{pr}

{\bf Proof: } We use the method of continuity as in the proof of 4.4.13 [DK].

First of all note that, by the Weitzenb\"ock formula, the elliptic
operator ${\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_{\gamma_s,0}$ is injective, hence ${\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_{\gamma,a}$  remains
injective when $(\gamma,a)$ is close enough to
$(\gamma_s,0)$ in the ${\cal C}^1\times {\cal C}^0$-topology.   Similarly the
operator $d_g^*+d^+_g$ is injective for $\gamma$ close enough to
$\gamma_s$.

Put $M:=\Gamma_\gamma(F_A^+)-\mu_{0,0}(\Psi)$, $N:= {\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_{\gamma,A,a}\Psi$,
and let $L$
be the pair $L:=(M,N)$. Consider the system $(SW_L)$ of equations  for a
pair $(B,\Phi)\in
A^1(su(2))\times A^0(S^+_s\otimes E_s)$
 given by
$$d_g^*(B)=0\ ,\ \Gamma_\gamma(F_{B}^+)-\mu_{0,0}(\Phi)=M,\ \ \ \
{\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_{\gamma,B,a}\Phi= N \ .\eqno{(SW_L)}
$$

This system can be written as
$$\left\{\begin{array}{lll}
(d_g^*+\Gamma_\gamma\circ d^+_g)(B)&=&M-\Gamma_\gamma((B\wedge
B)^+_g)+\mu_{0,0}(\Phi)\\
{\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_{\gamma,a}(\Phi)&=&N-\gamma(B)\Phi \ .
\end{array}\right.  \eqno{(SW'_L)}
$$

First  we prove that there  are constants $\alpha_0>0$,  $\psi_0>0$  such
that the Sobolev
norms of any solution $(B,\Phi)$ of $(SW_L)$ with
$\parallel B\parallel_{L^4}\leq \alpha_0$, $\parallel\Phi_0\parallel\leq\psi_0$   can be
\underbar{apriori}
\underbar{estimated}  in terms of
 Sobolev norms of $L$.  We start with an input-estimate for the
${L^2_1}$-norms.

The second equation in $(SW'_L)$ gives (by the injectivity and ellipticity
of the operator ${\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_{\gamma,a}$) an estimate of the form
$$\parallel\Phi\parallel_{L^2_1}\leq const(\parallel N\parallel_{L^2}+\parallel
B\parallel_{L^4}\parallel\Phi\parallel_{L^4})\leq const(\parallel N\parallel_{L^2}+\parallel
B\parallel_{L^4}\parallel\Phi\parallel_{L^2_1})$$
using the boundedness  of the Sobolev inclusion
$L^2_1\subset L^4$.
For a sufficiently small bound $\alpha_0$ for $\parallel B\parallel_{L^4}$ we get an
estimate
$$\parallel\Phi\parallel_{L^2_1}\leq const\parallel N\parallel_{L^2} \ .
$$
Using now the first equation,  the same method yields, for a small enough
$\alpha_0$, an
estimate of the form
$$\parallel B\parallel_{L^2_1}\leq const (\parallel M\parallel_{L^2}+\parallel N\parallel_{L^2}^2)\ .$$

We can estimate in a similar way the higher Sobolev norms by applying the
same procedure
as in the proof of the " Local estimates"-theorem  (Theorem 2.1.4). In this
way we obtain, for
sufficiently small bounds $\alpha_0$ of
$\parallel B\parallel_{L^4}$ and $\psi_0$ of $\parallel\Phi\parallel_{L^4}$, bounds for all Sobolev
norms of $B$
and $\Phi$ in terms of  the Sobolev norms of $M$ and $N$.

In a second step we use the continuity method to solve    the family of
equations
$(SW_{t L})$ for $0\leq t\leq 1$.

A solution  $(B,\Phi)$ of $(SW'_{tL})$ satisfies
$$\begin{array}{lllll}

\parallel\Phi\parallel_{L^4}&\leq& const\parallel \Phi\parallel_{L^2_1}&\leq&
const(\parallel N\parallel_{L^2}+\parallel B\parallel_{L^4}\parallel\Phi\parallel_{L^4})\\
\parallel B\parallel_{L^4}&\leq &const\parallel B\parallel_{L^2_1}&\leq& const(\parallel M\parallel_{L^2}+\parallel
B\parallel_{L^4}^2+\parallel\Phi\parallel_{L^4}^2)\ .
\end{array}
$$
Therefore, by the same argument as above, there are constants $\kappa>0$,
$C_1$,
$C_2$ such that for for any $t\in [0,1]$ and any solution
$(B,\Phi)$ of
$(SW_{tL})$ with $\parallel B\parallel_{L^4}<\kappa$ one has
$$\parallel \Phi\parallel_{L^4}\leq C_1\parallel N\parallel_{L^2}\ ,\ \ \ \parallel
B\parallel_{L^4}\leq C_2(\parallel M\parallel_{L^2}+\parallel N\parallel^2_{L^2}) \ .
$$

Now, if $0<\alpha\leq\kappa$, $\psi>0$, $\parallel
N\parallel_{L^2}\leq\frac{\psi}{2C_1}$, and
$\parallel M\parallel_{L^2}+\parallel N\parallel^2_{L^2}\leq\frac{\alpha}{2C_2}$, then we get the
implications
$$\parallel\Phi\parallel_{L^4}<\psi\Rightarrow
\parallel \Phi\parallel_{L^4}\leq\frac{\psi}{2}\ ,\ \  \parallel
B\parallel_{L^4}<\alpha\Rightarrow
\parallel B\parallel_{L^4}\leq\frac{\alpha}{2}\eqno{(I)}$$

Put $\alpha_1:=\min(\alpha_0,\kappa)$. Choose also the $L^2$
bounds $\mu_1$ and $\nu_1$ sufficiently small such that
$\mu_1+\nu^2_1<\frac{\alpha_1}{2C_2}$ and $\nu_1<\frac{\psi_0}{2C_1}$. With
these
choices, the  implications
 $$\parallel\Phi\parallel_{L^4}<\psi_0\Rightarrow\parallel\Phi\parallel_{L^4}<\frac{\psi_0}{2}\ ,\ \
\parallel B\parallel_{L^4}<\alpha_1\Rightarrow\parallel B\parallel_{L^4}\leq\frac{\alpha_1}{2}$$
hold  for pairs solving one of the systems $(SW_{tL}),\ 0\leq t\leq 1$, as
soon as the
$L^4$-norms of $M$ and $N$ are bounded by $\mu_1$ and $\nu_1$.

On the other hand, since $\alpha_1\leq\alpha_0$, this shows that for any
sequence $t_i\rightarrow t_0\in [0,1]$ and any sequence $(B_i,\Phi_i)_i$ of
smooth pairs such that $(B_i,\Phi_i)$ solves $(SW_{t_iL})$ with $\parallel
B_i\parallel_{L^4}<\alpha_1$, $\parallel\Phi_i\parallel_{L^4}<\psi_0$, we have {\sl uniform
apriori
bounds} for all Sobolev norms of the terms. Therefore there exists a
subsequence
$(B_{i_n},\Phi_{i_n})_{i_n}$ which converges in the ${\cal
C}^{\infty}$-topology to a
solution
$(B_0,\Phi_0)$ of $(SW_{t_0L})$ which still satisfies  $\parallel
B_0\parallel_{L^4}<\alpha_1$,
$\parallel\Phi_0\parallel_{L^4}<\psi_0$, since the stronger closed conditions $\parallel
B_0\parallel_{L^4}\leq\frac{\alpha_1}{2}$,
$\parallel\Phi_0\parallel_{L^4}\leq\frac{\psi_0}{2}$ must be satisfied at the limit.

Therefore the set of those $t$ for which there exists a smooth
solution $(B,\Phi)$ of
$(SW_{tL})$ satisfying $\parallel B\parallel_{L^4}<\alpha_1$, $\parallel\Phi\parallel_{L^4}<\psi_0$ is
closed in
$[0,1]$.

We prove now (compare with [DK] - proof of 4.4.13) that for an
appropriate choice of the bounds $\alpha$, $\psi$, the set
$$\{t\in [0,1]|\ (SW_{tL})\ {\rm has\ a\ smooth\ solution}\ (B,\Phi)\ {\rm
with}\ \parallel B\parallel_{L^4}<\alpha, \ \parallel\Phi\parallel_{L^4}<\psi\}\ $$
is also open in $[0,1]$.

Suppose that $(B_0,\Phi_0)$ is a smooth solution of $(SW_{t_0L})$, $0\leq
t_0\leq 1$.

Let $F$ be the map from   $L^2_3(\Lambda^1 (su(2)))\oplus
L^2_3(S_s^+\otimes E_s)\times [0,1]$  to the space
$ \left(L^2_2(\Lambda^0(su(2)))\oplus
L^2_2(\Lambda_{+g} (su(2)))\right)\oplus L^2_2(S_s^-\otimes E_s)$ given by
$$(B,\Phi,t)\stackrel{F}{\longmapsto}\left(d_g^*B+
\Gamma_\gamma((F_{B})^+_g)-\mu_{0,0}(\Phi)-tM,
{\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_{\gamma,B,a}\Phi- tN\right)\ .$$
Define the sesquilinear map
$$m_{0,0}:[S^+_s\otimes E_s]\times [S^+_s\otimes
E_s]\rightarrow
{\rm End}_0(S^+_s)\otimes {\rm End}_0(E_s)$$ by
$$m_{0,0}(\Psi_1,\Psi_2):={\rm Pr}_{[{\rm End}_0(S^+_s)\otimes
{\rm End}_0(E_s)]} [\Psi_1\otimes\bar\Psi_2]\ .$$
 The derivative $\delta_0$ of $F$ in $(B_0,\Phi_0,t_0)$ with respect to the
first two
coordinates  is
$$(\beta,\varphi)\stackrel{\delta_0}\longmapsto\left[
\matrix{d_g^*\beta+\Gamma_\gamma\left[d\beta+
(\beta\wedge B_0)+(B_0\wedge\beta)\right]^+_g
&-m_{0,0}(\varphi,\Phi_0)-m_{0,0}(\Phi_0,\varphi)\cr
\gamma(\beta)\Phi_0& {\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_{\gamma,a}\varphi+\gamma(B_0)\varphi   }\right]
 .$$
The same formula defines an operator   $L^2_1\longrightarrow L^2$ which
  is invertible if the $L^4$ norms of $B_0$ and $\Phi_0$ are sufficiently
small (say
smaller than  $\alpha_2>0$ and $\psi_2>0$)  because the limit of $\delta_0$
in the space of these operators for
$\parallel B_0\parallel_{L^4}\rightarrow 0$, $\parallel
\Phi_0\parallel_{L^4}\rightarrow 0$ is the isomorphism
$$(\beta,\varphi)\longmapsto
\left(\matrix{d_g^*\beta+\Gamma_\gamma(d_g^+\beta)&0\cr 0&
{\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_{\gamma,a}\varphi}\right)\ .$$
But this implies that
$$\delta_0:L^2_3(\Lambda^1 su(2))\oplus L^2_3(S_s^+\otimes E_s)\rightarrow
\left(L^2_2(\Lambda^0su(2))\oplus L^2_2(\Lambda_{+,g} su(2))\right)\oplus
L^2_2(S_s^-\otimes E_s)$$
 is also injective, hence
isomorphism, since it has vanishing index. By the implicit function
theorem, the
equations
$(SW_{tL})$ therefore have a solution in $L^2_3$ for all $t$ in a
neighbourhood of
$t_0$.

We take now $\alpha_2:=\min(\alpha_1,\alpha_2,\kappa)$,  and
$\mu_2$, $\nu_2$ such that $\mu_2+\nu^2_2<\frac{\alpha_2}{2C_2}$ and
$\nu_2<\frac{\min(\psi_0,\psi_2)}{2C_1}$. With this choices the set
$$\{t\in [0,1]|\ SW_{tL}\ {\rm has\ a\ smooth\ solution}\ (B,\Phi)\ {\rm
with}\ \parallel B\parallel_{L^4}<\alpha_2, \ \parallel\Phi\parallel_{L^4}<\psi_2\}$$
is closed (by the implications (I)) and open in $[0,1]$, if $M$ and  $N$
are $L^2$-bounded by
$\mu_2$ and
$\nu_2$ respectively.

Suppose now that we have chosen  constants $\alpha$, $\psi$, $\mu$, $\nu$
in the
theorem to be  smaller than
$\alpha_2$, $\psi_2$, $\mu_2$, $\nu_2$   respectively. Then we have a smooth
solution $(B,\Phi)$ and an   $L^2_1$-solution $(A,\Psi)$ of the same
equations $(SW_{L})$ both satisfying $\parallel B\parallel_{L^4}$,
$\parallel A\parallel_{L^4}<\alpha_2$,
 $\parallel\Phi\parallel_{L^4}$, $\parallel\Psi\parallel_{L^4}<\psi_2$.

Then
$$\left\{ \begin{array}{ll}
(d_g^* +\Gamma_\gamma\circ d^+_g)(A-B)&=
\Gamma_\gamma\left(((B-A)\wedge A)^+_g +(B\wedge(B-A))^+_g\right)+\\
&\ \ \ \ \ \ \ \ \ \ +m_{0,0}((\Phi-\Psi),\Psi)+m_{0,0}(\Phi,(\Phi-\Psi))\\
{\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_{\gamma,a}(\Phi-\Psi)&=\gamma(A-B)\Psi+\gamma(B)(\Psi-\Phi)\ .
\end{array}\right.$$

Using the second equation, we see that for a small enough bound $\alpha_3$
for
$\parallel B\parallel_{L^4}$,  an estimate  of the form
$$\parallel\Phi-\Psi\parallel_{L^4}\leq const\parallel\Phi-\Psi\parallel_{L^2_1}\leq const\parallel
A-B\parallel_{L^4}\parallel\Psi\parallel_{L^4}$$
holds.
But then the first equation gives for a sufficiently small bound
$\alpha_4\leq\alpha_3$ for $\parallel A\parallel_{L^4}$ and $\parallel B\parallel_{L^4}$ ,
an estimate of the form

$$\parallel A-B\parallel_{L^4}\leq const\parallel A-B\parallel_{L^2_1}\leq const
(\parallel\Psi\parallel_{L^4}^2+\parallel\Psi\parallel_{L^4}\parallel\Phi\parallel_{L^4})\parallel A-B\parallel_{L^4}\ ,$$
which  implies $A=B$ and $\Phi=\Psi$ if $\parallel\Phi\parallel_{L^4}$,
$\parallel\Psi\parallel_{L^4}$ are bounded by a sufficiently small constant $\psi_4$.

Defining $\alpha:=\min(\alpha_1,\alpha_2,\alpha_4,\kappa)$,
$\psi:=\min(\psi_0,\psi_2,\psi_4)$ and choosing $\mu$, $\nu$ such that
$$\mu+\nu^2<\frac{\alpha}{2C_2}\ , \ \ \nu<\frac{\psi}{2C_1}\ ,$$ the
assertions of
Proposition 2.1.12 hold.
\hfill\vrule height6pt width6pt depth0pt \bigskip
\subsection{Removable singularities}

The same patching arguments as in [DK] give a generalization of Corollary 2.1.5
(Estimates in
terms of the curvature) to strongly simply connected  Riemannian manifolds. In
 particular one
has:

\begin{lm} Let $(\Omega,g)$ be a strongly simply connected Riemannian 4-manifold
endowed with a $Spin^{U(2)}(4)$-structure $\sigma:P^u\longrightarrow  P_g$ with
trivialized $PU(2)$-bundle $\bar\delta(P^u))$, and let
$a\in {\cal A}(\det(P^u))$ be a fixed connection.

There exists a positive constant
$\varepsilon_\Omega$, and for every precompact interior domain
$\Omega'{\  {\subset}\hskip -2.2mm{\raisebox{.28ex\Omega$ there exists a positive constant
$M_{\Omega,\Omega'}$ such
that any solution $(A,\Psi)$
 of the $PU(2)$-monopole equations $SW^\sigma_a$ with $\parallel
F_A\parallel_{L^2}<\varepsilon_\Omega$ is gauge equivalent  to a solution $(A',\Psi')$
satisfying
$$\parallel A'\parallel_{L^4(\Omega')}<
M_{\Omega,\Omega'}\parallel F_A\parallel_{L^2}\ .$$
\end{lm}
\hfill\vrule height6pt width6pt depth0pt \bigskip

The next lemma is formulated in [DK] for the standard metric (Proposition
2.3.13).  In
Donaldson's approach this result is an intermediate step in proving the
gauge fixing theorem
(Theorem 1.1). The same proof can be given in the case of an arbitrary
metric with
 nonnegative
sectional curvature. Indeed, it suffices to note that Lemmas 2.3.10, 2.3.11
[DK]
need only the
injectivity of the operator $d_{g_s}^*+d^+_{g_s}$ (see Proposition 2.1.10).

\begin{lm} Let $g_s$ be a metric on the sphere $S^4$ which non-negative
sectional
curvature. Then there are constants $\varepsilon_s$, $M_s$ such that any
connection $A$ in
the trivial $SU(2)$-bundle $E_s$ with $\parallel F_A\parallel_{L^2}<\varepsilon_s$ is gauge
equivalent to a connection $\tilde A$ satisfying
$$d_s^*(A)=0\ ,\ \ \parallel \tilde A\parallel_{L^2_1}<M_s\parallel F_A\parallel_{L^2} \ .
$$
\end{lm}
\hfill\vrule height6pt width6pt depth0pt \bigskip

The problem we study now is the following: Given a  monopole $[(A,\Psi)]$
on the
punctured ball $B\setminus\{0\}$, does it extend to $B$ ? The "Removable
singularities"
theorem below asserts that, as in the instanton case, the answer is positive
 if the energy is finite. More precisely,
 $(A,\Psi)$ is gauge equivalent to a solution which extends, when  $\parallel
F_A\parallel_{L^2}^2<\infty$.

We identify
as usual the ball $\bar B$ with the upper hemisphere in order to transfer
the problem to the
compact sphere $S=S^4$.  We denote by
$S^{\bullet}$   the punctured sphere
$S^{\bullet}:=S\setminus\{0\}$, and by
$B^{\bullet}(r):=B(r)\setminus\{0\}$ the punctured
 ball
of radius $r$.

We begin with  estimates for cut-off
solutions on $S$: Suppose   we have a pair $(A,\Psi)$ on the punctured sphere
which solves the monopole equations on a punctured ball $B^{\bullet}(R)$.
The next
lemma asserts that by applying suitable gauge transformations to the pair
$(A,\Psi)$ restricted to small annuli, and then cutting-off  towards the
origin, we get
a family of global pairs on the ball $B(R)$ which are  "almost solutions".

 In the proof of the "Removable Singularities" theorem which
follows below, we shall see that, after modifying the initial pair by
cutting off towards
infinity,  this procedure leads to a
\underbar{converging} sequence of  pairs, and the limit is a solution in a
neighbourhood of
 $0$. By construction,  the limit will be gauge equivalent to the given
pair on a sufficiently
small punctured ball.

Let $\gamma:\Lambda^1_{S}\textmap{\simeq}{\rm Hom}(S^+_s,S^-_s)$
be a  Clifford map on the sphere,  $g$ the corresponding metric, $E_s$ the
trivial $SU(2)$-bundle, and $a\in A^1_{S}(u(1))$ a  connection in the
trivial line
bundle on the sphere. Let $\sigma$ be the corresponding
$Spin^{U(4)}(4)$-structure.
\begin{lm}
 Let
$(A,\Psi)\in {\cal A}( {\Bbb P}(E_s|_{S^{\bullet}}))\times
A^0(S^+_s|_{S^{\bullet}}\otimes E_s|_{S^{\bullet}})$ be a pair on the punctured
sphere whose restriction to the punctured ball $B^{\bullet}(R)$,   solves the
$PU(2)$-monopole equation for
$(\sigma|_{B^{\bullet}(R)} ,a|_{B^{\bullet}(R)})$  such
that $\int\limits_{S^{\bullet}}|F_A|^2< 8\pi^2$.

  For
any sufficiently small positive number
$r$, there exists a pair
$$(A_r,\Psi_r)\in{\cal A}({\Bbb P}(E))\times A^0(S^+_s\otimes
E)$$
   and   gauge transformations
$\rho_r\in SU(E_s|_{S^{\bullet}})$  such that
$$\rho_r^*\left((A,\Psi)|_{S \setminus\bar
B(r)}\right)=(A_r,\Psi_r)|_{S \setminus\bar B(r)}\ ,\eqno{(i)} $$
$$
\parallel  F_{A_r}^{\pm}\parallel_{L^2}\rightarrow \parallel F_A^{\pm}\parallel_{L^2}\ , \ \ \parallel
\Psi_r\parallel_{L^4}\rightarrow
\parallel\Psi\parallel_{L^4}\ ,
\parallel{\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_{A_r,a}\Psi_r\parallel_{L^2}\rightarrow\parallel{\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_{A,a}\Psi\parallel_{L^2},\\
$$
$$ \parallel\Gamma(F_{A_r}^+)-
\mu_{0,0}(\Psi_r)\parallel_{L^2}\rightarrow \parallel\Gamma (F_{A}^+)-
\mu_{0,0}(\Psi)\parallel^2\ \ {\rm as}\  r\rightarrow 0
 \eqno{(ii)}$$
$$\parallel\Gamma(F_{A_r}^+)-
\mu_{0,0}(\Psi_r)\parallel_{L^2(B(R))}\rightarrow
0\ ,\ \ \parallel{\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_{A_r,a}(\Psi_r) \parallel_{L^2(B(R))}\rightarrow 0\  {\sl as}\
r\rightarrow 0\ . \eqno{(iii)}$$
\end{lm}

{\bf Proof: }  The pairs $(A_r,\Psi_r)$ will be obtained in three steps: \\
1. We shall apply
gauge transformations to  the restriction of $(A,\Psi)$ to a small annulus
in order
to get   bounds for  the $L^4$ norms  of the restricted pairs. \\
2. We extend the gauge
transformations obtained in this way to the punctured sphere. \\
3. We cut off the
  pairs obtained in 2. towards the origin and estimate the cut-off pairs.

We use the  notations of [DK].  Let
${\cal N}$,
${\cal N}'$ be the annuli
$${\cal N}:=\{x\in B|\ \frac{1}{2}<|x|<1\}\ ,\ \ {\cal N}':=\{x\in B|\
\frac{4}{6}<|x|<\frac{5}{6}\}  \ . $$
1.  For $0<r\leq \frac{5}{6}$ put also ${\cal N}_r:=\frac{6r}{5}{\cal N}$,
${\cal
N}'_r=\frac{6r}{5}{\cal N}'$.  Note that the constants $\varepsilon_{{\cal
N}_r}$ in Lemma
2.1.13 are bounded below by a positive number $\varepsilon_0$ independent
of $r$, and the
constants
$M_{{\cal N}_r,{\cal N}'_r}$ are bounded  above by a constant $M_0$
independent of $r$.  This
follows by Remark 2.1.2 and by the conformal invariance of the $L^4$-norm
($L^2$-norm)
on 1-forms ( 2-forms).

Let $\varphi$ be  a $[0,1]$-valued cut-off function on $S$ which vanishes
on an open
ball containing the interior boundary of ${\cal N}'$ and is equal to 1 outside a
sphere contained in ${\cal N}'$. Put $\psi:=\varphi^2$,
$\varphi_r:=\varphi(\frac{5}{6r}\ \cdot)$, $\psi_r=\varphi_r^2 $. The cut-off
functions $\varphi_r$ and
$\psi_r$   vanish inside $B(\frac{4r}{5})$ and are identically 1 outside $B(r)$.

By the conformal invariance of the $L^4$-norm on 1-forms and Remark 2.1.2
it follows
that $\parallel d\psi_r\parallel_{L^4}$ converges  to the $L^4$-norm of $d\psi$
computed with the standard flat metric as $r\rightarrow 0$. Therefore there
is a uniform
bound
$$\parallel d\psi_r\parallel_{L^4}\leq c_0\ .$$

Choose now
$r_0$ sufficiently small so that
$\int\limits_{{\cal N}_{r_0}}
|F_{A}|^2<\varepsilon^2_0\leq\varepsilon^2_{{\cal N}_r}$.
 It follows from Lemma 2.1.13    that for all $r\leq r_0$ the exists a gauge
transformation
$f_r$ of $E_s|_{\bar {\cal N}_r}$ such
that the transformed connection $A'_r:=f_r^*(A|_{\bar {\cal N}'_r })$ in
$E_s|_{{\cal
N}'_r}$ satisfies
$$\parallel A'_r\parallel_{L^4}<{\cal M}_{{\cal N}_r,{\cal N}'_r}\parallel F_{A|_{{\cal
N}_r}}\parallel_{L^2}\leq{
M_0} \parallel F_{A|_{{\cal N}_r}}\parallel_{L^2}\ .\eqno{(14)} $$

We can  assume  that $f_r$ is defined on the whole pointed ball
$B(\frac{6r}{5})^{\bullet}$. Since $\pi_3(SU(2))={\Bbb Z}$, there is apriori an
obstruction against the extension of   $f_r|_{{\cal N}'_r}$  towards
infinity to a gauge
transformation on the pointed sphere.

2.  However,  we can always extend   $f_r|_{{\cal N}'_r}$ as an isomorphism of
$SU(2)$-bundles
$\tilde f_r: E_r|_{S^{\bullet}}\longrightarrow E_s|_{S^{\bullet}}$\ , where $E_r$ is a
(possibly
non-trivial) $SU(2)$-bundle on the sphere with a fixed trivialisation over
$S\setminus\{\infty\}$.
 The second Chern class
$c_2(E_r)$ coincides up to sign with the obstruction (which belongs to
$\pi_3(SU(2))$) to
 the extension of $f_r|_{{\cal N}'_r}$.

3. Let $\tilde A'_r:={\tilde f_r}^*(A)$ be the pull back connection. We
cut-off $\tilde
A'_r$ towards the origin by setting
$$A_r:=\psi_r\tilde A'_r \ .$$
Then $A_r$ is a connection in the bundle $E_r$
which is flat on $B(\frac{4r}{5})$ and  coincides with $A'_r$ in a
neighbourhood of the
exterior boundary of
${\cal N}'_r$.   We have
$$F_{A_r}=\psi_r F_{\tilde  A'_r}+d\psi_r\wedge\tilde
A'_r+(\psi^2_r-\psi_r)(\tilde
A'_r\wedge\tilde  A'_r)\ .\eqno{(15)}$$

Therefore, since the last two terms are supported in ${\cal N}'_r$ we get
by (14) the
estimate
$$\parallel F_{A_r}^{\pm}-\psi_r F_{\tilde  A'_r}^{\pm}\parallel_{L^2}\leq c_0 M_0\parallel
F_{ {A}|_{{\cal
N}_r}}\parallel_{L^2} + M^2_0\parallel F_{{A}|_{{\cal N}_r}}\parallel^2_{L^2} \ .
\eqno{(16)}$$
But
$$\parallel F_{A|_{S\setminus B(r)}}^{\pm}\parallel_{L^2}\leq\parallel \psi_r F_{\tilde
A'_r}\parallel_{L^2}^{\pm}\leq\parallel F_{A|_{S\setminus
B(\frac{4r}{5})}}^{\pm}\parallel_{L^2}\ ,$$
 and
the right hand term in (16) tends to 0, as $r\rightarrow 0$, because we
have assumed
$$\parallel F_A \parallel_{L^2}^2< 8 \pi^2 \ .
$$
Therefore
$\lim\limits_{r\rightarrow 0}\parallel F_{A_r}^{\pm}\parallel_{L^2}= \parallel
F_A^{\pm}\parallel_{L^2}$  as claimed. In particular, the second Chern class of
$E_r$ must
vanish if $r$ is sufficiently small, since it is an integer of module bounded by
$\frac{1}{8\pi^2}\parallel F_{A_r}^{\pm}\parallel_{L^2}^2$.  Therefore we may suppose that
$E_r=E_s$, and that $\tilde f_r$ are gauge transformations of $E_s$.

Let $\tilde\Psi'_r:=\tilde f_r^*(\Psi)$ be the pull-back spinor, and
$\Psi_r:=\varphi_r\Psi'_r$ the corresponding cut-off spinor. Note that
outside the ball
$B(r)$, the pair $(A_r,\Psi_r)$ is gauge equivalent to $(A,\Psi)$, so that
the restriction
$(A_r,\Psi_r)|_{B(R)\setminus\bar B(r)}$ solves the  monopole equations.

The assertions $ii)$, $iii)$ follow now easily by similar computations.
\hfill\vrule height6pt width6pt depth0pt \bigskip

Let $\gamma$ be a Clifford map on $\bar B$, $g:=g_{\gamma}$ the associated
metric, and
fix $a\in A^1_B(u(1))$.

\begin{thry} (Removable singularities)  Let $(A_0,\Psi_0)$
be a solution of the $PU(2)$-monopole equations for the pair
$({\germ u}_{\gamma}|_{B^{\bullet}},a|_{B^{\bullet}})$  such that
$$\int_{B^{\bullet}}|F_{A_0}|^2<\infty\ .$$
There exists  a solution
$(\tilde A_0,\tilde\Psi_0)$ of the monopole equation for
$({\germ u}_{\gamma},a)$ and
   a gauge transformation
$\rho\in SU(E_0|_{B^{\bullet}})$ such that $\rho^*(\tilde
A_0,\tilde\Psi_0)=(A_0,\Psi_0)$.
\end{thry}

{\bf Proof: }  We use similar arguments as in the proof of the "Removable
singularities" theorem for
the instanton equation (Theorem 4.4.12 [DK]).

Identify as in the proof of Theorem 2.1.4. the unit ball $\bar B$ with the upper
hemisphere of $S$, fix trivializations of $S^{\pm}_s|_{\bar B}$, and endow
the sphere with
a  $Spin(4)$-structure
$\gamma_s:\Lambda^1_{S}\textmap{\simeq}{\rm Hom}(S_s^+,S^-_s)$
  extending  the standard Clifford map $\gamma_0$ on the flat ball,
and so that the associated metric  $g_s:=g_{\gamma_s}$ has nonnegative sectional
curvature.  Extend the pair $(\gamma,a)$ to the   sphere. Since the problem
is local, we may
suppose that the obtained pair $(\gamma',a')$ is sufficiently close to
$(\gamma_s,0)$, such
that 2.1.12 applies.

The solution $(\tilde A_0,\tilde \Psi_0)$ will be obtained in seven steps :\\ \\
1. Restrict $(A_0,\Psi_0)$ to   ${\cal N}_R$, and apply a gauge
transformation  on the
annulus in order to get (by Lemma 2.1.14) a  solution with a bound of the
$L^4$ norm of the
connection component restricted to ${\cal N}'_R$ in terms of the curvature.
This gauge
transformation can be chosen so that it extends to the ball
$B(\frac{6R}{5})^{\bullet}$
bounded by the exterior boundary of ${\cal N}_R$. In this way we obtain a
pair on
the whole pointed ball $B(\frac{6R}{5})^{\bullet}$  with a bound of the
connection
component restricted to ${\cal N}'_R$.
 \\ \\
2. Cut  the obtained solution  towards $\infty$
using   cut-off   functions of the form $\lambda_R:=1-\varphi_R$ for the
connection
component and $\lambda_R^2$ for the spinor component.   This yields a pair
$(C_R,\Phi_R)$ which extends to the whole pointed sphere $S^{\bullet}$  and
vanishes on
$S^{\bullet}\setminus B(R)$.

Computations similar to the ones in
Lemma 2.15 show that
$$\begin{array}{c}
\parallel F_{C_R}\parallel_{L^2}\rightarrow 0 , \ \parallel\Phi_R\parallel_{L^4}\rightarrow 0 \
,\phantom{i_{i_{i_{i_{i_{i_i}}}}}}\\
\parallel\Gamma ( F_{C_R})^+ -\mu_{0,0}(\Phi_R)\parallel_{L^2}\rightarrow 0 , \
\parallel{\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_{C_R,a}\Phi_R\parallel_{L^2}\rightarrow 0
\end{array}\eqno{(19)}$$
as $R\rightarrow 0$.\\ \\
3. Now fix a sufficiently small $R>0$, and   apply Lemma 2.1.15 to the pair
$(C_R,\Phi_R)$.  Denote by $(C_{R,r},\Phi_{R,r})$, $r\ll R$ the family of
pairs given by this
lemma.\\
\\
4. Once $R$ is sufficiently small,
the $L^2$- norm of the curvature  $F_{C_{R,r}}$ is smaller than the constant
$\varepsilon_s$ in the gauge fixing theorem for the sphere ( Lemma 2.1.14.)
Therefore $C_{R,r}$ is gauge equivalent
on $S$ to a connection $\tilde C_{R,r}$  satisfying
$$d_{g_s}^*(C_{R,r})=0\ ,\ \ \parallel \tilde C_{R,r}\parallel_{L^2_1}< M_s\parallel
F_{C_{R,r}}\parallel_{L^2}\ .
\eqno{(20)}
$$
Let $\tilde\Phi_{R,r}$ be the spinor obtained by applying the same gauge
transformation to $\Phi_{R,r}$. Since the
$L^2_1$ norm of 1-forms controls the
$L^4$-norm, we can assume, by choosing a sufficiently small $R$ and using
(ii), 2.1.15 and
(19), that we have uniform (with respect to $r$) estimates
$$\begin{array}{c}
\parallel \tilde C_{R,r}\parallel_{L^4}<\min(\eta ,\alpha) ,\ \
\parallel\tilde\Phi_{R,r}\parallel_{L^4}<\psi,\\
\parallel\Gamma((F_{\tilde C_{R,r} })^+)
-\mu_{0,0}(\tilde\Phi_{R,r})\parallel_{L^2}\leq \mu, \ \
\parallel{\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_{[\tilde C_{R,r}],a'}\tilde\Phi_{R,r}\parallel_{L^2}\leq \nu\ .
\end{array}\eqno{(21)}$$
Here $\eta$ is the constant given by    Corollary  2.1.11,  and $\alpha$,
$\psi$, $\mu$
and $\nu$ are the constants given by   the "Regularity" theorem   2.1.12. Now
we fix
$R$ with all these properties and let
$r$ tend to 0.\\ \\
5. For a decreasing sequence $r_i\rightarrow
0$, the sequence $(\tilde C_{R,r_i},\tilde\Phi_{R,r_i})|_{S^{\bullet}}$
converges in the
Fr\'echet topology to a smooth pair $(C',\Phi')$ .\\

Indeed,   by (21) we have $\parallel \tilde C_{R,r}\parallel_{L^4}< \eta$, and by (20)
$d_s^*(\tilde C_{R,r}|_B)=0$ so we can make use of  Corollary
2.1.11.  By the scale invariance of the
$L^4$-norm and  Remark 2.1.2, the conclusion on this corollary holds for
any  sufficiently
small  ball $D\subset B^{\bullet}\subset S^{\bullet}$ ).

But once $D\subset B^{\bullet}$ is fixed, the pair
$(\tilde C_{R,r}|_D,\tilde\Phi_{R,r}|_D)$ becomes gauge equivalent to the
fixed pair
$(C_R,\Phi_R)$, if $r$ is sufficiently small (use Lemma 2.1.15, (i) ).
Therefore the
numbers $Q_l(\tilde C_{R,r})$, $\parallel\nabla_{\tilde
C_{R,r}}^{(i)}\tilde\Phi_{R,r}|_D\parallel_{L^2}$ occurring in Corollary 2.1.11
are uniformly
bounded, by the gauge invariance property of these numbers.

Now  2.1.11 gives bounds   for all Sobolev norms of the restrictions $\tilde
C_{R,r}|_D$, $\tilde\Phi_{R,r}|_D$.  Therefore  we can find a decreasing
convergent
sequence $r_i\rightarrow 0$ of positive numbers, such that $(\tilde
C_{R,r_i},\tilde\Phi_{R,r_i})|_{S^{\bullet}}$ converges in the Fr\'echet ${\cal
C}^{\infty}$-topology on the punctured  sphere to a pair
$(C',\Phi')$. \\ \\
6. This pair $(C',\Phi')$ has an   $L^2_1$ extension  to the whole sphere.

Indeed, the  $L^2_1$-estimates given by the "Regularity" theorem
 2.1.12 give  a uniform bound for the $L^2_1$-norms of $(\tilde
C_{R,r},\tilde\Phi_{R,r})$.
Therefore  we can   suppose,  after taking a subsequence if necessary,  that
$(\tilde C_{R,r_i},\tilde\Phi_{R,r_i})$ converges weakly  in $L^2_1$ to an
$L^2_1$-pair $(\tilde C,\tilde\Phi)$.\\ \\
7. The pair $(\tilde C,\tilde\Phi)$ is  smooth.

Indeed, the restriction to the punctured sphere
coincides with $(C',\Phi')$, hence this restriction must be smooth.

By the semicontinuity of the norm with respect to the  weak topology in
reflexive Banach
spaces [La]
it follows that the obtained pair  $(\tilde C,\tilde\Phi)$ still satisfies
the inequalities
(20). Moreover, the
$L^2$-sections $\Gamma(F_{\tilde C }^+)
-\mu_{0,0}(\tilde\Phi)$, ${\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_{\tilde C,a'}\tilde\Phi$ are
bounded by the constants $\mu$, $\nu$ and are smooth on the whole sphere,
because they
are smooth on $S^{\bullet}$ (  where by 5. the
convergence is in ${\cal C}^{\infty}$) and they vanish in $B(\frac{4R}{5})$
by Lemma
2.1.15 $(iii)$. Therefore, by the "Regularity" theorem 2.1.12  it follows
that $(\tilde
C,\tilde\Phi)$ is smooth. Its restriction to the ball
$B(\frac{4R}{5})$ solves the monopole equations and  its restriction to the
punctured ball
is gauge equivalent to the given pair.
\hfill\vrule height6pt width6pt depth0pt \bigskip

\subsection{Compactified moduli spaces}

Let $(X,g)$ be a closed oriented Riemannian 4-manifold. Fix a line bundle
$L$ of Chern class
$c$ on $X$. By Proposition 1.1.10, the data of a
$Spin^{U(2)}(4)$-structure  $P^u\longrightarrow P_g$ on $X$ with $\det(P^u)\simeq L$ is
equivalent via the map $\bar\delta$ to the data of $PU(2)$-bundle $\bar P$
whose Pontrjagin
class satisfies
$$p_1(\bar P)\equiv (w_2(X)+\bar c)^2\  {\rm mod}\ 4 \ .
$$

For a $Spin^{U(2)}(4)$-structure $\sigma$ and a  connection $a\in{\cal
A}(L)$, denote
by ${\cal M}^\sigma_a$ the moduli space of pairs $(A,\Psi)\in{\cal
A}(\bar\delta(P^u))\times
A^0(\Sigma^+(P^G))$  solving the $PU(2)$-monopole equations $SW^\sigma_a$
(see sect. 1.2.2,
1.2.4).
\begin{dt} Fix a $Spin^{U(2)}(4)$-structure $\sigma:P^u\longrightarrow P_g$ in $(X,g)$ with
$\det(P^u)=L$ and a connection $a\in{\cal A}(L)$. An ideal
$PU(2)$-monopole of type $(\sigma,a)$ is a pair
$([A',\Psi'],\{x_1,\dots,x_l\})$ consisting of
an element $\{x_1,\dots,x_l\})$    in the $l$-th-symmetric power $S^l(X)$ of
$X$, and  a monopole $[A',\Psi']\in{\cal M}^{\sigma'}_a$, where
$\sigma':P'^u\longrightarrow P_g$ is the
$Spin^{U(2)}(4)$-structure on $X$ characterized by
$$\det (P'^u)=L\ ,\ \
l=\frac{1}{4}\left(p_1(\bar\delta(P'^u))-(p_1(\bar\delta(P^u))\right) \ .
$$
\end{dt}

We denote by  $I{\cal M}^\sigma_a$ the space of ideal monopoles of type
$(\sigma,a)$.

 Let $\delta_{x}$ be  the Dirac measure associated with a point $x\in X$.
\begin{lm}  The map $F:I{\cal
M}^\sigma_a\longrightarrow [{\cal C}^0(X,{\Bbb R})]^*$, defined  by
$$F([A',\Psi'],\{x_1,\dots,x_l\})=|F_{A'}|^2+8\pi^2\sum\limits_{i=1}^l\delta
_{x_i} \ ,
$$
is bounded with respect to the strong topology in the dual space $ [{\cal
C}^0(X,{\Bbb R})]^*$.
\end{lm}
{\bf Proof: } Let $\varphi\in{\cal C}^0(X,{\Bbb R})$ with $\sup\limits_X|\varphi|\leq 1$. Then
$$\begin{array}{ll}|\langle
F([A',\Psi'],\{x_1,\dots,x_l\}),\varphi\rangle|\leq& \left[\parallel
F_{A'}^-\parallel_{L^2}^2-
\parallel F_{A'}^+\parallel_{L^2}^2\right] +2\parallel F_{A'}^+\parallel_{L^2}^2 +8\pi^2 l
 \\ \\
&=-2\pi^2 p_1(\bar\delta(P^u))+2 C\parallel \Psi'\parallel_{L^4}^4\ ,
\end{array}$$
where $C$ is a universal positive constant. The assertion follows from the
${\cal
C}^0$-boundedness of the spinor proved in Proposition 2.1.8.
\hfill\vrule height6pt width6pt depth0pt \bigskip

Let ${\germ m}'=([A',\Psi'],s')$ be an ideal monopole of type $(\sigma,a)$ with
$s'\in S^l(X)$ and
$[A',\Psi']\in{\cal M}^{\sigma'}_a$. For a positive number $\varepsilon$ we
define
$U({\germ m}',\varepsilon)$ to be the set of ideal monopoles ${\germ m}''=([A'',\Psi''],s'')$ of type
$(\sigma,a)$ satisfying $s''\subset s'$, which have the following property:

  Denoting by $\sigma':P'^u\longrightarrow  P_g$, $\sigma'':P''^u\longrightarrow  P_g$ the
$Spin^{U(2)}(4)$-structures corresponding to $(A',\Psi')$,  $(A'',\Psi'')$,
there exists an
isomorphism of $Spin^{U(2)}(4)$-structures
$\varphi:P''^u|_{X\setminus s'}\longrightarrow P'^u|_{X\setminus s'}$ such that

$$d_1(\varphi^*(A',\Psi'),(A'',\Psi''))<\varepsilon\ ,$$
 where $d_1$ is a metric defining the
Fr\'echet ${\cal C}^{\infty}$-topology in   ${\cal
A}(\bar\delta(P''^u|_{X\setminus
s'}))\times A^0(\Sigma^+(P''^u|_{X\setminus s'})))$.

Let $M>0$ be a bound  for the map $F$ defined above. The weak topology in
the ball of radius
$M$ in
$[{\cal C}^0(X,{\Bbb R})]^*$ is metrisable (see [La], Theorem 9.4.2). Let $d_2$
be a metric
defining this topology.

We endow    the set $I{\cal M}^\sigma_a$  with a metric topology by taking
as basis of
open neighbourhoods for  an ideal monopole ${\germ m}'$ the sets
$U({\germ m}',\varepsilon)\cap
F^{-1}(B_{d_2}(F({\germ m}'),\varepsilon))$.

\begin{thry}
With respect to the metric topology defined above the moduli space  $ {\cal
M}^\sigma_a\subset I{\cal M}^\sigma_a$  is an open subspace with compact closure
$\overline{{\cal M}^\sigma_a}$.
\end{thry}
{\bf Proof: } The first assertion is obvious. For the second, we use the same
argument as in the
instanton case, but we make use in an essential way of the ${\cal
C}^0$-boundedness of the
spinor:

Let ${\germ m}_n$ a sequence of ideal monopoles. It is easy to see that we can
reduce the general
case to the case  where ${\germ m}_n=[A_n,\Psi_n]\in {\cal M}^\sigma_a$. By Lemma
2.1.18, the
sequence of measures $\mu_n:=F({\germ m}_n)$ is bounded, so  after replacing
${\germ m}_n$ by a
subsequence, if necessary,  it converges weakly to a (positive) measure
$\mu$ of total
volume
$\mu(1)\leq M$. The set
$$S_\varepsilon:=\{x\in X|\exists n\in{\Bbb N}\ \forall m\geq n\  (\mu_m(D)\geq
\varepsilon^2\ {\rm for\ every}\ {\rm geodesic\ ball}\ D\ni x)\} $$
contains at most $\frac{M}{\varepsilon^2}$ points, so it is finite for
every positive number
$\varepsilon$. Choosing the constant $\varepsilon$ provided by  the   "Global
compactness" theorem  (Corollary 2.1.7), it follows by a standard diagonal
procedure that
there exists a subsequence $({\germ m}_{n_m})_m$ and gauge transformations
$f_m$ on $X\setminus S_\varepsilon$, such that  $f_m^*({\germ m}_{n_m})$
converges to a
 solution
$(A_0,\Psi_0)$ of the  monopole equations $SW^{\sigma}_a$ restricted   to
$X\setminus
S_\varepsilon$. By the "Removable Singularities" theorem, we can extend
this solution to a
global solution $(\tilde A_0,\tilde \Psi_0)$ of the monopole equations
$SW^{\sigma'}_a$  for a
suitable $Spin^{U(2)}(4)$-structure
$\sigma':P'^u\longrightarrow P_g$ with $\det (P'^u)=L$. We have
$$ |F_{\tilde A_0}|^2 =|F_{A_0}|^2=\mu-8\pi^2\sum_{x\in
S_\varepsilon}\lambda_x\delta_x
$$
with positive numbers $\lambda_x$. It remains to prove that the $\lambda_x$
are integers.
 Since $F({\germ m}_{n_m})\rightarrow\mu$, we have for small enough $r>0$
$$\lambda_x=\lim\limits_{m\rightarrow\infty}\frac{1}{8\pi^2}
\int\limits_{B(x,r)}|F_{A_{n_m}}|^2-|F_{\tilde A_0}|^2=
$$
$$\lim\limits_{m\rightarrow\infty}\frac{1}{8\pi^2}
\int\limits_{B(x,r)}-{\rm Tr} (F_{A_{n_m}}^2)+{\rm Tr} (F_{\tilde A_0})^2+
2 \left(|\mu_{0,0}(\Psi_{n_m})|^2-|\mu_{0,0}(\tilde \Psi_0)|^2\right) \ .
$$

As in the instanton case we get
$$\lim\limits_{m\rightarrow\infty}\frac{1}{8\pi^2}
\left[\int\limits_{B(x,r)}-{\rm Tr} (F_{A_{n_m}}^2)+{\rm Tr} (F_{\tilde
A_0})^2\right]_{{\rm mod}\
{\Bbb Z}}=\lim\limits_{m\rightarrow\infty}(\tau_{S(x,r)}(A_{n_m})-\tau_{S(x,r)}({\
tilde A_0}))$$
$$=0\ {\rm in}\
\qmod{{\Bbb R}}{{\Bbb Z}}
$$
by the convergence $f_m^*(A_{n_m}|_{X\setminus S_\varepsilon})\rightarrow
{\tilde
A_0}|_{X\setminus S_\varepsilon}$. Here
$\tau_S(B)$ denotes the Chern-Simons invariant of the connection $B$ on a
3-manifold $S$
([DK]).

On the other hand, by  the apriori
${\cal C}^0$-bound of the spinor component on the space of monopoles, the term
$\int\limits_{B(x,r)}2
\left(|\mu_{0,0}(\Psi_{n_m})|^2-|\mu_{0,0}(\tilde \Psi_0)|^2\right)$ can be
made as small as
we please by choosing $r$ sufficiently small.  This shows that the
$\lambda_x$ are integers,
and that
$$\sum_{x\in S_\varepsilon}\lambda_x=
\frac{1}{4}\left(p_1(\bar\delta(P'^u))-(p_1(\bar\delta(P^u))\right)\ ,
$$
which completes  the proof.
\hfill\vrule height6pt width6pt depth0pt \bigskip

\section{Transversality Results and Compactification of Perturbed Moduli Spaces}

Equations for pairs $(A,\Psi)$, where $A$ is a unitary connection with fixed
determinant connection and
$\Psi$ a non-abelian  Dirac spinor   have been already considered   [PT1],
[PT2]. For
instance, the definition of $Spin^c$-polynomial invariants  starts with the
construction of the moduli space of solutions of the   $(ASD-Spin^c)$-equations
$$\left\{\begin{array}{lll}
{\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_{A} \Psi&=&0 \ ,\ \ \ \Psi\ne 0\\
F_A^+&=&0\ .
\end{array}\right.$$

The proofs of the corresponding transversality results are incomplete. They
are based
on the following false assertion ([PT2], [PT1]):\\ \\
{\bf (A)} Let $P^c\longrightarrow P_g$ be a $Spin^c(4)$-structure with spinor bundles
$\Sigma^{\pm}(P^c)$ on a Riemannian 4-manifold $(X,g)$,   $E$ a Hermitian
2-bundle
on $X$, and  $A$   a unitary connection in  $E$. If the
${\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_{A}$-harmonic non-vanishing positive spinor
$\Psi\in A^0(\Sigma^+(P^c)\otimes E)$ is fibrewise degenerate   considered
as morphism
$E^{\vee}\longrightarrow \Sigma^+$, then
$A$ is reducible.
 \\

(In the proof of this assertion  ([PT1] p . 277) is used the fact that, in
the presence of
a $Spin^c(4)$-structure, the  Clifford pairing $(\alpha,\sigma)\longmapsto
\gamma(\alpha)\sigma$  between 1-forms and positive spinors   has no
  divisors of zero. This is  true for real 1-forms, but not for complex ones. )

Counterexamples are easy to find:

Every holomorphic section in a holomorphic Hermitian 2-vector bundle ${\cal E}$
on a K\"ahler surface can be regarded as  a degenerate  harmonic  positive
spinor in
$\Sigma^+_{\rm can}\otimes {\cal E}$, where $\Sigma^+_{\rm
can}=\Lambda^{00}\oplus\Lambda^{02}$ is the positive spinor bundle of the
canonical $Spin^c(4)$-structure in $X$, if we endow ${\cal E}$ with the Chern
connection given by the holomorphic structure. Therefore any indecomposable
holomorphic 2-bundle ${\cal E}$ with $H^0({\cal E})\ne 0$ gives a
counterexample to
the  assertion {\bf (A)}.\\

Note that counterexamples occur precisely in the K\"ahler framework, where all
explicit computations of moduli spaces and invariants were carried out.

Below we shall prove a complete $S^1$-equivariant transversality theorem for the
$PU(2)$-monopole equations, which can be extended to handle also the
simpler case of
the $(ASD-Spin^c)$-equations.
The terms containing derivatives are no longer linear, and the symbol of the
linearized equations does not have diagonal form.   Note also that for our
perturbations
of the
$PU(2)$-monopole equations  there is no way to obtain an
apriori ${\cal C}^0$-bound for the spinor with help of the Weitzenb\"ock
formula.

\subsection{Generic transversality results}

Let $\sigma:P^u\longrightarrow  P_g$ be a $Spin^{U(2)}(4)$-structure on the closed oriented
Riemannian
4-manifold $(X,g)$.
Recall that there is a natural embedding
$$P^u\times_\pi{\Bbb C}^4\longrightarrow{\rm Hom}(\Sigma^+(P^u),\Sigma^-(P^u))\ ,$$
so that any section $\lambda\in A^0(P^u\times_\pi{\Bbb C}^4)$ in this complex
4-bundle can be
regarded as a morphism $A^0(\Sigma^+(P^u))\longrightarrow  A^0(\Sigma^-(P^u))$
commuting with the
${\Bbb G}(P^u)$-actions in the two spinor bundles.

Consider the parameter space
$${\cal P}:=A^0(P^u\times_\pi{\Bbb C}^4)
\times A^0({\rm GL}({\rm ad}_+(P^u)))\times \prod_i
A^0({\rm End}({\rm ad}_+(P^u)))
$$
and set
$${\cal A}:= {\cal P}\times {\cal A}(\bar\delta(P^u))
\times A^0(\Sigma^+(P^u))\ .
$$

Fix    a finite system $(Y_i)$ of generators for the $A^0$-module
${\cal X}(X)$ of vector fields on $X$, and define a map
$$SW:{\cal A}\longrightarrow
A^0({\rm ad}_+(P^u)\otimes
{\scriptscriptstyle|}\hskip -4pt{\g}_0(P^u))\times A^0(\Sigma^-(P^u)) $$
  by
$$SW(\lambda,K,K_i,A,\Psi)= \left(
\matrix {\Gamma (F_A^+)-
(K\otimes{\rm id}_{{\scriptscriptstyle|}\hskip -4pt{\g}_0})\mu_{0,0}(\Psi) + \cr
\ \ \ \  +\sum\limits_i (K_i\otimes{\rm id}_{{\scriptscriptstyle|}\hskip -4pt{\g}_0})\left [\nabla_{A,Y_i}
(\mu_{0,0}(\Psi)) \right]\
 \cr \cr
({\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_{A}+\lambda)\Psi
}\right)\ .
$$

Complete the   configuration space ${\cal A}$ with respect to suitable
Sobolev norm
$L^2_k$ and  $A^0(\Sigma^-(P^u))\otimes A^0({\rm ad}_+(P^u)\otimes
{\scriptscriptstyle|}\hskip -4pt{\g}_0(P^u))$ with respect to  $L^2_{k-1}$.

According to sect. 1.2.3, a pair $(A,\Psi)\in {\cal A}(\bar\delta(P^u))\times
A^0(\Sigma^+(P^u))$ will be called abelian if there exists a reduction
$P^c_1\times_X P^c_2\textmap{\rho} P^u$ of $P^u$ to $S^1\times S^1$  such
that $(A,\Psi)$
is induced by a triple $(A_1,A_2,\Phi)$ with $A_i\in{\cal A}(\det(P^c_i))$,
$A_1\otimes
A_2=a^{\otimes 2}$ and $\Phi\in A^0(\Sigma^+(P^c_1))$.
\begin{thry}  Let $p=(\lambda,K,K_i,A,\Psi)\in Z(SW)$ with $\Psi\ne 0$.
If  $SW$ is not a submersion in $p$, then there exists a non-empty open set
$U\subset X$
such that $(A,\Psi)|_U$ is an abelian pair.
\end{thry}

{\bf Proof: }  Let $(S,\Phi)\in A^0({\rm ad}_+(P^u)\otimes
{\scriptscriptstyle|}\hskip -4pt{\g}_0(P^u))\times  A^0(\Sigma^-(P^u))$ be a smooth pair orthogonal to
${\rm im}(d_p(SW))$
with $(S,\Phi)\ne 0$.

First we show\\ \\
{\bf Assertion 1.} Either $S= 0$, or  the solution $(A,\Psi)$ is abelian on
a  non-empty open
set.\\

{\bf Proof: } Suppose that $\{x\in X|\ S_x\ne 0\}\ne \emptyset$.

Let $X_i\subset X$ be the set of points where $\Psi$ has rank $i$. Since
$\Psi$ is the solution of
a second order elliptic equation with scalar symbol, it follows that the
open set   $X_1\mathop{\bigcup}
X_2=\{\Psi\ne 0\}$ is dense.

Using the variation of $K$, we get immediately that fibrewise
$$S(a)\bot
\mu_{0,0}(\Psi) (a)\ {\rm in}\ {\scriptscriptstyle|}\hskip -4pt{\g}_0 ,\ {\rm for\ every}\ a\in
{\rm ad}_+(P^u)^{\vee}\ .$$

This implies  that the open set $\{S\ne 0\}$ is contained in $X_0\mathop{\bigcup}
X_1$, hence
there exists a nonempty contractible open   set
$$U\subset (X_1\cap\{S\ne 0\})\ .$$
Choose a trivialization of $P^u|_U$; it induces
a $Spin(4)$-structure
$$\gamma_U:\Lambda^1_U\longrightarrow{\rm Hom}_{\Bbb H} (S^+,S^-)$$
 in $U$ compatible with the metric $g|_U$, and identifications
$\Sigma^{\pm}(P^u)|_U=S^{\pm}\otimes E$,  where
$S^{\pm}$,
$E$ are trivial
$SU(2)-$, respectively $U(2)-$bundles.  If we denote by $A_U$, $a_U$ the
connection matrices
of $A$ and $a$, we see that the   connection $\nabla_{A}$ on $\Sigma^+(P^u)|_U$
corresponds to the connection induced by the Levi-Civita connection
$\nabla_{\gamma_U}$ in
$S^+$ and the connection $B=A_U+\frac{1}{2}a_U$ in $E$.

Choose a frame  $(e_1,e_2)$ in $E$ and put $L_i= X\times{\Bbb C} e_i$. We  may
suppose    that
$\Psi|_U$ is a section in $S^+\otimes L_1$. Let $\sigma\in A^0(S^+)$ be the
nowhere
vanishing spinor $\sigma\in A^0(S^{\pm})$  defined by
$$\Psi|_U=\sigma\otimes e_1 \ .
$$
We get
 %
$$(\Psi\bar\Psi)_0=\frac{(\sigma\bar \sigma)_0}{2}\otimes\left(\matrix{1&0\cr
0&-1\cr}\right)\ .$$
Decompose the connection $B$ as
$$d_B=\left(\matrix{d_1&-\bar b\cr b&d_2\cr}\right)\ ,$$
where
$ b\ e_2\otimes\bar e_1\in A^1(L_2\otimes\bar L_1)$ is the second
fundamental  form of
$L_1$ with respect to the connection $B$. Write $b=u+i\ v\in A^1_{\Bbb C}(U)$.
Using the orthogonal decomposition
${\scriptscriptstyle|}\hskip -4pt{\g}_0(P^u|_U)={\Bbb R}\left(\matrix{1&0\cr0&-1}\right)\oplus(L_2\otimes\bar
L_1)$, we  get
$${\rm ad}_+(P^u|_U)\otimes {\scriptscriptstyle|}\hskip -4pt{\g}_0(P^u|_U)= su(S^+)\oplus \left[{\rm End}_0(S^+)\otimes_{\Bbb C}
(L_2\otimes\bar L_1)\right]\ \eqno{(1)}$$
Since   $S(a)\bot \mu_{0,0}(a)$ for   $a\in su(S^+)^{\vee}$, we get that,
with respect to this decomposition,  $S|_U$ has the form
$$S|_U=s\otimes (e_2\otimes\bar e_1)=\left(\matrix{0&-s^*\cr
s&0}\right) \eqno{(2)}$$
with $s\in{\rm End}_0(S^+)$.
The component of $\nabla_A(\mu_{0,0}(\Psi))$ in the second summand
occurring in (1) is
$$\frac{(\sigma\bar\sigma)_0}{2} \otimes\nabla_A
 \left(\matrix{1&0\cr0&-1\cr} \right)=
(\sigma\bar\sigma)_0 \otimes \left(\matrix{0 &-\bar b\cr b&0}\right)\ .
$$
By the orthogonality condition $(\Phi,S)\bot{\rm im} d_p(SW)$,  we get     for
any compactly
supported variation $k_i\in A^0({\rm End}(su(S^+)))$ of $K_i$  the identity
$$0=Re\left\langle k_i(\sigma\bar\sigma)_0
\otimes b(Y_i)(e_2\otimes\bar e_1),
s\otimes(e_2\otimes\bar e_1)\right\rangle=
Re\left[\left\langle k_i(\sigma\bar\sigma)_0,
\overline{b(Y_i)}\ s\right\rangle\right]\ .
$$
Writing $s=g+ih$, with Hermitian endomorphisms $g$ and $h$,  this means
$$\langle k_i(\sigma\bar\sigma)_0, u(Y_i)g+v(Y_i)h \rangle=0 $$
for all compactly supported $ k_i\in  A^0({\rm End}(su(S^+)))$.

It follows that $[u(Y_i)g+v(Y_i)h]\otimes[(\sigma\bar\sigma)_0]^{\vee}=0$
as current,
hence as section as well, where $(\cdot)^{\vee}$ denotes  the duality
morphism defined
by  the   inner product. Therefore, since $\sigma$ is nowhere vanishing on
$U$, we must
have
$$u(Y_i)g+v(Y_i)h=0\ \ \forall\ i\ ,\  \eqno{(3)}
$$
hence $ u\otimes g+v\otimes h=0$.\\ \\
{\bf Claim 1:} The 1-forms $u$ and $v$ have the same norm and are
orthogonal in every point $x\in U$.
\\ \\
 Indeed,  the
component in $S^-\otimes L_2$ of $({\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_{\gamma,A}+\lambda)\Psi$ is
$\beta(u)(\sigma) e_2$, and this component must vanish. It follows that
$\det(\gamma(b))=0$,
hence $g_\gamma^{\Bbb C}(b,b)=0 $. Writing   $b=u+iv$, this means that
$|u|_{g_\gamma}=|v|_{g_\gamma}$, and  $(u,v)_{g_\gamma}=0$. This proves the
claim.\\

Now let $x\in U$ be point where $b\ne 0$. Then $u_x$ and $v_x$ are linearly
independent
by Claim 1, hence they remain linearly independent for all $y$ in a sufficiently
small neighbourhood $V$ of
$x$.  Taking the contraction of the left hand term in (3) with a compactly
supported $m\in
A^0(su(S^+)^{\vee}))$,  we find that $m(h)=m(g)=0$ for every such
endomorphism $m$.
Therefore
  $g=h=0$, which contradicts the choice of $x$. It follows that $b=0$ on
$U$, hence
$(A,\Psi)|_U$ is abelian. This completes the proof of Assertion 1.

Now we prove\\ \\
{\bf Assertion 2.} If $(\Phi,S)\ne 0$, then the solution $(A,\Psi)$ is
abelian on a
non-empty open set.\\ \\
{\bf Proof: }

We may assume that $S=0$.

Consider the contraction map
$$({\Bbb H}_+\otimes_{\Bbb C} {\Bbb C}^2) \times ({\Bbb H}_-\otimes_{\Bbb C} {\Bbb C}^2)\longrightarrow
{\rm Hom}_{\Bbb C}({\Bbb H}_+,{\Bbb H}_-)$$
$$ (h_+\otimes v,h_-\otimes w) \longmapsto \langle v,w \rangle
h_-\otimes  \bar h_+\ ;$$
 it defines   a sesquilinear  map $q:\Sigma^+(P^u)\times \Sigma^-(P^u)\longrightarrow
P^u\times_\pi{\Bbb C}^4$.
Using variations of the parameter $\lambda$, we get easily\\ \\
{\bf Claim 2:} $q(\Psi,\Phi)=0$ \\

Suppose $\Phi\ne 0$, and  let $X^i$ be the set of points where $\Phi$ has
rank $i$. From
Claim 2  it follows that
$\Psi|_{X^2}=0$, hence by Aronzajin's theorem the harmonic spinor $\Psi$ must
vanish if $X^2$ was nonempty. Since we have assumed
$\Psi\ne 0$, it follows that $X^2=\emptyset$, and therefore the set
$X^1= X \setminus Z(\Phi)$ is open and non-empty.   By Claim 2 again it follows
that ${\rm rk}(\Psi|_{X^1})\leq 1$ and by  Aronzajin's theorem $ X^1\setminus
Z(\Psi)$ is dense
in $X^1$.  On this non-empty open set
$\Psi$ and
$\Phi$ have both rank 1.  Fix as above a trivialisation of $P^u$ on a
contractible open set
$U\subset X^1 \setminus Z(\Psi)$, and define in the same way the objects
$S^{\pm}$, $E$, $B$, $(e_1,e_2)$, $(L_1, L_2)$.  Using Claim 2, we see that
it is possible
choose the framing of
$(e_1,e_2)$ such that
$\Phi|_U$,  $\Psi|_U$ have
the form
$$\Phi|_U=\sigma_-\otimes e_1\ ,\ \Psi|_U=\sigma_+\otimes e_2\ ,$$
where $\sigma_{\pm}$  are nowhere vanishing
sections in $S^{\pm}$.

Using variations $\alpha\in A^1(su(E))$  of the connection $A$ which have
the form
$\alpha=\beta\otimes\varphi$ with $\beta\in A^1(U),$ $\varphi\in
A^0(su(E))$, we obtain
$$Re\left\langle\gamma(\beta)(\sigma_+)\varphi(e_2),\sigma_-\otimes
e_1\right\rangle=0   \eqno{(*)}$$
for every compactly supported $\beta,\ \varphi$.  Now choose    $\tilde
\beta \in A^1(U)$
and $\tilde\varphi \in A^0(su(E))$ such that
$\gamma(\tilde \beta)(\sigma_+)=\sigma_-$\ ,\  $\tilde\varphi(e_2)=e_1$
(use $e_1\bot
e_2$), and set  $\beta:=\rho\tilde \beta$, $\varphi:=\rho\tilde \varphi$,
where $\rho$ is a
non-vanishing $[0,1]$-valued cut-off function. Then   $(*)$ gives   a
contradiction.
\hfill\vrule height6pt width6pt depth0pt \bigskip
\\  \\

The same continuation method as in [DK] gives
\begin{pr} A $PU(2)$-monopole $(A,\Psi)$ with $\Psi\ne 0$, whose restriction to a
non-empty open set is abelian, is globally abelian.
\end{pr}

Note that we do not need to assume that $X$ is simply connected: The
non-vanishing
spinor
$\Psi$ provides a well defined continuation of the abelian reduction.

Theorem 2.1.20 shows that ${\cal A}^{SW}\subset{\cal A}$ is smooth in the
complement of
  the   abelian locus and of the locus $\Psi=0$.
 Our next aim is to study ${\cal A}^{SW}$ around the abelian locus.

\subsection{Transversality around the abelian locus}

We fix an abelian reduction $\rho:(P^c_1\times_X P^c_2)\longrightarrow P^u$ of $P^u$. The
corresponding
abelian solutions of $SW^{\sigma}_a$ are up to gauge equivalence  twisted
abelian
monopoles $(B_1,\Phi_1)$ for  $P^c_1\longrightarrow P_g$ satisfying
$$\left\{\begin{array}{ccc}
{\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_{B_1}\Phi_1&=&0\\
\Gamma(F_{B_1}^+)&=&(\Phi_1\bar\Phi_1)_0+\Gamma(F_a^+) \   .
\end{array}\right.  \eqno{(SW^{{\germ c}_1}_{\Gamma(F_a^+)})}$$

In particular   no solution of the form $(B_1,0)$ can occur if the class
$c_1(\det(P_1^c))-c_1(\det(P^u))$ is not $g$-anti-selfdual. Note that the
same is valid
for any
$\rho$-reducible solution of the perturbed equations. Denote by ${\cal
A}_\rho$ the
configuration space
$${\cal A}_\rho:=\{(\lambda,K,K_i, A,\Psi)\in{\cal A}|\ (A,\Psi)\ {\rm is}\
\rho-{\rm
reducible}\}.
$$

\begin{thry} Consider an abelian reduction $\rho:(P^c_1\times_X P^c_2)\longrightarrow
P^u$ of
$P^u$ such
that  $c_1(\det(P_1^c))-c_1(\det(P^u))$ is not $g_\gamma$-anti-selfdual. There
 exists an open
dense subset ${\cal P}_0$ in the set of parameters  ${\cal P}$ such that ${\cal
A}_\rho^{SW}\cap p_{\cal P}^{-1}({\germ p})$  is smooth  for any ${\germ p}\in{\cal P}_0$.
\end{thry}

 We may suppose that
$\sigma:P^u\longrightarrow P_g$ is associated with the pair $(P^c,E)$ as in
Proposition 1.1.11; then
$\rho$ defines a splitting
$E^{\vee}\otimes\det(P^u)=L \oplus M$. Denote by $\Sigma^{\pm}$ the spinor
bundles of $P^c$,
and fix a connection $b\in{\cal A}(\det(\Sigma^{\pm})$.

Let ${\germ m}=(A,\Psi)$ be an abelian $PU(2)$-monopole of type $(\sigma,a)$
associated
with
the reduction $\rho$. This means that
$\Psi\in A^0(L\otimes\Sigma^+)\subset A^0(\Sigma^+(P^u))$ and the splitting
$\Sigma^{\pm}(P^u)=(\Sigma^{\pm}\otimes L)\oplus (\Sigma^{\pm}\otimes M)$ is
$ A$-parallel.
The $PU(2)$-bundle $\bar\delta(P ^u)$ reduces to $L^{\vee} \otimes M$. Let
$c_A$ be the
connection in this line bundle associated with $A$. Then $A$ gives rise to
 connection $A_1$, $A_2$ in the summands $L$, $M$ characterized by the
conditions
$$A_1^{-1}\otimes A_2=c_A\ , \ \ A_1 \otimes A_2=
a\otimes b^{-1}\in{\cal A}(\det(P^u)
\otimes (\det\Sigma)^{\vee})\ .
$$

We compute the cohomology group ${\Bbb H}^2_{{\germ m}}$ of the elliptic complex
associated with
${\germ m}$ (see sect. 1.2.4).  Denote by ${\Bbb H}^2_{{\germ m},ab}$ the second harmonic
space of the
elliptic
complex associated  to ${\germ m}$ regarded as element in the abelian moduli space
associated to the
$Spin^c(4)$-structure $P^c_1\longrightarrow P_g$. We have

$$H^2_{\germ m}={\Bbb H}^2_{{\germ m},ab}\oplus\{(\alpha,\varphi)\in A^2_+(L^{\vee}\otimes
M)\oplus
A^0(\Sigma^-\otimes M)|\  (N^1_{A,\Psi})^*(\alpha,\varphi)=0\} \ ,
$$
where  $N^1_{A,\Psi}$ is the operator
$$
A^1(L^{\vee}\otimes M)\oplus A^0(\Sigma^+ \otimes M)\textmap{N^1_{A,\Psi}}
 A^2_+(L^{\vee}\otimes M)\oplus
A^0(\Sigma^-
\otimes M)\ ,$$
$$N^1_{A,\Psi}(\alpha,\varphi):=\left(\begin{array}{ll} d^+_{c_A}\alpha-
\Gamma^{-1}(\varphi\bar\Psi)_0\\
{\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_{c_A}\varphi+\gamma(\alpha)\Psi
\end{array}\right) \ .
$$
The subscript $(\ \cdot\ )_0$ in the right hand term means the trace-free
component with
respect to ${\rm End}(\Sigma^+)$.
Consider also the operator
$$  N^0_{A,\Psi}: A^0(L^{\vee}\otimes M)\longrightarrow A^1(L^{\vee}\otimes M)\oplus
A^0(\Sigma^+\otimes M)\ ,\ \ N^0_{A,\Psi}(h)=\left(\begin{array}{ll}\
-d_{c_A} h\\
 h(\Psi)
\end{array}\right)
$$
whose formal adjoint is
$$(\alpha,\varphi)\ {\raisebox{0.25ex}{$\scriptscriptstyle\vert$}}\hskip -0.8mm
{\textmap{(N^0_{A,\Psi})^*}}  -d_{c_A}^*\alpha+\varphi\bar\Psi \ .
$$
Let ${\germ p}=(\lambda,K,\kappa)$, $\kappa:=(K_i)_i$ be a point in the space of
perturbations.
If ${\germ m}$ is a solution of the  ${\germ p}$-perturbed equations, we get a similar
decomposition
of
${\Bbb H}^2 ({\germ m})$ , but the corresponding formula for
$N^1_{A,\Psi}$   will be
$$N^1_{ A,\Psi,{\germ p}}(\alpha,\varphi)=
\left(\begin{array}{ll} d^+_{c_A}\alpha -\Gamma^{-1}\left[K(\varphi\bar\Psi)_0 +
\sum\limits_i  K_i\left[\alpha(Y_i)(\Psi\bar\Psi)_0)-
\nabla_{{c_A},Y_i}(\varphi\bar\Psi)\right]\right]\\
{\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_{{c_A}}\varphi+\lambda\varphi+\gamma(\alpha)\Psi
\end{array}\right).
$$

The operators $N^0_{A,\Psi,{\germ p}}$, $N^1_{ A,\Psi,{\germ p}}$ form    an elliptic
complex
$N_{{\germ p}, A,\Psi}$, which will  be called the \underbar{normal}
\underbar{elliptic}
\underbar{complex} of the abelian solution ${\germ m}$.

Let $\chi(N_{A,\Psi,{\germ p}})$ be the index of this complex.
\begin{pr} Let ${\cal
M}_\rho$ be the moduli space
$${\cal M}_\rho:=\{[({\germ p}, A,\Psi )]|\ ({\germ p}, A,\Psi )\in {\cal A}_\rho\ ,\
SW({\germ p},A,\Psi)=0\}
 $$
of $\rho$-reducible abelian solutions.  \\
(i) ${\cal M}_\rho$ is smooth and the
projection $p_{\cal P}$    on the  perturbation-argument
${\germ p}:=(\lambda,K,\kappa)$-argument is proper.  The dimension of the fibre
over a regular
 value of
$p_{\cal P}$ is the expected dimension
$w_{\det(\Sigma^+)\otimes L^2} $ of the corresponding twisted Seiberg-Witten
equations.\\
(ii) There exists a dense open set of parameters ${\cal P}_0\subset{\cal
P}$ such
that for every  ${\germ p}\in {\cal P}_0$ the complement of the zero section in
the fibrations
$$
\mathop{\bigcup}\limits_{({\germ p},{\germ m})\in{\cal M}_\rho }H^i_{{\germ m}}(N_{{\germ m},{\germ p}})\longrightarrow
 {\cal M}_\rho^{{\germ p}}:=p_{\cal
P}^{-1}({\germ p})
  \ ,\\  i=1, 2
 $$
is smooth  of the expected dimensions $w_{\det(\Sigma^+)\otimes L^2}\mp
index_{\Bbb R}((N_{{\germ m},{\germ p}}))$.
 \end{pr}

{\bf Proof: } The first assertion follows as in [KM].  Note that only
$K$-perturbations are
sufficient for this transversality result.

To prove the existence  of the open set
${\cal P}_0$ with the above properties, consider first
 the product
$$\Pi_\rho:=[{\cal A}(M\otimes L^{\vee})\times A^0(\Sigma^+\otimes
L)]\times [{\cal
A}^1(M\otimes\bar L)\times A^0(\Sigma^+\otimes M)\setminus\{(0,0)\}]\times{\cal
P}_1\ ,$$
where ${\cal P}_1\subset {\cal P}$ is a dense open set of regular values
for the map
${\cal M}_\rho\longrightarrow {\cal P}$. This means in particular that ${\cal
M}^1_\rho:= {\cal
M}_\rho\cap p_{\cal P}^{-1}({\cal P}_1)\longrightarrow {\cal P}_1$ is a locally
trivial fibre bundle.

We denote by $A_c$ the $PU(2)$-connection in ${\Bbb P}(E)$ corresponding to a
connection
$c\in{\cal A}(M\otimes L^{\vee})$.

The abelian gauge group ${\cal G}_\rho={\cal C}^\infty(X,S^1)$ acts on
$\Pi_\rho$ from
the left by the formula
$$\left(c,\Phi,\alpha,\varphi ,{\germ p}\right)\cdot g= \left((c^{g^{ 2}},\ g
\Phi,\ g^{-2} \alpha,\
g^{-1}\varphi ,{\germ p}\right)\ .
$$
This action is induced from the  action of the whole gauge group $SU(E)$
via the inclusion
${\cal G}_\rho=U(L)\subset SU(E)$ associated with the splitting $\rho$.

On $\Pi_\rho$ we consider the following equations, which ask  that the
pair $(A_c,\Phi)$ is a $\rho$-reducible abelian solution of the ${\germ p}$-perturbed
$PU(2)$-monopole equations  and that $(\alpha,\varphi)$ lies in
${\Bbb H}^1(N_{A_b,\Phi,{\germ p}})\setminus\{0\}$. In other words $(\alpha,\varphi)$
is a non-trivial
infinitesimal deformation in the normal- (with respect to ${\cal M}_\rho$)
direction of
the abelian
$PU(2)$-monopole
$(A_c,\Phi)$.
$$\left\{\begin{array}{ll}
{\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_{c}\Phi+\lambda\Phi&= 0\\
- F_{c} -\Gamma^{-1} ( K(\Phi\bar\Phi)_0)&=0\\
({\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_{c}+\lambda)\varphi+\gamma(\alpha)\Phi&=0 \\
  d^+_{c}\alpha-\Gamma^{-1}\left[K(\varphi\bar\Phi)_0-
\sum\limits_iK_i\left[\alpha(X_i)(\Phi\bar\Phi)_0+
\nabla_{{c},X_i}(\varphi\bar\Psi)\right]\right]&=0\\
d^*_{c}\alpha-\varphi\bar\Phi&=0  \ .

\end{array}\right.      \eqno{(N_\rho)}
$$
In the second   equation the notations $(\ \cdot\ )_0$ means the
projection of the tensor
product
$(\Sigma^+\otimes\bar\Sigma^+)\otimes(L\otimes\bar L)$ on
$(\Sigma^+\otimes\bar\Sigma^+)_0$; in the  fourth equation it means the
projection of
$(\Sigma^+\otimes\bar\Sigma^+)\otimes(M\otimes\bar L)$ on
$(\Sigma^+\otimes\bar\Sigma^+)_0\otimes(M\otimes\bar L)$ .

The effect of a gauge transformation $g$ is:
$$b\longmapsto (b)^{g^{-2}}=g^2\circ d_{b}\circ g^{-2}\ ,\ \ \bar\Phi\longmapsto
\bar\Phi g\ ,\ \ \varphi\bar\Phi \longmapsto g^2\varphi\bar\Phi\ ,\ \
d^+_{b}\alpha
\longmapsto g^2d^+_{b}\alpha\ ,$$
hence the  system is gauge invariant. The (finite dimensional) quotient
$${\cal M}^{N_\rho}:=\qmod{ \{{\rm  solutions\ of\ } (N_\rho)\}}{{\cal G}}$$
analytic space which fibres over the smooth  manifold ${\cal M}_\rho^1$, via
$(c,\Phi,\alpha,\varphi,{\germ p})\longmapsto(A_c,\Phi,{\germ p})$. The fibre over
$[(A,\Phi,{\germ p})]\in  {\cal M}_\rho^1$ is the pointed harmonic vector space
${\Bbb H}^1(N_{A_b,\Phi,{\germ p}})\setminus\{0\}$.

The same method  as in the proof of Theorem 2.1.20 give\\ \\
{\bf Claim.} The function defined by the left hand side of the system
$N_\rho$ is a
submersion, in particular ${\cal M}^{N_\rho}$ is a smooth manifold.\\  \\
This ends the proof   of Proposition 2.2.4.

\hfill\vrule height6pt width6pt depth0pt \bigskip

\begin{co}
1. If the index  $\chi(N_{A,\Psi,{\germ p}})$ of the normal
elliptic complex  $N_{A,\Psi,{\germ p}}$ is negative,  and
$w_{\det(\Sigma^+)\otimes L^2}=0$
then
${\cal A}^{SW}_\rho\cap p_{\cal P}^{-1}({\germ p})$ is smooth, closed and open in
${\cal
A}^{SW}$ for any ${\germ p}\in {\cal P}_0$.

In this case the  dimension of any abelian moduli spaces ${\cal M}_\rho\cap
p_{\cal
P}^{-1}({\germ p})$ with
${\germ p}\in{\cal P}_0$ is larger than the expected dimension of the
$PU(2)$-monopole equation.
\\
2. If the rank of  of the normal elliptic complex $N_{{\germ p},A,\Psi}$ is
non-negative and
$w_{\det(\Sigma^+)\otimes L^2=0}$, then  ${\cal A}^{SW}\cap
p_{\cal P}^{-1}({\germ p}) $  is also smooth  in the points of
${\cal A}_\rho\cap p_{\cal P}^{-1}({\germ p})$ and ${\cal A}_\rho^{SW}\cap
p_{\cal P}^{-1}({\germ p})$
is a smooth closed submanifold of
${\cal A}^{SW}\cap p_{\cal P}^{-1}({\germ p})$,  for any ${\germ p}\in {\cal P}_0$.
\end{co}

For a perturbation parameter ${\germ p}\in{\cal P}$, denote by ${\cal
N}^\sigma_{a,{\germ p}}$ the
moduli space of solutions of the equation
$SW({\germ p},\cdot,\cdot)=0$.

Using  all the results,  we can conclude with the following

\begin{thry}   Suppose that the Donaldson moduli space
${\cal D}\subset{\cal M}^\sigma_a$ of
  contains no abelian solutions.

Then for any parameter ${\germ p}$ in a dense second category subset of the parameter
space ${\cal P}$, the perturbed moduli space $ {\cal
N}^\sigma_{a,{\germ p}}\setminus{\cal D}$    is smooth of the expected dimension
in the
complement of the abelian locus, and it is smooth in the abelian points
which have abelian
expected dimension $0$.\\
\\
 \end{thry}

\subsection{The Uhlenbeck compactification of the perturbed moduli spaces}
\subsubsection{Estimates for solutions of the  perturbed equations}

Because of the   term
$\sum_i(K_i\otimes{\rm id})\nabla_{A,Y_i}(\mu_{00}(\Psi))$ in the second
perturbed equation  it is
no longer possible to get an apriori ${\cal C}^0$-bound of the spinor
component on the space of
solutions, even for   perturbations $(\lambda,K,K_i)$ close to
$(0,{\rm id}_{{\rm ad}_+},0)$. However one
can   easily prove the following result.

Fix a closed oriented  Riemannian 4-manifold $(X,g)$,   $L$ a Hermitian
line bundle on $X$ and
$a\in{\cal A}(L)$.
\begin{pr} Let $M=M_{(g,a)}$ be the apriori ${\cal C}^0$-bound for the
spinor component on the
non-perturbed moduli space given by the Weitzenb\"ock formula (Proposition
2.1.8), and let
$\varepsilon$ be a positive number.  If
$\lambda$ is small enough in the
${\cal C}^1$-topology, $K-{\rm id}_{{\rm ad}_+}$,
$K_i$ are sufficiently small in the ${\cal C}^0$-topology,  then for every
$Spin^{U(2)}(4)$-structure $\sigma$ of determinant $L$ in $X$ and any solution
$(A,\Psi)$ of the perturbed   $PU(2)$-equations
$SW^{\sigma}_{a,(\lambda,K,K_i)}$ with
$$\sup\limits_X|\Psi| <
M+\varepsilon \eqno{(o)}$$
 satisfies
$$\sup\limits_X|\Psi|\leq M+\frac{\varepsilon}{2}\eqno{(c)}$$
\end{pr}

{\bf Proof: }
Suppose for simplicity that $\det(P^u)$ is trivial, and $a\in{\cal
A}(\det(P^u)$ is the flat
connection.

For arbitrary sections $\mu\in A^0(P^u\times_\pi{\Bbb C}^4)\subset{\rm End}_{\Bbb C}(\Sigma(P^u))$,
$\Phi\in\Sigma^+(P^u)$ and arbitrary connection $B\in{\cal
A}(\bar(\delta(P^u))$ we have
the
identity
$${\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_{\gamma,B}(\mu\Phi)=-\mu
{\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_{\gamma,B}\Phi+{\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_\gamma(\mu)\Psi-2(\mu\cdot\gamma(\nabla_{\gamma,A}\Phi))
$$
where  on the right  ${\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_\gamma$ denotes a Dirac type operator
$${\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}:A^0(P^u\times_\pi{\Bbb C}^4)\longrightarrow  A^0({\rm ad}_+\otimes{\Bbb C})\subset
{\rm End}(\Sigma^+(P^u))$$
 and
$\cdot$ denotes the contraction with the complexified Riemannian metric in
$P^u\times_\pi{\Bbb C}^4$.

If $(A,\Psi)$  is a solution of the perturbed
equations associated to the perturbing parameter ${\germ p}=(\lambda,K,(K_i))$, then
$$\begin{array}{ll}\left({\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_{\gamma,A}^2(\Psi),\Psi\right)&=
\left({\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_{\gamma,A}(- \lambda \Psi ),\Psi\right)=\\
&=\left( \lambda ({\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_{\gamma,A}\Psi),\Psi\right)-
\left({\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_{\gamma}(\lambda)
(\Psi),\Psi\right)+2((\lambda\cdot\nabla_{\gamma,A}\Psi),\Psi)
\end{array}$$
and
$$(\Gamma(F_A^+)(\Psi),\Psi)=
\left(K\mu_{00}(\Psi)(\Psi) , \Psi\right)-
\left(\sum\limits_iK_i[\nabla_{\gamma,A,Y_i}\mu_{00}(\Psi)] (\Psi),\Psi\right)
$$

Using again the  Weitzenb\"ock formula
$$({\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_{\gamma,A}^2(\Psi),\Psi)=(\nabla_A^*\nabla_A\Psi,\Psi)+(\Gamma(F_A^+)
(\Psi),\Psi)+
\frac{s}{4}|\Psi|^2 \ ,
$$
we see that pointwise
$$\begin{array}{ll}(\nabla_A^*\nabla_A\Psi,\Psi)&\leq
-|(\Psi\bar\Psi)_0|^2-\frac{s}{4}|\Psi|^2+\\ &+c_1|K-{\rm id}_{{\rm ad}_+}|\ |\Psi|^4+
c_2\left(\sum\limits_i|K_i|\right) |\Psi|^3|\nabla_A\Psi|+\\&+c_3|\lambda|\
|\Psi||\nabla_A\Psi|+c_4|\nabla_\gamma(\lambda)|\ |\Psi|^2 \leq\\ & \\
&\leq-|(\Psi\bar\Psi)_0|^2-\frac{s}{4}|\Psi|^2+\\
&+c_1|K-{\rm id}_{su(\Sigma^+)}|\ |\Psi|^4+
\frac{c_2}{2}\left(\sum\limits_i|K_i|\right)
|\Psi|^2(|\Psi|^2+|\nabla_A\Psi|^2)+ \\ &+\frac{c_3}{2}|\lambda|\
\left(|\Psi|^2+|\nabla_A\Psi|^2\right)+ c_4|\nabla_\gamma(\lambda)|\ |\Psi|^2
\end{array}
$$
where $c_i$ are positive constants.
Recall (see the proof of Proposition 2.1.8) that it exists a positive
constant $C'$
such that
$|(\Psi\bar\Psi)_0|\geq C' |\Psi|^2$. Put
$$u:=|\lambda|\ ,\ \ v:=|K-{\rm id}_{su(E)}|\ ,\ \ w:=\sum\limits_i|K_i|\ ,\ \
z:=|\nabla_\gamma\lambda|\ .$$

The inequality above gives now
$$\begin{array}{ll}
(\nabla_A^*\nabla_A\Psi,\Psi)\leq&-(C'-c_1v-\frac{c_2}{2}w)|\Psi|^4-(\frac{s
}{4}-
\frac{c_3}{2}\
u -c_4\ z)|\Psi|^2+\\ &+
(\frac{c_2}{2}\ w\ |\Psi|^2+ \frac{c_3}{2}\ u ) |\nabla_A\Psi|^2
\end{array}
$$

Recall that the bound $M$ of the spinor component on the non-perturbed
moduli space
given by
Proposition 2.1.8 is $M:=\max (0,\frac{\sup(-s)}{4C'})$.

Using the identity
$$\frac{1}{2}\Delta |\Psi|^2=(\nabla_A^*\nabla_A\Psi,\Psi)-|\nabla_A
\Psi|^2\ ,$$
the inequality above gives then
$$\begin{array}{cl}
\Delta|\Psi|^2  \leq&  -|\nabla_A\Psi|^2(1-\frac{c_3}{2}\ u-\frac{c_2}{2}\
w\ |\Psi|^2)-\\ \\

&-([C'-c_1v-\frac{c_2}{2}w)|\Psi|^4-(\frac{s}{4}-\frac{c_3}{2}\ u -c_4\
z)|\Psi|^2
\end{array} \eqno{(i)}
$$
  In a point $x$ where the absolute maximum $M_\Psi$ of $|\Psi|^2$ is
obtained, we get by the maximum principle:
$$\begin{array}{c}
0\leq \frac{1}{2}(\Delta|\Psi|^2)(x) \leq  \\
-|\nabla_A\Psi(x)|^2\left(1-\frac{c_3}{2}  u(x)-\frac{c_2}{2}  w(x)\
M_\Psi\right)-\\
-\left(C'-c_1v(x)-\frac{c_2}{2}w(x)\right) M_\Psi^2-
\left(\frac{s(x)}{4}-\frac{c_3}{2}
u(x)
-c_4 z(x)\right) M_\Psi
\end{array}
$$

We have assumed  $M_\Psi< M+\varepsilon$, hence the bracket
$$\left(1-\frac{c_3}{2}  u(x)-\frac{c_2}{2}  w(x)\ M_\Psi\right)$$
becomes a   positive number as soon as
$\sup u$ and  $\sup w$ are sufficiently small, such that
$$1-\frac{c_3}{2}  \sup u-\frac{c_2}{2}  \sup w(M+\varepsilon) >0\ .
$$
 But then we get as in the non-perturbed
case
$$\left(C'-c_1v(x)-\frac{c_2}{2}w(x)\right) M_\Psi\leq  -
\left(\frac{s(x)}{4}-\frac{c_3}{2}  u(x) -c_4 z(x)\right)
$$
For any $\varepsilon>0$ we get the estimate
$$  M_\Psi \leq -\frac{s(x)}{4C'}+\frac{\varepsilon}{2}\leq \frac{\sup(-s)}{4C'}
+\frac{\varepsilon}{2}\eqno{(c)}
$$
as soon as  $\sup  u$,  $\sup v$, $\sup w$ and $\sup z$ are sufficiently
small with respect
to
$\varepsilon$ and with respect to the constants $c_i^{-1}$.
\hfill\vrule height6pt width6pt depth0pt \bigskip

Fix a positive number $\varepsilon$. \underbar{By}  \underbar{definition}
the perturbed
moduli
space ${\cal M}^\sigma_{a,{\germ p}}$  is defined to be  the moduli space of solutions
$(A,\Psi)$ of $SW^{\sigma}_{a,{\germ p}}$ satisfying the non-strict inequality $(o)$,
where the perturbation ${\germ p}=(\lambda,K,K_i)$ is sufficiently close to
$(0,{\rm id}_{{\rm ad}_+},0)$,
such that the two inequalities $(o)$ and $(c)$ become equivalent.  We   use
the first
  inequality
(which is an open condition) to endow ${\cal M}^\sigma_{a,{\germ p},\varepsilon}$
with the
 structure
of an analytic space (smooth outside the locus $\Psi=0$, for generic
perturbation
${\germ p}$) and we use the second inequality to construct an
  Uhlenbeck compactification of ${\cal M}^\sigma_{a,{\germ p}}$.
\\ \\

We say for simplicity that a perturbation-parameter ${\germ p}=(\lambda,K,K_i)$
is "small" with
respect to a Sobolev or a ${\cal C}^k$-topology if
$(\lambda,K-{\rm id}_{ad_+},(K_i))$ is "small"
with
respect to that topology.

\begin{thry} (Local estimates)     There is a
positive constant $\varepsilon_2>0$  such that for any Clifford map
$\gamma'$ on $\bar B$ which is sufficiently
${\cal C}^{3}$-close to $\gamma_0$ and every $a\in A^1(u(1))$
sufficiently small in the ${\cal C}^2$-topology, the following holds:

For any solution $(A,\Psi)$  of the perturbed $PU(2)$-monopole equation for
the triple $({\germ u}_{\gamma'},a',{\germ p}')$ over the open ball $B$  satisfying \\
i) $d^*_0  A=0$, \\
ii) $\parallel A\parallel_{L^4}\leq\varepsilon_2$, $\sup_B|\Psi|<\infty$\\
 and any interior domain
$D{\  {\subset}\hskip -2.2mm{\raisebox{.28ex B$, one has   estimates of the form :
$$\begin{array}{l}\parallel A\parallel _{L^2_l(D)}\leq
p_{D,l,\gamma',a',{\germ p}'}(\sup_B|\Psi|,\parallel A\parallel_{L^4})\ ,\\
\parallel\Psi\parallel_{L^2_l(D)}\leq q_{D,l,\gamma',a',{\germ p}'}(\sup_B|\Psi|,
\parallel A\parallel_{L^4}) \end{array}$$
where $p_{D,l,\gamma',a',{\germ p}'}$, $q_{D,l,\gamma',a',{\germ p}'}$ are polynomials with
$$p_{D,l,\gamma',a',{\germ p}'}(0,0)=0\ , \ \ q_{D,l,\gamma',a',{\germ p}'}(0,\cdot)=0\ .$$
\end{thry}
{\bf Proof: }

We use the same notations and method as in the proof of 2.1.4 but a
different order of
obtaining the estimates. We treat only the case of
${L^2_1(D)}$-norms:

Let $\varphi$ be a cut-off function supported in the open ball $B$ which
is identically 1 in a neighbourhood of $\bar D$. Then $\psi:=\varphi\Psi$
extends as section in the spinor bundle $S^+_s\otimes E_s$ and
$\alpha:=\varphi A$ extends as a $su(2)$-valued 1-form on the sphere.
$\Psi$ solves the first perturbed Seiberg-Witten equation, hence
$${\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_{\gamma',a',A}+\lambda={\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_{\gamma',a'}+\gamma'(A)+\lambda\ ,$$
 we get
$${\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_{\gamma,a}(\varphi\Psi)=\gamma(d\varphi)\Psi+
\varphi{\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_{\gamma,a}\Psi=
\gamma(d\varphi)\Psi-\varphi(\gamma(A)+\lambda)\Psi  \eqno{(sw)} $$
We obtain an elliptic estimate of
the form
$$\begin{array}{ll}
\parallel\psi\parallel_{L^2_1}\leq
const\parallel({\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_{\gamma,a}+\lambda)\psi\parallel_{L^2}
&\leq const_D(\sup|\Psi| + \parallel A\parallel_{L^4}\parallel\psi\parallel_{L^4})\\
&\leq   const'_D(\sup|\Psi| + \parallel A\parallel_{L^4}\parallel\psi\parallel_{L^2_1})
\end{array}$$
where $const_D$ depends on $\lambda$ and  on $D$ via $d\varphi$.  Here we
have used the
bounded inclusion $L^2_1\subset L^4$.

If $\parallel A\parallel_{L^4}$ is sufficiently small, we get an estimate of the form
$$\parallel\psi\parallel_{L^2_1}\leq const_D \left[ \sup|\Psi|\right] \eqno{(1)}
$$

Since, by hypothesis, the $su(2)$-valued
1-form $A$ on $\bar B$ is a solution in Coulomb gauge of the second
monopole equation, it
satisfies the following non-linear equation
$$\delta_\gamma A+( A\wedge
A)^+_\gamma=\Gamma_{\gamma}^{-1}\left(K\mu_{00}(\Psi)-
\sum\limits_i K_i\nabla_{A,a,Y_i}[\mu_{00}(\Psi)]\right)\ .$$
On the other hand
$d^*_s(\alpha)=\varphi d_s^*A-A(\nabla \varphi)=-A(\nabla\varphi)$, where
$\nabla$ means the gradient with respect to the fixed metric  $g_s$.
Therefore
$$\begin{array}{lll}\delta_\gamma\alpha&=&- A(\nabla\varphi)+\varphi
d^+_\gamma A+(d\varphi\wedge A)^+_\gamma \\
 &=&-  A(\nabla\varphi)+(d\varphi\wedge A)^+_\gamma
+\varphi\left[\Gamma_\gamma^{-1}[K\mu_{00}(\Psi)-
\sum\limits_iK_i\nabla_{A,a,Y_i}[\mu_{00}(\Psi)]]\right.\\ &&\ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \
-\left.(A\wedge A)^+_\gamma\phantom{\matrix{a\cr a}}\right].
\end{array} \eqno{(sw')}$$
Writing $\nabla_{A,a,Y_i}=\nabla_{0,a,Y_i}+A$, and one gets  an estimate of
the form
$$
\begin{array}{ll}\parallel\alpha\parallel_{L^2_1}\leq const_D&\left(\parallel A\parallel_{L^4}+
\sup|\Psi|^2+
\sup|\Psi|\parallel\Psi\parallel_{L^2_1}+\sup|\Psi|^2\parallel A\parallel _{L^2}  \right.\\
&\left.\ +\parallel
A\parallel_{L^4}\parallel\alpha\parallel_{L^2_1}
\right)\ ,
\end{array}$$
which gives, for a sufficiently small bound for
$\parallel A\parallel_{L^4}$, an estimate of the type
$$\parallel\alpha\parallel_{L^2_1}\leq const_D\left(\parallel
A\parallel_{L^4}+\sup|\Psi|^2+\sup|\Psi|\parallel\Psi\parallel_{L^2_1}+\sup|\Psi|^2\parallel A\parallel _{L^2}
\right)\ .
$$

Combined with (1), this gives
$$\parallel \alpha\parallel_{L^2_1}\leq   p_{D,1}(\sup|\Psi|,\parallel A\parallel_{L^4}) \eqno{(1')}
$$
for a polynomial $p_{D,1}$ with $p_{D,1}(0,0)=0$.

\hfill\vrule height6pt width6pt depth0pt \bigskip

Combining the "Local estimates" theorem with "Gauge fixing Theorem" we get,
as in section
2.1

\begin{co} (Estimates in terms of the curvature) There exists a constant
$\varepsilon>0$,
and a neighbourhood $N\subset{\cal C}l_B\times A^1_B(u(1))$ of
$(\gamma_0,0)$ with respect to the
${\cal C}^3\times{\cal C}^2$-topology with the following property:

For  any interior ball $D{\  {\subset}\hskip -2.2mm{\raisebox{.28ex B$ and any
$l\geq 1$ there exist two polynomials $P_{D,l,\gamma,a,{\germ p}}$,
$Q_{D,l,\gamma,a,{\germ p}}$
without constant terms, such that every solution
$(A,\Psi)$ of the $PU(2)$-monopole equations for the triple
$({\germ u}_\gamma,a,{\germ p})$ with
$(\gamma,a)\in N$ and
$\parallel F_A\parallel_{L^2}\leq\varepsilon$, is gauge equivalent on $B$ to a pair $(\tilde
A,\tilde\Psi)$ satisfying the estimates
$$\parallel\tilde A\parallel_{L^2_l(D)}\leq P_{D,l,\gamma,a,{\germ p}}(\sup\limits_{\bar
B}|\Psi|,\parallel
F_A\parallel_{L^2})\ ,\ $$
\
$$\parallel\tilde
\Psi\parallel_{L^2_l(D)}\leq Q_{D,l,\gamma,a,{\germ p}}(\sup\limits_{\bar B}|\Psi|, \parallel
F_A\parallel_{L^2}) \ .
$$
\end{co}

This implies (compare with Corollary 2.1.7)

\begin{co} (Global compactness)  Let $(\Omega,g)$ be an oriented Riemannian
manifold
with a
$Spin^{U(2)}(4)$-structure $\sigma:P^u\longrightarrow  P_g$ and a fix a connection
$a\in\det(P^u)$.    Let
$(A_n,\Psi_n)$ be a sequence of solutions of the $PU(2)$-monopole equations
associated
with the triple $(\sigma,a,{\germ p})$ with $\sup_{\Omega}|\Psi_n|$ uniformly
bounded. Suppose
that
any point
$x\in\Omega$ has a geodesic ball neighbourhood
$D_x$ such that for all large enough $n$,
$$\int_{D_x}| F_{A_n}|^2< \varepsilon^2$$
where $\varepsilon$ is the constant in Corollary 3.3.3 . Then there is a
subsequence
$(n_m)\subset{\Bbb N}$ and gauge transformations $u_{m}\in{\cal G}_0$ such that
$u_m^*\left[(A_{m_n},\Psi_{m_n})\right]$ converges in the ${\cal
C}^{\infty}$-topology on
$\Omega$.

\end{co}

\subsubsection{Regularity results for the perturbed equations}

We begin by stating the following slightly stronger form of the regularity
result for almost
anti-self-dual connections in 4.4.13 [DK]. It  can be obtained by precisely
the same
continuity
method.

\begin{lm} (Regularity for small $L^2_1$-connections in Coulomb gauge)\\
Let $g_s$ be a metric with nonnegative sectional curvature on the sphere.
There exists a
positive number $\varepsilon>0$,   such that any
$L^2_1$-connection  $A\in L^2_1(\Lambda^1(su(2)))$ with\\
1. $d^*A=0$ \\
2. $\parallel A\parallel_{L^2_1}<\varepsilon$.\\
3.  $F_A^+\in L^2_l$.\\
belongs to $L^2_{l+1}$.
There are polynomials $p_l$ with non-negative coefficients  and without constant
term    such that
$$\parallel A\parallel_{L^2_{l+1}}\leq p_l (\parallel F_A^+\parallel_{L^2_l})
$$
 for any $l\geq 1$ and any connection $A$ satisfying 1.-3.
\end{lm}
\hfill\vrule height6pt width6pt depth0pt \bigskip

Let $\gamma$ be  the Clifford map of a $Spin^{U(2)}(4)$-structure $P^u\longrightarrow
P_g$,
  ${\germ p}$ a perturbation parameter and  $a\in{\cal
A}(\det P^u)$. Put

$$sw^1_{\gamma,a,{\germ p}}(A,\Psi):={\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_{\gamma,a,A}\Psi+\lambda\Psi\ ,
$$
$$sw^2_{\gamma,a,{\germ p}}(A,\Psi):=\Gamma_\gamma(F_A^+)-
K\mu_{0,0}(\Psi)+\sum\limits_iK_i\nabla_{A,a,Y_i}[\mu_{0,0}(\Psi)]
$$

\begin{pr} (Regularity  of
$L^2_1$-almost solutions with $L^2_1$-small connection component and $L^{\infty}$-
bounded
spinor component)

 Let $\gamma$ be a Clifford map on the sphere
which is sufficiently ${\cal C}^1$-close to
$\gamma_s$,   let $a\in A^1(u(1))$   sufficiently ${\cal C}^0$-small,
and endow the sphere with the metric
$g:=g_\gamma$   associated with $\gamma$.

There exists a positive number $\varepsilon>0$ such that any pair
$$(A,\Psi)\in
L^2_1\left(\Lambda^1(su(2))\right)\times\left[ L^2_1\left(\Lambda^0(S^+_s\otimes
E_s)\right)\cap L^{\infty}\left(\Lambda^0(S^+_s\otimes
E_s)\right)\right]$$
 with\\
1.  $\parallel A\parallel_{L^2_1}<\varepsilon$, $d^*A=0$,\\
2. $sw^i_{\gamma,a,{\germ p}}(A,\Psi)$   smooth  ($i=1,\ 2$).\\
is also smooth.
\end{pr}

{\bf Proof: } Put $\sigma_i:= sw^i_{\gamma,a,{\germ p}}(A,\Psi)$.

Note first  that for $B\in A^1({\rm End}(E_s))$, $\Phi\in A^0(S^+\otimes E_s)$
we have the
identity
$${\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_\gamma[\gamma(B)\Phi]=[\Gamma_\gamma(d^+B)+d^*B]\Phi
-\gamma(B){\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_\gamma\Phi- 2 (B\cdot\nabla_\gamma\Phi)
$$
where  the contraction $\cdot$ in the last term acts by
$$(\beta\otimes \theta)\cdot
(\alpha\otimes s^+\otimes\varphi)=g(\beta,\alpha) s^+\otimes\theta(\varphi)\ ,
$$
for $\beta,\ \alpha\in  \Lambda^1$, $s^+\in S^+$, $\theta\in {\rm End}(E_s)$.

Write the operator ${\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_{\gamma,A,a}$ as
${\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_{\gamma}+\gamma(A)+\gamma(\alpha)$,
where $\alpha:=\gamma^{-1}(\lambda)+\frac{1}{2}a$.  Using the elliptic
equation,
$${\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_\gamma(\Psi)=-[\gamma(A)+\gamma(\alpha)]\Psi+\sigma_1 \eqno{(1)}
$$
and the condition  $A\in L^2_1\subset L^4$, $\Psi\in L^{\infty}$, we get
$\Psi\in L^4_1$.

Apply the operator
${\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_\gamma$ to the equation (1),
and use the Coulomb-gauge condition. We get that $\Psi$ is a
distribution-solution of the
equation
$$
\begin{array}{ll} {\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_\gamma^2(\Psi) = &-\Gamma(d^+A)(\Psi)
-[\Gamma(d^+(\alpha))+
d^*(\alpha)]\Psi + [\gamma(A)+
\gamma(\alpha)] {\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_\gamma\Psi \\ &\ \ \ \ \
\ + 2\left[(A+\alpha) \cdot\nabla_\gamma\Psi\right]= \\ \\
&=-\Gamma(d^+A)(\Psi)-[\Gamma(d^+(\alpha))+d^*(\alpha)]\Psi - [\gamma(A)+
\gamma(\alpha)] ^2\Psi \\  &\ \ \ \ \ \ [\gamma(A)+
\gamma(\alpha)]  \sigma_1 + 2\left[(A+\alpha) \cdot \nabla_\gamma \Psi\right]
\end{array}\eqno{(2)}$$

Since $\Psi\in L^4_1\cap L^{\infty}$ and $A\in L^2_1\subset L^4$ one can
easily check that
the right-hand term belongs to $L^2$, hence $\Psi\in L^2_2$.
$$\Gamma_\gamma(F_A^+)=
K\mu_{0,0}(\Psi)- \sum\limits_i K_i\nabla_{A,a,Y_i}[\mu_{0,0}(\Psi)]
+\sigma_2 \eqno{(3)}
$$

Writing
$\nabla_{A,a,Y_i}[\mu_{0,0}(\Psi)]=
\nabla_{Y_i}[\mu_{0,0}(\Psi)]+ \left[(A+\frac{a}{2})(Y_i),
\mu_{0,0}(\Psi)\right]$, we get now easily (using $\Psi\in L^{\infty}\cap
L^2_2$, $A\in
L^2_1\subset L^4$)   that the right hand term in (3) belongs to
$L^2_1$. We choose now $\varepsilon$ as given by  Lemma 3.3.5 and get $A\in
L^2_2$.  Now
estimate the first derivatives of the second term in (1). Taking into
account that
$L^2_2\subset L^p$ for every $p\geq 1$, we get that this derivatives belong to
$L^{4-\lambda}$, hence $\Psi\in L^{4-\lambda}_2$, for every
$\lambda>0$.

Now we can prove that right hand term in (2) belongs to $L^2_1$. For
instance we can
estimate the term  $A\cdot\nabla_\gamma\Psi$ as follows
$$\parallel  \nabla_\gamma( A\cdot\nabla_\gamma\Psi)
\parallel_{L^2}\leq \parallel A\parallel_{L^4_1}\parallel
\Psi\parallel_{L^4_1}+\parallel A\parallel_{L^p}\parallel \Psi\parallel _{L^{4-\lambda}_2}\
$$
for every $p$ such that $\frac{1}{p}+\frac{1}{4-\lambda}\leq \frac{1}{2}$.
Therefore
$\Psi\in L^2_3$, which is a Banach algebra. It follows that  the right hand
term in (3) is in
$L^2_2$, which implies  $A\in L^2_3$, by Lemma 3.3.5. Since we are now in
the  "stable
range", we can prove easily by induction that $\Psi,\ A\in L^2_k$, for
every $k>3$.
\hfill\vrule height6pt width6pt depth0pt \bigskip

\subsubsection{Removable singularities and Uhlenbeck compactification for
the perturbed
moduli spaces}

We mention without proof the following result, which is analogous to of
Lemma 2.2.3.

\begin{re}  The conclusion of Lemma 2.2.3 holds also for pairs $(A,\Psi)\in{\cal
A}({\Bbb P}(E_s|_{S^{\bullet}}))\times A^0(S^+_s\otimes E_s|_{S^{\bullet}})$
which solve the
\underbar{perturbed} $PU(2)$-monopole equation in a punctured neighbourhood of $0$ if we
assume
$$\int\limits_{S^{\bullet}}|F_A|^2<8\pi^2\ , \ \
\sup\limits_{S^{\bullet}}|\Psi|<\infty\ .$$
\end{re}

Let $a\in A^1(B,u(1))$, and ${\germ p}$ a perturbing parameter for the
$PU(2)$-monopole
equations
on the ball.
\begin{thry} (Removable singularities)  Let $(A_0,\Psi_0)$
be a solution of the perturbed $PU(2)$-monopole equations for the triple
$({\germ u}_{\gamma}|_{B^{\bullet}},a|_{B^{\bullet}},{\germ p}|_{B^{\bullet}})$  such that
$$\int_{B^{\bullet}}|F_{A_0}|^2<\infty\ ,\   \
\sup\limits_{S^{\bullet}}|\Psi|<\infty\ .\ .$$
There exists  a solution
$(\tilde A_0,\tilde\Psi_0)$ of the perturbed monopole equation for  the triple
$({\germ u}_{\gamma},a,{\germ p})$ and
   a gauge transformation
$\rho\in SU(E_0|_{B^{\bullet}})$ such that
$$\rho^*(\tilde
A_0|_{B^{\bullet}} , \tilde\Psi_0|_{B^{\bullet}}) = (A_0,\Psi_0)\ . $$
\end{thry}

{\bf Proof: } The proof is completely similar to the proof of Theorem 2.2.4. The only
difference is the
way in which one gets a uniform bound for the $L^2_1$-norms of the pairs
$(\tilde
C_{R,r},\tilde\Phi_{R,r})$ as $r\rightarrow 0$. For the connection
component this follows
immediately from (the result, which is analogous to of) formulas (19), (20)
in the proof of
Theorem 2.2.4. For the
spinor component this follows immediately from (19) and the elliptic
estimates associated
 to
the equation
$$({\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_{0,a}+\lambda)\tilde\Phi_{R,r}=
({\raisebox{0.15ex}{$\not$}}{\hskip -1pt {D}}_{\tilde C_{R,r},a}+ \lambda) \tilde\Phi_{R,r}-
\gamma(\tilde C_{R,r}) \tilde\Phi_{R,r}\
.$$
\hfill\vrule height6pt width6pt depth0pt \bigskip
\vspace{0.5cm}

Let $(X,g)$ be a compact oriented Riemannian  4-manifold and $L$ a
Hermitian line bundle
on $X$ and $a\in{\cal A}(L)$.  Since we are going to work with different
$Spin^{U(2)}(4)$-structures in
$X$ it is more convenient to identify the space ${\cal P}$ of perturbations with
$\Lambda^1_{\Bbb C}\times GL(\Lambda^2_+)\times\prod_i {\rm End}(\Lambda^2_+)$, which
depends
only on the base  manifold $(X,g)$. Fix    a positive number
$\varepsilon>0$ and denote by ${\cal P}_\varepsilon\subset {\cal P}$ the
open ${\cal
C}^1$-neighbourhood of $(0,{\rm id}_{\Lambda^2_+},0)$ in the space of
perturbations ${\cal
P}:=\Lambda^1_{\Bbb C}\times GL(\Lambda^2_+)\times\prod_i {\rm End}(\Lambda^2_+)$
which is given
by Proposition 3.3.1.  ${\cal P}_\varepsilon$ has the property   that the
conditions $(o)$
and $(c)$ in 3.3.1.  become equivalent, for any
$Spin^{(U(2)}(4)$-structure of determinant $L$, as soon as the perturbation
parameter
belongs to ${\cal P}_\varepsilon$.

We recall that for a  $Spin^{(U(2)}(4)$-structure $\sigma$ of determinant
$L$, and
perturbation parameter  ${\germ p}\in{\cal P}_\varepsilon$ the perturbed moduli  space
${\cal M}^\sigma_{a,{\germ p}}$ was defined to be the space of solutions
$[(A,\Psi)]$ of the
corresponding perturbed equations which satisfy one of the two equivalent
inequalities $(o)$,
$(c)$.

\begin{dt}  For ${\germ p}\in{\cal P}_\varepsilon$ we define the moduli  space
$I{\cal M}^\sigma_{a,{\germ p}}$ of ideal monopoles of type $(\sigma,a,{\germ p})$ to
be the space of
pairs $([A',\Psi'],s)$ consisting of an element $s=\{x_1,\dots,x_l\})$    in the
$l$-th-symmetric power $S^l(X)$ of
$X$, and  a monopole $[A',\Psi']\in{\cal M}^{\sigma'}_{a,{\germ p}}$, where
$\sigma':P'^u\longrightarrow P_g$ is
the $Spin^{U(2)}(4)$-structure on $X$ characterized by
$$\det (P'^u)=L\ ,\ \ l= \frac{1}{4}\left(p_1(\bar\delta(P'^u)) -
(p_1(\bar\delta(P^u))\right)
\ .
$$
\end{dt}

An essential point needed in the definition of the metric topology on the
non-perturbed space of
ideal monopoles was the existence of an apriori bound for the measures
$|F_A|^2+8\pi^2\sum_i\delta_{x_i}$ associated to ideal solutions. This
follows immediately
in the non-perturbed case from the ${\cal C}^0$-boundedness of the spinor
component, which
gives a bound of the measure $|F_A^+|$. Because of the term $\sum\limits_i
K_i(\nabla_{a,A,Y_i}(\mu_{00}(\Psi))$ in the second perturbed equation, the
boundedness of
the measure $|F_A^+|^2$ is no longer obvious, but follows easily from the
following
\begin{lm} Let ${\germ p}\in{\cal P}_\varepsilon$. There is an apriori bound of
the total
volume of the measure $|\nabla_{A}\Psi|^2$ on the space $I{\cal
M}^\sigma_{a,{\germ p}}$ of ideal
monopoles of type $(\sigma,a,{\germ p})$.
\end{lm}

{\bf Proof: }  Integrate the inequality $(i)$ in the proof of Proposition 3.3.1, and
recall that by the
definition of   ${\cal P}_\varepsilon$,  the bracket
$(1-\frac{c_3}{2}\ u-\frac{c_2}{2}\ w\ (M+\varepsilon)^2)$ which bounds
from below the
coefficient of   $|\nabla_{A}\Psi|^2$ in this inequality, is positive. Then
use the boundedness
of the spinor.
\hfill\vrule height6pt width6pt depth0pt \bigskip

Put now a metric topology on the space $I{\cal M}^\sigma_{a,{\germ p}}$ of ideal
monopoles of type $(\sigma,a,{\germ p})$
precisely as in the non-perturbed
case. With this definition one has

\begin{thry}
Let ${\germ p}\in{\cal P}_\varepsilon$ The moduli space  ${\cal
M}^\sigma_{a,{\germ p}}\subset I{\cal M}^\sigma_{a,{\germ p}}$  is an open subspace
with compact
closure $\overline{{\cal M}^\sigma_a}$.
\end{thry}

{\bf Proof: } Consider a sequence $[(A_n,\Psi_n)]\in {\cal
M}^\sigma_{a,{\germ p}}$. As in the non-perturbed case, we show using the
compactness results
and the "Removable Singularities" theorem above, that there exists a subsequence
$(n_m)\subset {\Bbb N}$, a finite set $\{x_1,\dots.x_l\}$ and  gauge
transformations $f_m$ such
that \\
1. The sequence of measures $|F_{A_{n_m}}|^2$ converges weakly to a measure
$\mu$.\\
2. $(B_m,\Phi_m):= f_m^*(A_{n_m},\Psi_{n_m})$ converges on
$X\setminus\{x_1,\dots,x_l\}$ in the
${\cal C}^{\infty}$-topology to a solution $(A_0,\Psi_0)$ which can be
extended to a solution
$(\tilde A_0,\tilde\Psi_0)$ of the perturbed  equations
$SW^{\sigma'}_{a,{\germ p}}$ associated to
a $Spin^{U(2)}(4)$ structure $\sigma'$.

Moreover one has
$$\mu=|F_{\tilde A_0}|^2+ 8\pi^2\sum  \lambda_i\delta_{x_i}
$$

It only remains to prove that the $\lambda_i$'s are integers and that
$\sum\limits_i\lambda_i= \frac{1}{4}\left(p_1 (\bar\delta(P'^u)-
p_1(\bar\delta(P^u))\right)$.

This follows with the same method as  in the non-perturbed case, but one
needs the below
lemma, which shows that the self-dual component of the curvature-sequence
associated to a
sequence of smooth solutions converging to a singular solution cannot
bubble in the
singularities. This is of course obvious in the non-perturbed case.
\hfill\vrule height6pt width6pt depth0pt \bigskip
\begin{lm} Let $x\in\{x_1,\dots, x_l\}$. There exists a positive constant
$C$ with the following
property: For every $r>0$, $\eta>0$ there exists $n_{r,\eta}\in{\Bbb N}$ such that
 %
$$  \int\limits_{B(x,r)}  |F_{B_m}^+|^2  < CVol(B(x,r))+\eta\ ,\
\ \forall m\geq n_{r,\eta}
$$
\end{lm}
{\bf Proof: } Indeed, by the second monopole equation and the apriori boundedness of
the spinor
component one gets an uniform  estimate of the form
$$\int\limits_{B(x,r)}  |F_{B_m}^+|^2\leq C_1 Vol(B(x,r)) +
C_2\int\limits_{B(x,r)}|\nabla_{B_m}\Phi_m|^2   \ .
$$

Using the same argument as in Lemma 3.3.10 and the notations of Proposition
3.3.1,
we get
further
$$
\int\limits_{B(x,r)}  |F_{B_m}^+|^2\leq C_1 Vol(B(x,r)) +
C_3\int\limits_{B(x,r)}(1 - \frac{c_3}{2}\ u-\frac{c_2}{2}\ w\
(\Phi_m+\varepsilon)^2)| \nabla_{B_m}\Phi_m|^2
$$

Now we integrate the inequality $(i)$ in the proof of Proposition 3.3.1
using   Stokes
Theorem on the left. By the boundedness of the spinor component again, we
get an estimate
of the form
$$ \int\limits_{B(x,r)}  (1-\frac{c_3}{2}\ u-\frac{c_2}{2}\ w\
(\Phi_m+\varepsilon)^2)| \nabla_{B_m}\Phi_m|^2\leq C_4
Vol(B(x,r)) + \left| \int\limits_{S(x,r)}*  d|\Phi_n|^2 \right|
$$

But in a neighbourhood of  ${S(x,r)}$  we have $\Phi_m\rightarrow \tilde
\Psi_0$ in the
Fr\'echet ${\cal C}^{\infty}$-topology, hence we can find $n_{r,\eta}\in{\Bbb N}$
such that
$$\left|\int\limits_{S(x,r)}* d|\Phi_n|^2\right| \leq
  \left|\int\limits_{B(x,r)}| \Delta \tilde \Psi_0|^2 \right| +
\eta\leq C_4 Vol(B(x,r))
+\eta\  \ ,\ \ \forall m\geq n_{r,\eta} \ ,
$$
for  a constant $C_4$ depending on $\tilde \Psi_0$.

\hfill\vrule height6pt width6pt depth0pt \bigskip

\vspace{1cm}

\hfill\vrule height6pt width6pt depth0pt \bigskip

\newpage

\centerline{\large{\bf References}}
\vspace{10 mm}
\parindent 0 cm

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\vspace{1cm}\\
Author's address : \\

Mathematisches Institut, Universit\"at Z\"urich,  Winterthurerstr. 190, \\
CH-8057 Z\"urich,  {  e-mail}:  teleman@math.unizh.ch

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