%---------------------------------
%
% This is an AMSTeX document, for version 2.1 of the AMSPPT style
%
% "The genus of embedded surfaces in the projective plane"
%			P. B. Kronheimer and T.S. Mrowka
%			Version: 14/11/94

\input amstex
\magnification=\magstephalf
\documentstyle{amsppt}
\hoffset = .4 truein
\voffset = .6 truein
\refstyle{C}
\define\CPtwo{\Bbb C\Bbb P^2}
\define\CPtwobar{\rlap{$\bar{\phantom{\Bbb C\Bbb P}}$}\CPtwo}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\topmatter
\title
		The Genus of Embedded Surfaces in\\
		the Projective Plane%
\endtitle 
\rightheadtext	
		{EMBEDDED SURFACES IN $\CPtwo$}
\author		
		P. B. Kronheimer and T. S. Mrowka
\endauthor
\address	
		Mathematical Institute, 
		24--29 St.~Giles, Oxford OX1 3LB
\endaddress
\email		
		kronheim\@maths.oxford.ac.uk
\endemail
\address
		California Institute of Technology,
		Pasadena,
		CA 91125
\endaddress
\email
		mrowka\@cco.caltech.edu
\endemail
\subjclass
		Primary 57R57, 57N13, 57R55; Secondary 57R40, 14J60
\endsubjclass
\thanks		
		First author supported by an Advanced Fellowship
		from the EPSRC.	Second author partially supported by NSF/NYI
		grant number DMS-9357641 and a Sloan Fellowship.
\endthanks
\abstract
		We show how the new invariants of 4-manifolds resulting
		from the Seiberg-Witten monopole equation lead quickly
		to a proof of the `Thom conjecture'.
\endabstract
\endtopmatter

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\document

\head			1. Statement of the result
\endhead
%%
The genus of a smooth algebraic curve of degree $d$ in $\CPtwo$ is given by
the formula $g=(d-1)(d-2)/2$.  A conjecture sometimes attributed to Thom
states that the genus of the algebraic curve is a lower bound for the
genus of any smooth $2$-manifold representing the same homology class. The
conjecture has previously been proved for $d\le4$ and for $d=6$, and
less sharp lower bounds for the genus are known for all degrees
\cite{5,6,7,10,14}.  In this note we confirm the conjecture.

\proclaim{Theorem 1} Let $\Sigma$ be an oriented $2$-manifold smoothly
embedded in $\CPtwo$ so as to represent the same homology class as
an algebraic curve of degree $d$.  Then the genus $g$ of\/ $\Sigma$ 
satisfies $g\ge (d-1)(d-2)/2$. 
\endproclaim

Very recently, Seiberg and Witten \cite{12,13,15} introduced new invariants of
4-manifolds, closely related to Donaldson's polynomial invariants
\cite{2}, but in many respects much simpler to work with.
The new techniques have led to more elementary proofs of many theorems
in the area.  Given the monopole equation and the vanishing theorem
which holds when the scalar curvature is positive (something which was
pointed out by Witten), the rest of the argument presented here is not
hard to come by.  A slightly different proof of the Theorem, based on
the same techniques, has been found by Morgan, Szabo and Taubes.

It is also possible to prove a version of Theorem 1 for other complex
surfaces, without much additional work, so reproducing and extending
results of \cite{8} and \cite{9}.  This and various other applications
will be treated in a later paper, with joint authors.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\head	2. The monopole equation and the Seiberg-Witten invariants
\endhead
%%
Let $X$ be an oriented, closed Riemannian 4-manifold.  Let a spin$^c$
structure on $X$ be given.  We write $c$ for the spin$^c$ structure and
write $W^+=W^+_c$ and $W^-=W^-_c$ for the associated 
spin$^c$ bundles.  Thus $W^+$ is a $U(2)$ bundle and Clifford multiplication
leads to a linear isomorphism $\rho=\rho_c$ from $\Lambda^+(X)\otimes{\Bbb
C}$ to the associated $PSL(2,{\Bbb C})$ bundle: 
 $$
	\rho:  \Lambda^+\otimes{\Bbb C} \to \frak{sl}(W^+).
 $$
The usual conventions ensure that $\rho$ carries the real forms to
$\frak{su}(W^+)$ and that if $e_i$ ($i=1,\ldots,4$) are an oriented
orthonormal frame at a point then $\rho(e_1\wedge e_2 + e_3\wedge e_4)$ has
eigenvalues $\pm 2i$.  There is a pairing
 $$
	\sigma: W^+ \times\bar W^+ \to \frak{sl}(W^+)
 $$
modelled on the map ${\Bbb C}^2 \times \bar{\Bbb C}^2 \to \frak{sl}(2)$
given by 
 $$
	(v,w) \mapsto  i (v \bar w^t)_o,
 $$
where the subscript means the trace-free part. Let $L$ denote the
determinant of $W^+$, a line bundle on $X$.  The equations introduced by
Seiberg and Witten are the following pair of equations for a unitary
connection $A$ in $L$ and a section $\Phi$ of $W^+$:
 $$
	\eqalign{
		D_A \Phi &= 0 \cr
		\rho( F_A^+) &=  i\sigma(\Phi,\Phi).\cr} \eqno (*)
 $$
Here $D_A:\Gamma(W^+)\to\Gamma(W^-)$ is the Dirac operator on the spin$^c$
bundle, and $F_A^+$ is the self-dual part of the curvature of $A$. The gauge
group $\text{Map}(X,S^1)$ acts on the set of solutions via its action as
scalar automorphisms of $W^+$, and we write $M_c$ for the moduli space, the 
quotient of the set of solutions by the gauge group. We call a solution {\it
irreducible} if $\Phi$ is not identically zero. Such solutions form free
orbits of the gauge group, while the reducible solutions ($\Phi=0$) have
stabilizer $S^1$. 

If the homology of $X$ has no $2$-torsion, the spin$^c$ structure $c$ is
entirely determined by the topological type of $L$, which may be any line
bundle whose first Chern class is an integral lift of $w_2$.  In general,
when $L$ is fixed, the spin$^c$ structures with determinant $L$ are a
principal space for the $2$-torsion part of $H^2(X;\Bbb Z)$. When
necessary, we write $M_c(g)$ to indicate the dependence on the Riemannian
metric $g$.

\proclaim{Lemma 2} Any solution $(A,\Phi)$ satisfies the $C^0$
bound 
 $$
	|\Phi|^2 \le \max(0,-s)
 $$
at the points where $|\Phi|$ is maximum.  Here $s$ is the scalar curvature
of $X$.
\endproclaim

\demo{Proof} The Weitzenb\"ock formula for the Dirac operator reads
 $$
	D_A^* D_A \Phi = \nabla^*_A\nabla_A\Phi + \frac s4\Phi -
	\frac12\rho(F_A^+)\Phi. 
 $$
At points where $|\Phi|^2$ is maximum we calculate
 $$
	\eqalign{
	      0 &\le \Delta|\Phi|^2 \cr
		&= 2 \langle \nabla^*_A\nabla_A\Phi,\Phi\rangle 
			- 2 \langle \nabla_A\Phi,\nabla_A\Phi\rangle\cr
		&\le 2 \langle \nabla^*_A\nabla_A\Phi,\Phi\rangle \cr
		&= -\frac s2|\Phi|^2 + \langle
		   i\sigma(\Phi,\Phi)\Phi,\Phi\rangle\cr 
		&= -\frac s2|\Phi|^2 - \frac12|\Phi|^4. \cr }
 $$
The identity used in the last line can be verified from the model of
$\sigma$ given above.  If $|\Phi|^2$ is non-zero at the
maximum, we may divide by $|\Phi|^2$ to obtain the bound $|\Phi|^2\le-s$.
\qed \enddemo

\proclaim{Corollary 3} If $(A_i,\Phi_i)$
is a sequence of solutions on $X$, then there is a subsequence
$\{i'\}\subset\{i\}$ and gauge transformations $g_{i'}$ such that the
sequence $g_{i'}(A_{i'}, \Phi_{i'})$ converges in $C^\infty$.
Thus $M_c$ is compact. \endproclaim

\demo{Proof} The $C^0$ bound on $\Phi$ makes this entirely straightforward.
Choose any smooth connection $B$ in $L$.  We can find gauge transformations
$g^o_i$ in the identity component of $\text{Map}(X,S^1)$ so that the 1-form
$g^o_i(A_i)-B$ is co-closed.  The component group of $\text{Map}(X,S^1)$ is
isomorphic to $H^1(X;\Bbb Z)$, and by choosing a gauge transformation $g_i$
from an appropriate component, we can arrange further that the harmonic part
of $g_i(A_i)-B$ is bounded, independent of $i$, because $H^1(X;\Bbb
R)/H^1(X;\Bbb Z)$ is compact. The equations and the $C^0$ bound on $\Phi$
then give an $L^p_1$ bound on $g_i(A_i)-B$ as well as on $g_i(\Phi_i)$ for
any $p$.  By a bootstrapping argument, one obtains bounds on all higher
derivatives (depending on $B$). There is therefore a convergent subsequence.
\qed\enddemo

On a manifold with boundary, a suitable gauge-fixing condition for a 1-form
$g^o(A)-B$ is that it be co-closed and annihilate the normal vectors at the
boundary.  The argument above then proves:

\proclaim{Lemma 4} If $Z$ is a compact, oriented Riemannian $4$-manifold with
boundary, equipped with a spin$^c$ structure, and if $(A_i,\Phi_i)$ is a
sequence of solutions on $Z$ with $|\Phi_i|$ uniformly bounded, then there
is a subsequence $\{i'\}\subset\{i\}$ and gauge transformations $g_{i'}$
such that the sequence $g_{i'}(A_{i'}, \Phi_{i'})$ converges in
$C^\infty$.\qed\endproclaim

The monopole equations $(*)$ can be perturbed in a gauge-invariant fashion
by adding a form to the right-hand side of the second equation: if
$\delta\in\Omega^+(X)$, we consider the equations
 $$
	\eqalign{
		D_A \Phi &= 0 \cr
		\rho( F_A^++i\delta) &=  i\sigma(\Phi,\Phi).\cr} \eqno
		(*_\delta) 
 $$
Let $M_{c,\delta}$ denote the moduli space of solutions.  A $C^0$ bound on
$|\Phi|^2$ can be obtained from the equations as before (one only has to
replace $-s$ in Lemma 2 by $-s+2|\rho(\delta)|$).  Let ${\Cal M}_c$ denote
the parametrized moduli space, that is:
 $$
	\eqalign{
		{\Cal M}_c &= \bigcup_{\delta\in\Omega^+}
		M_{c,\delta}\times\{\delta\} \cr
			   &\subset \bigl({\Cal
			   A}\times\Gamma(W^+)\times\Omega^+\bigr)\big/
			   \text{Map}(X,S^1), \cr}
 $$
where $\Cal A$ is the space of unitary connections in $L$.  It is also
convenient to allow $A$ and $\Phi$ to be of class $L^2_k$, the form $\delta$
to be of class $L^2_{k-1}$ and the gauge transformations to be of class
$L^2_{k+1}$ for some large $k$, so that ${\Cal M}^*_c$ is a subset of a
Banach manifold. Here we are writing $M^*_c$ and $\Cal M^*_c$ for the moduli
space and parametrized moduli space of irreducible solutions. We omit
mention of $k$ from our notation.   

\proclaim{Lemma 5} The irreducible part of the parametrized moduli space
$\Cal M^*_c$ is a smooth manifold.   The projection $\pi:\Cal
M^*_c\to\Omega^+$ is Fredholm of index
 $$
	d(L)=\frac14 \bigl( c_1(L)^2 - (2\chi + 3\sigma)\bigr),
 $$
where $\chi$ and $\sigma$ are the Euler number and signature of $X$.
The projection of the whole space $\Cal M_c$ to $\Omega^+$ is proper. 
\endproclaim

\demo{Proof} Let $(A,\Phi,\delta)$ be a solution of the equations with
$\Phi$ non-zero.  The linearization of the equations at this point is the
operator 
 $$
	P : \Omega^1(i\Bbb R) \times \Gamma(W^+) \times \Omega^+
		\to \Gamma(W^-) \times i\frak{su}(W^+)
 $$
given by
 $$
	P(a,\phi,\epsilon) = \left(D_A\phi + a\Phi,\ \rho(d^+a + 
	i\epsilon) + 2\text 
	{Im}\sigma(\phi,\Phi) \right).
 $$
To establish surjectivity, it is sufficient to show that no element
$(\psi,\kappa)$ in the range can be $L^2$-orthogonal to the image of $P$.
But if $(\psi,\kappa)$ is orthogonal, then by varying $\epsilon$ alone one
can first see that $\kappa$ is zero.  Then by varying $a$ alone and using
the fact that $\Phi$, being a solution of the Dirac equation, cannot vanish
on an open set, one sees that $\psi$ is zero also.  So the set of
irreducible solutions is a manifold, by the implicit function theorem.  The
gauge group acts freely, and the quotient $\Cal M^*_c$ is a
manifold also, as slices for the gauge group action can be constructed using
the Coulomb gauge fixing condition.

To calculate the Fredholm index taking account of the Coulomb condition, one
must calculate the index of a linear operator $P+d^*$ which is a compact
perturbation of the sum of the two operators
 $$
	\displaylines{
	D_A : \Gamma(W^+)\to\Gamma(W^-)\cr
	d^* + d^+ : \Omega^1 \to \Omega^0\oplus \Omega^+.\cr}
 $$
The real index of the first is $\frac14 c_1(L)^2 - \frac14 \sigma$, and the
index of the second is $-\frac12 (\chi + \sigma)$.  The sum of these two
indices is $d(L)$. The last clause of the Lemma is, like Corollary 3, a
consequence of Lemma 2.\qed \enddemo

We can now define a simple mod 2 version of the invariants of Seiberg and
Witten.  Suppose we have a spin$^c$ structure $c$ such that the determinant
bundle $L$ satisfies $d(L)=0$.  (Such a spin$^c$ structure may or may not
exist: the condition is the same as that for $X$ to have an almost complex
structure). Call a Riemannian metric $g$ `good' if the image of $c_1(L)$ in
real cohomology is not represented by an anti-self-dual form.  If $g$ is
good, then there exists a positive $\epsilon$ such that the equation
$F^+_A+i\delta=0$ has no solutions when $|\delta|<\epsilon$ (we may take the
$C^0$ norm).  Let $\delta$ be a regular point of the projection $\pi$ in the
Lemma above with $|\delta|<\epsilon$, so that the fibre
$M^*_{c,\delta}=M_{c,\delta}$ is a compact zero-manifold. Define
$n_c(g)$ to be the number of points in this moduli space,
counted mod 2:
 $$
	n_c(g)  = \# M_{c,\delta}(g) \pmod 2.
 $$
If $\delta_1$ and $\delta_2$ are both regular values with $C^0$ norm less
than $\epsilon$, we can join them by a path which is transverse to $\pi$;
the inverse image of this path is a compact 1-manifold, with boundary the
union of the two moduli spaces, so $n_c(g)$ is independent of $\delta$.
Similarly, if $g_1$ and $g_2$ are two good metrics which can be joined by a
path of good metrics, then $n_c(g_1)=n_c(g_2)$.

As shown in \cite{1}, if $b^+$ is bigger than 1 and $L$ is non-trivial, the
space of good metrics is path-connected.  In this case, $n_c$ is an
invariant of the pair $(X,c)$ alone, independent of the metric.  Our
application of the invariants, however, deals with a manifold with $b^+=1$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\head	3. The invariants of $\CPtwo\# n\CPtwobar$
\endhead
%%
Let $\CPtwo$ be the complex projective plane with its standard orientation,
and let $X$ be the manifold $\CPtwo\# n\CPtwobar$.  As a basis for the
second cohomology, we take in $\CPtwo$ the Poincar\'e dual $H$ of the class
represented by the projective line with its complex orientation, and in the
$i$th copy of $\CPtwobar$ a similar class which we call $E_i$.  The
quadratic form on $H^2(X;\Bbb R)$ has signature $(1,n)$, and we write $C^+$
for the component of the open positive cone in $H^2(X;\Bbb R)$ which
contains $H$.  For each Riemannian metric on $g$, there is a self-dual
harmonic form $\omega_g$ whose cohomology class lies in $C^+$.  This form is
unique up to positive scaling. The condition that a metric $g$ be good can
be rephrased as the condition $c_1(L)\smallsmile[\omega_g]\ne0$.

The manifold $X$ is a complex manifold in a standard way, as the blow-up of
the projective plane at $n$ points.  There is therefore a preferred spin$^c$
structure $c$.  The corresponding line bundle $L$ has $c_1(L)=c_1(X)$ and
$d(L)=0$. In terms of the basis introduced above, and writing $E=\sum E_i$,
we have
 $$
	c_1(L) = 3H - E.
 $$
Note that since $[\omega_g]$ is a class of positive square, we cannot have
$c_1(L)\smallsmile[\omega_g]=0$ unless $c_1(L)$ has negative square; that
is, unless $n>9$.

\proclaim{Proposition 6} 
For this standard spin$^c$ structure, the invariant $n_c(g)$ for a good
metric $g$ is $0$ mod\/ $2$ if $c_1(L)\smallsmile[\omega_g]$ is positive,
and\/ $1$ mod\/ $2$ if $c_1(L)\smallsmile[\omega_g]$ is negative.
\endproclaim

\demo{Proof} Let $g_t$ ($t\in[\,-1,1\,]$) be a smooth path of Riemannian
metrics whose endpoints are good.  By the argument of \cite{1}, we can
perturb the path so that the path $[\omega_{g_t}]$ in $C^+$ is
transverse to the hyperplane $c_1(L)^\bot$.  Let $k$ be the number of times
$[\omega_{g_t}]$ meets the hyperplane (which is the number of times that
$c_1(L)\smallsmile[\omega_{g_t}]$ changes sign).  We first show that
$n_c(g_{-1})$ and $n_c(g_1)$ differ by the parity of $k$.

Without loss of generality, we assume that $c_1(L)\smallsmile[\omega_{g_t}]$
changes sign just once, at $t=0$. The parametrized moduli space
 $$
	\bold M_c = \bigcup_t M_c(g_t) \times \{t\}
 $$
then contains a single reducible point, in the fibre over $0$.  Choose a
$\delta\in\Omega^+$ so as to perturb the equations, and make it sufficiently
small that the expression
 $$
	f(t) =c_1(L)\smallsmile [\omega_{g_t}] + 2\pi\int
	\delta\wedge\omega_{g_t}
 $$
changes sign once, transversely.  We may assume this happens still at $t=0$.
There is then a single reducible solution of the perturbed equations at
$t=0$, and at all other points the perturbed space $\bold
M_{c,\delta}$ is a smooth 1-manifold, with boundary the union of
$M_{c,\delta}(g_1)$ and $M_{c,\delta}(g_{-1})$.  We must show that an odd
number of arcs emerge from the reducible point.

Let us examine the Kuranishi model of $\bold M_{c,\delta}$ in the
neighbourhood of the reducible solution $(A_0,0)$.  The model is
$\psi^{-1}(0)/S^1$, where $\psi$ is (the germ of) an $S^1$-equivariant map 
 $$
	\psi : {\Bbb R} \oplus \text{Ker}(D_{A_0}) \to \text{Coker}(d^+)
	\oplus \text{Coker}(D_{A_0})
 $$
and the $S^1$ action is the standard scalar multiplication on the kernel and
cokernel of $D_{A_0}$.  The ${\Bbb R}$ on the left-hand side represents the
$t$ coordinate.  On the right-hand side, $\text{Coker}(d^+)$ is
$1$-dimensional, and if $\text{Coker}(D_{A_0})$ has complex dimension $r$
then $\text{Ker}(D_{A_0})$ has dimension $r+1$ by the index theorem.  If we
identify $\text{Coker}(d^+)$ with $\Bbb R$ via the map $\theta\mapsto \int
\theta\wedge\omega_{g_0}$, then we can calculate
 $$
	{\partial\psi\over\partial t}|_{(0,0)} = {i\over2\pi}(f'(0), 0),
 $$
following \cite{1}.  We have arranged that $f'(0)$ is non-zero, so the local
model can be reduced to $\psi_1^{-1}(0)/S^1$, for some $\psi_1: \Bbb C^{r+1}
\to \Bbb C^r$.  Removing the singular point, we obtain a model for a deleted
neighbourhood of the reducible in $\bold M_{c,\delta}$ as the zero set of a
section $\bar\psi_1$ of the bundle $\Bbb C^r\otimes\Cal O(1)$ on
$\Bbb{CP}^r\times\Bbb R^+$.  Here $\Cal O(1)$ is the line bundle of
degree 1 on $\Bbb{CP}^r$, pulled back to the punctured cone
 $$
  \Bbb{CP}^r\times\Bbb R^+ = (\Bbb C^{r+1}\setminus\{0\})/S^1.
 $$  

The zero set of $\bar\psi_1$ is transverse, by our assumption on the
smoothness of $\bold M^*_{c,\delta}$.  For an open dense set of numbers
$\epsilon$, $\bar\psi_1^{-1}(0)$ meets $\Bbb{CP}^r\times\{\epsilon\}$ in
an odd number of points, because the Euler number of the bundle $\Bbb
C^r\otimes\Cal O(1)$ on $\Bbb{CP}^r$ is $1$.  This completes the proof
that $n_c(g_{-1})$ and $n_c(g_1)$ differ in parity as claimed.

To complete the proof of Proposition 6, it is sufficient to exhibit a metric
$g_+$ for which $c_1(L)\smallsmile[\omega_{g_+}]$ is positive and for which
$M_{c}(g_+)$ is empty.  In \cite{4} it is shown that $\CPtwo\#n\CPtwobar$
admits K\"ahler metrics $g_+$ of positive scalar curvature.  In the K\"ahler
situation, $\omega_g$ is a positive multiple of the K\"ahler form, and its
cup-product with $c_1(L)$ has the same sign as the scalar curvature. The
moduli space for this metric is empty by Lemma~2, so both of our
requirements are satsified. An alternative way to obtain a metric $g_+$ with
the required properties is to use the constructions of \cite{11} and
\cite{3}. \qed\enddemo

\proclaim{Corollary 7} If $g$ is a Riemannian metric on
$X=\CPtwo\#n\CPtwobar$ with $c_1(L)\smallsmile[\omega_g]$ negative, then the
moduli space $M_c(g)$ for the line bundle $L$ with $c_1(L)=c_1(X)$ is
non-empty.\qed \endproclaim

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\head	4. Solutions on tubes
\endhead
%%
We return to a general oriented Riemannian 4-manifold having a spin$^c$
structure $c$ of determinant $L$.  Suppose $X$ contains an oriented 3-manifold
$Y$ which separates $X$ into two components $X_-$ and $X_+$ with common
boundary, and suppose that the Riemannian metric is a product
$[-\epsilon,\epsilon]\times Y$ in a collar of $Y$.   Let $(X_R, g_R)$ be the
Riemannian manifold obtained from $X$ by cutting open along $Y$ and
inserting a cylinder $[-R,R]\times Y$. (So $g_0$ is the original metric.)
We identify $X_R$ with $X$ and write $M_c(g_R)$ for the moduli space.

\proclaim {Proposition 8} Suppose the moduli space $M_c(g_R)$ is non-empty
for all sufficiently large $R$.  Then there exists a solution of the
equations on the cylinder $\Bbb R\times Y$ which is translation-invariant in
a temporal gauge. \rom(The spin$^c$ structure on the cylinder is the
restriction of the spin$^c$ structure on $X$.\rom) \endproclaim

\noindent
(After identifying the line bundle $L$ on $\Bbb R\times Y$ with the
pull-back of a line bundle on $Y$, we say that a solution $(A,\Phi)$ is in
temporal gauge if the $dt$ component of $A$ vanishes.)

\demo {Proof} On the cylinder $\Bbb R\times Y$, the two spin$^c$ bundles
$W^+$ and $W^-$ can be identified using Clifford multiplication by $dt$, and
both can be identified with the pull-back of the $U(2)$ bundle $W_3\to Y$
associated to the spin$^c$ structure which $Y$ obtains.  Let $A(t)$ be the
path of connections in $L\to Y$ obtained by restricting $A$ to the slices
$\{t\}\times Y$, and let $\Phi(t)$ be the path in $\Gamma(W_3)$ obtained
similarly via the above identifications.  Let
$\partial_A:\Gamma(W_3)\to\Gamma(W_3)$ be the self-adjoint Dirac operator on
the 3-manifold.  If $A$ is in temporal gauge, it can be recovered from the
path $A(t)$ in the space $\Cal A_3$ of connections on $Y$.

In temporal gauge, the equations $(*)$ become the following equations for
$A(t)$ and $\Phi(t)$:
 $$
    \eqalign{
		dA/dt &= -*F_A + i\tau(\Phi,\Phi)\cr
		d\Phi/dt &= -\partial_A \Phi.\cr }
 $$
Here $\tau$ is a pairing from $W_3\times\bar W_3$ to $\Lambda^1(Y)$,
obtained from Clifford multiplication by using the hermitian metric on
$W_3$.  These are the downward gradient flow equations for the functional
$\tfrac12C$ on $\Cal A_3\times\Gamma(W_3)$ given by
 $$
	C(A,\Phi) = \int (A-B)\wedge F_A + 
		\int\langle\Phi,\partial_A\Phi\rangle.
 $$
Here $B$ is any chosen connection on $L\to Y$, made necessary by the fact
that $\Cal A_3$ has no preferred base-point in general.

This function is invariant under gauge transformations $h$ belonging to the
identity component of the gauge group $\text{Map}(Y,S^1)$, but if $h$
has a non-trivial class $[h]\in H^1(X;\Bbb Z)$ then
 $$
	C(h(A), h(\Phi)) = C(A,\Phi)  + 8\pi^2 c_1(L)\smallsmile[h].
 $$
Let $\gamma$ be a 1-dimensional cycle in $Y$ dual to $c_1(L)$. As $Y$ sits
in $X$, this cycle is the intersection with $Y$ of a 2-cycle $\Gamma$ in $X$
dual to $c_1(L)$ on $X$.  Thus $\gamma$ bounds relative cycles $\Gamma_+$
and $\Gamma_-$ in $X_+$ and $X_-$.
	 
Now suppose that $(A_R,\Phi_R)$ is a solution of the equations on
$(X_R,g_R)$, defined for all sufficiently large $R$. Take a reference
connection $B_R$ on $L\to X_R$ obtained from  fixed connections $B_\pm$
over the pieces $X_{\pm}$ (independent of $R$) and a connection
$[-R,R]\times B$ on the cylinder (after again choosing an identification of
$L$ on the cylinder with the pull-back of a bundle on $Y$).  Without loss of
generality, $A_R$ is in temporal gauge with respect to the chosen product
structure over the cylinder.

Let $l(R)$ be the change in $C$ along the cylinder:
 $$
	l(R) = C\bigl(A_R(R), \Phi_R(R)\bigr) - 
		C\bigl(A_R(-R), \Phi_R(-R)\bigr).
 $$
This quantity is negative and independent of all choices. We claim that
$l(R)$ satisfies a uniform bound, independent of $R$.  To see this, note
first that the argument of Corollary~3 and Lemma~4 means that we can choose
gauge transformations $h^+_R$ and $h^-_R$ over $X_+$ and $X_-$ so that
$h^\pm_R(A_R)-B_\pm$ and its first derivatives are uniformly bounded.  
We therefore have uniform bounds on each of the two quantities
 $$
	C\bigl(h^\pm_R(A_R)(\pm R), h^\pm_R(\Phi_R)(\pm R)\bigr).
 $$
However, since $\gamma$ is the boundary of a chain $\Gamma_\pm$ in $X_\pm$,
the degree of $h^\pm_R$ on $\gamma\subset Y$  is zero.  From the formula
above for how $C$ changes under gauge transformation, it follows that
 $$ 
  C\bigl(h^\pm_R(A_R)(\pm R), h^\pm_R(\Phi_R)(\pm R)\bigr)
	= C\bigl(A_R(\pm R), \Phi_R(\pm R)\bigr).
 $$
This gives the desired bound on $l(R)$.

Since the equations are of a gradient flow, the value of $C$ changes
monotonically along the cylinder.  Taking $R$ to be an integer $N$, we see
that there must be, for each $N$, a solution of the equations on
$[0,1]\times Y$ for which the change in $C$ along the interval is at most
$1/N$.  Furthermore, since the infimum of the scalar curvature of $X_R$ is
independent of $R$, these solutions satisfy a uniform $C^0$ bound, by
Lemma~2. Taking a sequence of such solutions with $N$ tending to $\infty$,
we deduce from Lemma~4 that there is a solution on $[0,1]\times Y$ for
which $C$ is constant.  Such a solution is translation-invariant in a
temporal gauge, and can be extended to a translation-invariant solution on
$\Bbb R\times Y$.\qed \enddemo

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\head	5. A vanishing argument
\endhead
%%
We continue to consider a cylinder $\Bbb R\times Y$, but we now take $Y$ to
be a Riemannian product $S^1\times \Sigma$, where $\Sigma$ is a 2-manifold
of constant scalar curvature.  We choose to make $\Sigma$ of unit area, so
that its scalar curvature is $-2\pi(4g-4)$. (The scalar curvature is twice
the Gaussian curvature).  We assume the genus $g$ is at least 1.

\proclaim{Lemma 9} If there exists a solution to the equations $(*)$ on
$\Bbb R\times Y$ which is translation-invariant in a temporal gauge, then
$c_1(L)[\Sigma]$ is not greater than $2g-2$ in absolute value. \endproclaim

\demo{Proof} Suppose that $(A,\Phi)$ is a solution which is
translation-invariant in a temporal gauge. Because $Y$ is compact, $|\Phi|$
achieves its supremum, so from Lemma 2 we obtain the $C^0$ bound
$|\Phi|^2\le 2\pi(4g-4)$.  This translates into the bound
$|\sigma(\Phi,\Phi)|\le \sqrt 2\pi(4g-4)$, where the norm on
$\frak{su}(W^+)$ is the one coming from the inner product $\text{tr}(a^*b)$
(see the model for $\sigma$ in section 2.)  From the first equation of $(*)$
we then obtain
 $$
	|F^+_A| \le\sqrt 2\pi(2g-2),
 $$
because $\rho$ increases lengths by a factor of $2$.  Because $A$ is
translation-invariant in a temporal gauge, the curvature $F_A$ is a 2-form
pulled back from $Y$.  From this it follows that $F_A^+$ and $F_A^-$ have
the same magnitude, and we therefore have
 $$
	|F_A| \le 2 \pi(2g-2),
 $$
and hence
 $$
 \eqalign{
	\bigl | c_1(L)[\Sigma]\bigr| &= 
		\Bigl|{i\over2\pi}\int_\Sigma F_A\Bigr| \cr
				     &\le {1\over2\pi} 
				       \sup|F_A|\,\text{Area}(\Sigma)\cr
				     &\le 2g-2. \qed\cr}
 $$
\enddemo

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\head	6. Proof of the Theorem
\endhead
%%
We now turn to the proof of Theorem 1.  Let $\Sigma$ be a smoothly embedded
oriented surface, dual to the class $dH$ in $\CPtwo$.  If $d$ is $1$ or $2$,
the inequality of the Theorem is trivial, while if $d$ is 3 the result
was known to Kervaire and Milnor \cite{5}.  We therefore assume that $d$ is
greater than 3. In this range we may suppose, without loss of generality,
that $\Sigma$ is not a sphere.

Regard $\Sigma$ now as living in the connected sum $X=\CPtwo\#d^2\CPtwobar$.
Let $S_i$ be a sphere in the $i\,$th copy of $\CPtwobar$, dual to the class
$-E$, and let $\tilde\Sigma\subset X$ be the surface formed by taking an
internal connected sum of $\Sigma$ with all the $S_i$.  This surface is dual
to the class $dH - E$, where $E=\sum E_i$ as before.  It has the same genus
as $\Sigma$ and has trivial normal bundle.  We take the canonical spin$^c$
structure $c$ on $X$, so that $c_1(L)=3H-E$ as in section 3.

Put a Riemannian metric $g_0$ on $X$ so that the boundary $Y$ of a tubular
neighbourhood of $\tilde\Sigma$ has a product metric.  Arrange that the
metric on the factor $Y \cong S^1\times\Sigma$ is of the sort considered
above. Let $X_-$ and $X_+$ be respectively the closed tubular neighbourhood
and the closure of its complement, and let $g_R$ be the Riemannian metric
obtained by inserting a cylinder $[-R,R]\times Y$ between $X_-$ and $X_+$.

\proclaim{Lemma 10} For $R$ sufficiently large,
$c_1(L)\smallsmile[\omega_{g_R}]$ is negative.\endproclaim

\demo{Proof} Let $R_i$ be a sequence increasing to infinity, and normalize
the forms $\omega_i=\omega_{g_{R_i}}$ so that $H\smallsmile[\omega_i]=1$.
The forms $\omega_i$ then have $\|\omega_i\|_{L^2}\le 1$ and there is a
subsequence converging on compact sets to an $L^2$ form $\omega$ on the
disjoint union of the two manifolds with cylindrical ends, $X_-\cup
\left([\,0,\infty)\times Y\right)$ and $\left((-\infty,0\,]\times
Y\right)\cup X_+$.  (The convergence is understood by identifying a compact
subset of the cylindrical-end manifold with a compact subset of $(X_R,g_R)$
once $R$ is large enough.) However, the space of closed and co-closed $L^2$
2-forms on $X_-\cup \left([\,0,\infty)\times Y\right)$ is empty, being the
image of the compactly-supported cohomology in the ordinary cohomology. 
So $[\omega_i]\cdot[\tilde \Sigma]$ goes to zero as $i\to\infty$. Because of
our normalization, we have
 $$
   \eqalign{
	[\omega_i]\smallsmile c_1(L) &= [\omega_i]\cdot[\tilde\Sigma]
					- (d-3)[\omega_i]\smallsmile H\cr
				     &=[\omega_i]\cdot[\tilde\Sigma]
					- (d-3).\cr }
 $$
Since $d$ is greater than $3$, this quantity is eventually negative as
claimed. \qed
\enddemo

Corollary 7 now tells us that the moduli space $M_c(g_R)$ is
non-empty for all sufficiently large $R$, and from Proposition~8 it follows
that there is a solution on the cylinder $\Bbb R \times Y$ which is
translation-invariant in a temporal gauge.  Using Lemma~9 then, we obtain
 $$
	c_1(L)[\tilde\Sigma] \ge -(2g-2).
 $$
The left-hand side is $(3H-E)(dH-E)$, which is $3d-d^2$. This inequality is
therefore equivalent to $2g-2\ge d^2-3d$, which is equivalent to the
inequality we wished to prove.	\qed

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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\enddocument

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