THE GENUS OF EMBEDDED SURFACES IN

THE PROJECTIVE PLANE

P. B. Kronheimer and T. S. Mrowka

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’.

1. Statement of the result

The genus of a smooth algebraic curve of degree d in ℂℙ^{2} 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 ≤ 4 and for d = 6, and less sharp lower bounds for the genus are known for all degrees [5,6,7,10,14]. In this note we confirm the conjecture.

Theorem 1. Let Σ be an oriented 2-manifold smoothly embedded in ℂℙ^{2} so as to represent the same homology class as an algebraic curve of degree d. Then the genus g of Σ satisfies g ≥ (d − 1)(d − 2)/2.

Very recently, Seiberg and Witten [12,13,15] introduced new invariants of 4manifolds, closely related to Donaldson’s polynomial invariants [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 [8] and [9]. This and various other applications will be treated in a later paper, with joint authors.

2. The monopole equation and the Seiberg-Witten invariants

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

Typeset by AMS-TEX

1991 Mathematics Subject Classification. Primary 57R57, 57N13, 57R55; Secondary 57R40, 14J60.

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.

leads to a linear isomorphism ρ = ρ_{c} from Λ^{+}(X) ⊗ ℂ to the associated P SL(2, ℂ) bundle:

ρ : Λ^{+} ⊗ ℂ → 𝔰𝔩(W^{+}).

The usual conventions ensure that ρ carries the real forms to 𝔰𝔲(W^{+}) and that if e_{i} (i = 1, …, 4) are an oriented orthonormal frame at a point then ρ(e_{1} ∧ e_{2} + e_{3} ∧ e_{4}) has eigenvalues ±2i. There is a pairing

σ : W^{+} × W̄^{+} → 𝔰𝔩(W^{+})

modelled on the map ℂ^{2} × ℂ̄^{2} → 𝔰𝔩(2) given by

(v, w) ↦ i(vw̄^{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 Φ of W^{+}:

\begin{aligned}D_{A}Φ & = 0 \\ ρ(F_{A}^{+}) & = iσ(Φ, Φ). \end{aligned}  (∗)

Here D_{A} : Γ(W^{+}) → Γ(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 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 irreducible if Φ is not identically zero. Such solutions form free orbits of the gauge group, while the reducible solutions (Φ = 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; ℤ). When necessary, we write M_{c}(g) to indicate the dependence on the Riemannian metric g.

Lemma 2. Any solution (A, Φ) satisfies the C^{0} bound

|Φ|^{2} ≤ max(0, −s)

at the points where |Φ| is maximum. Here s is the scalar curvature of X.

Proof. The Weitzenböck formula for the Dirac operator reads

D_{A}^{∗}D_{A}Φ = ∇_{A}^{∗}∇_{A}Φ + \frac{s}{4}Φ − \frac{1}{2}ρ(F_{A}^{+})Φ.

At points where |Φ|^{2} is maximum we calculate

0 ≤ Δ|Φ|^{2}

= 2⟨∇_{A}^{∗}∇_{A}Φ, Φ⟩ − 2⟨∇_{A}Φ, ∇_{A}Φ⟩

≤ 2⟨∇_{A}^{∗}∇_{A}Φ, Φ⟩

= −\frac{s}{2}|Φ|^{2} + ⟨iσ(Φ, Φ)Φ, Φ⟩

= −\frac{s}{2}|Φ|^{2} − \frac{1}{2}|Φ|^{4}.

The identity used in the last line can be verified from the model of σ given above. If |Φ|^{2} is non-zero at the maximum, we may divide by |Φ|^{2} to obtain the bound |Φ|^{2} ≤ −s. □

EMBEDDED SURFACES IN ℂℙ^{2} 3

Corollary 3. If (A_{i}, Φ_{i}) is a sequence of solutions on X, then there is a subsequence {i′} ⊂ {i} and gauge transformations g_{i′} such that the sequence g_{i′}(A_{i′}, Φ_{i′}) converges in C^{∞}. Thus M_{c} is compact.

Proof. The C^{0} bound on Φ makes this entirely straightforward. Choose any smooth connection B in L. We can find gauge transformations g_{i}^{o} in the identity component of Map(X, S^{1}) so that the 1-form g_{i}^{o}(A_{i}) − B is co-closed. The component group of Map(X, S^{1}) is isomorphic to H^{1}(X; 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; ℝ)/H^{1}(X; ℤ) is compact. The equations and the C^{0} bound on Φ then give an L_{1}^{p} bound on g_{i}(A_{i})−B as well as on g_{i}(Φ_{i}) for any p. By a bootstrapping argument, one obtains bounds on all higher derivatives (depending on B). There is therefore a convergent subsequence. □

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:

Lemma 4. If Z is a compact, oriented Riemannian 4-manifold with boundary, equipped with a spin^{c} structure, and if (A_{i}, Φ_{i}) is a sequence of solutions on Z with |Φ_{i}| uniformly bounded, then there is a subsequence {i′} ⊂ {i} and gauge transformations g_{i′} such that the sequence g_{i′}(A_{i′}, Φ_{i′}) converges in C^{∞}. □

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 δ ∈ Ω^{+}(X), we consider the equations

\begin{aligned}D_{A}Φ & = 0 \\ ρ(F_{A}^{+} + iδ) & = iσ(Φ, Φ). \end{aligned}  (∗_{δ})

Let M_{c,δ} denote the moduli space of solutions. A C^{0} bound on |Φ|^{2} can be obtained from the equations as before (one only has to replace −s in Lemma 2 by −s + 2|ρ(δ)|). Let M_{c} denote the parametrized moduli space, that is:

\begin{aligned}M_{c} & = ⋃_{δ∈Ω^{+}} M_{c,δ} × \{δ\} \\ & ⊂ (A × Γ(W^{+}) × Ω^{+})/Map(X, S^{1}), \end{aligned}

where A is the space of unitary connections in L. It is also convenient to allow A and Φ to be of class L_{k}^{2}, the form δ to be of class L_{k−1}^{2} and the gauge transformations to be of class L_{k+1}^{2} for some large k, so that M_{c}^{∗} is a subset of a Banach manifold. Here we are writing M_{c}^{∗} and M_{c}^{∗} for the moduli space and parametrized moduli space of irreducible solutions. We omit mention of k from our notation.

Lemma 5. The irreducible part of the parametrized moduli space M_{c}^{∗} is a smooth manifold. The projection π : M_{c}^{∗} → Ω^{+} is Fredholm of index

d(L) = \frac{1}{4}(c_{1}(L)^{2} − (2χ + 3σ)),

where χ and σ are the Euler number and signature of X. The projection of the whole space M_{c} to Ω^{+} is proper.

Proof. Let (A, Φ, δ) be a solution of the equations with Φ non-zero. The linearization of the equations at this point is the operator

P : Ω^{1}(iℝ) × Γ(W^{+}) × Ω^{+} → Γ(W^{−}) × i𝔰𝔲(W^{+})

given by

P (a, ϕ, ϵ) = (D_{A}ϕ + aΦ, ρ(d^{+}a + iϵ) + 2Imσ(ϕ, Φ)) .

To establish surjectivity, it is sufficient to show that no element (ψ, κ) in the range can be L^{2}-orthogonal to the image of P. But if (ψ, κ) is orthogonal, then by varying ϵ alone one can first see that κ is zero. Then by varying a alone and using the fact that Φ, being a solution of the Dirac equation, cannot vanish on an open set, one sees that ψ 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 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

D_{A} : Γ(W^{+}) → Γ(W^{−})
d^{∗} + d^{+} : Ω^{1} → Ω^{0} ⊕ Ω^{+}.

The real index of the first is \frac{1}{4}c_{1}(L)^{2} − \frac{1}{4}σ, and the index of the second is −\frac{1}{2}(χ+σ). The sum of these two indices is d(L). The last clause of the Lemma is, like Corollary 3, a consequence of Lemma 2. □

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 ϵ such that the equation F_{A}^{+} + iδ = 0 has no solutions when |δ| < ϵ (we may take the C^{0} norm). Let δ be a regular point of the projection π in the Lemma above with |δ| < ϵ, so that the fibre M_{c,δ}^{∗} = M_{c,δ} 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,δ}(g) (mod 2).

If δ_{1} and δ_{2} are both regular values with C^{0} norm less than ϵ, we can join them by a path which is transverse to π; 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 δ. 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 [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.

3. The invariants of ℂℙ^{2}#n ¯ℂℙ^{2}

Let ℂℙ^{2} be the complex projective plane with its standard orientation, and let X be the manifold ℂℙ^{2}#n ¯ℂℙ^{2}. As a basis for the second cohomology, we take in ℂℙ^{2} the Poincaré dual H of the class represented by the projective line with its complex orientation, and in the ith copy of ¯ℂℙ^{2} a similar class which we call E_{i}. The quadratic form on H^{2}(X; ℝ) has signature (1, n), and we write C^{+} for the component of the open positive cone in H^{2}(X; ℝ) which contains H. For each Riemannian metric on g, there is a self-dual harmonic form ω_{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) ⌣ [ω_{g}] ≠ 0.

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 = ∑ E_{i}, we have

c_{1}(L) = 3H − E.

Note that since [ω_{g}] is a class of positive square, we cannot have c_{1}(L) ⌣ [ω_{g}] = 0 unless c_{1}(L) has negative square; that is, unless n > 9.

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) ⌣ [ω_{g}] is positive, and 1 mod 2 if c_{1}(L) ⌣ [ω_{g}] is negative.

Proof. Let g_{t} (t ∈ [ −1, 1 ]) be a smooth path of Riemannian metrics whose endpoints are good. By the argument of [1], we can perturb the path so that the path [ω_{g_{t}}] in C^{+} is transverse to the hyperplane c_{1}(L)^{⊥}. Let k be the number of times [ω_{g_{t}}] meets the hyperplane (which is the number of times that c_{1}(L) ⌣ [ω_{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) ⌣ [ω_{g_{t}}] changes sign just once, at t = 0. The parametrized moduli space

M_{c} = ⋃_{t} M_{c}(g_{t}) × {t}

then contains a single reducible point, in the fibre over 0. Choose a δ ∈ Ω^{+} so as to perturb the equations, and make it sufficiently small that the expression

f(t) = c_{1}(L) ⌣ [ω_{g_{t}}] + 2π ∫ δ ∧ ω_{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 M_{c,δ} is a smooth 1-manifold, with boundary the union of M_{c,δ}(g_{1}) and M_{c,δ}(g_{−1}). We must show that an odd number of arcs emerge from the reducible point.

Let us examine the Kuranishi model of M_{c,δ} in the neighbourhood of the reducible solution (A_{0}, 0). The model is ψ^{−1}(0)/S^{1}, where ψ is (the germ of) an S^{1}-equivariant map

ψ : ℝ ⊕ Ker(D_{A_{0}}) → Coker(d^{+}) ⊕ Coker(D_{A_{0}})

and the S^{1} action is the standard scalar multiplication on the kernel and cokernel of D_{A_{0}}. The ℝ on the left-hand side represents the t coordinate. On the right-hand side, Coker(d^{+}) is 1-dimensional, and if Coker(D_{A_{0}}) has complex dimension r then Ker(D_{A_{0}}) has dimension r + 1 by the index theorem. If we identify Coker(d^{+}) with ℝ via the map θ ↦ ∫ θ ∧ ω_{g_{0}}, then we can calculate

\frac{∂ψ}{∂t} |_{(0,0)} = \frac{i}{2π} (f′(0), 0),

following [1]. We have arranged that f′(0) is non-zero, so the local model can be reduced to ψ_{1}^{−1}(0)/S^{1}, for some ψ_{1} : ℂ^{r+1} → ℂ^{r}. Removing the singular point, we obtain a model for a deleted neighbourhood of the reducible in M_{c,δ} as the zero set of a section ψ̄_{1} of the bundle ℂ^{r} ⊗ O(1) on ℂℙ^{r} × ℝ^{+}. Here O(1) is the line bundle of degree 1 on ℂℙ^{r}, pulled back to the punctured cone

ℂℙ^{r} × ℝ^{+} = (ℂ^{r+1} \ {0})/S^{1}.

The zero set of ψ̄_{1} is transverse, by our assumption on the smoothness of M_{c,δ}^{∗}. For an open dense set of numbers ϵ, ψ̄_{1}^{−1}(0) meets ℂℙ^{r} × {ϵ} in an odd number of points, because the Euler number of the bundle ℂ^{r} ⊗ O(1) on 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) ⌣ [ω_{g_{+}}] is positive and for which M_{c}(g_{+}) is empty. In [4] it is shown that ℂℙ^{2}#n ¯ℂℙ^{2} admits Kähler metrics g_{+} of positive scalar curvature. In the Kähler situation, ω_{g} is a positive multiple of the Kähler form, and its cupproduct 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 [11] and [3]. □

Corollary 7. If g is a Riemannian metric on X = ℂℙ^{2}#n ¯ℂℙ^{2} with c_{1}(L) ⌣ [ω_{g}] negative, then the moduli space M_{c}(g) for the line bundle L with c_{1}(L) = c_{1}(X) is non-empty. □

4. Solutions on tubes

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 [−ϵ, ϵ] × 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] × Y. (So g_{0} is the original metric.) We identify X_{R} with X and write M_{c}(g_{R}) for the moduli space.

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 ℝ × Y which is translation-invariant in a temporal gauge. (The spin^{c} structure on the cylinder is the restriction of the spin^{c} structure on X.)

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

Proof. On the cylinder ℝ×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} → Y associated to the spin^{c} structure which Y obtains. Let A(t) be the path of connections in L → Y obtained by restricting A to the slices {t} × Y, and let Φ(t) be the path in Γ(W_{3}) obtained similarly via the above identifications. Let ∂_{A} : Γ(W_{3}) → Γ(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 A_{3} of connections on Y.

In temporal gauge, the equations (∗) become the following equations for A(t) and Φ(t):

\begin{aligned}dA/dt & = − ∗ F_{A} + iτ(Φ, Φ) \\ dΦ/dt & = −∂_{A}Φ. \end{aligned}

Here τ is a pairing from W_{3}×W̄_{3} to Λ^{1}(Y), obtained from Clifford multiplication by using the hermitian metric on W_{3}. These are the downward gradient flow equations for the functional \frac{1}{2}C on A_{3} × Γ(W_{3}) given by

C(A, Φ) = ∫ (A − B) ∧ F_{A} + ∫ ⟨Φ, ∂_{A}Φ⟩.

Here B is any chosen connection on L → Y, made necessary by the fact that 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 Map(Y, S^{1}), but if h has a non-trivial class [h] ∈ H^{1}(X; ℤ) then

C(h(A), h(Φ)) = C(A, Φ) + 8π^{2}c_{1}(L) ⌣ [h].

Let γ 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 Γ in X dual to c_{1}(L) on X. Thus γ bounds relative cycles Γ_{+} and Γ_{−} in X_{+} and X_{−}.

Now suppose that (A_{R}, Φ_{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 → X_{R} obtained from fixed connections B_{±} over the pieces X_{±} (independent of R) and a connection [−R, R] × 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(A_{R}(R), Φ_{R}(R)) − C(A_{R}(−R), Φ_{R}(−R)).

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_{R}^{±}(A_{R}) − B_{±} and its first derivatives are uniformly bounded. We therefore have uniform bounds on each of the two quantities

C(h_{R}^{±}(A_{R})(±R), h_{R}^{±}(Φ_{R})(±R)).

However, since γ is the boundary of a chain Γ_{±} in X_{±}, the degree of h_{R}^{±} on γ ⊂ Y is zero. From the formula above for how C changes under gauge transformation, it follows that

C(h_{R}^{±}(A_{R})(±R), h_{R}^{±}(Φ_{R})(±R)) = C(A_{R}(±R), Φ_{R}(±R)).

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] × 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 ∞, we deduce from Lemma 4 that there is a solution on [0, 1] × 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 ℝ × Y. □

5. A vanishing argument

We continue to consider a cylinder ℝ×Y, but we now take Y to be a Riemannian product S^{1} × Σ, where Σ is a 2-manifold of constant scalar curvature. We choose to make Σ of unit area, so that its scalar curvature is −2π(4g − 4). (The scalar curvature is twice the Gaussian curvature). We assume the genus g is at least 1.

Lemma 9. If there exists a solution to the equations (∗) on ℝ × Y which is translation-invariant in a temporal gauge, then c_{1}(L)[Σ] is not greater than 2g − 2 in absolute value.

Proof. Suppose that (A, Φ) is a solution which is translation-invariant in a temporal gauge. Because Y is compact, |Φ| achieves its supremum, so from Lemma 2 we obtain the C^{0} bound |Φ|^{2} ≤ 2π(4g − 4). This translates into the bound |σ(Φ, Φ)| ≤ \sqrt{2}π(4g − 4), where the norm on 𝔰𝔲(W^{+}) is the one coming from the inner product tr(a^{∗}b) (see the model for σ in section 2.) From the first equation of (∗) we then obtain

|F_{A}^{+}| ≤ \sqrt{2}π(2g − 2),

because ρ 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}| ≤ 2π(2g − 2),

and hence

|c_{1}(L)[Σ]| = |\frac{i}{2π} ∫_{Σ} F_{A}|
≤ \frac{1}{2π} sup |F_{A}| Area(Σ)
≤ 2g − 2. □

6. Proof of the Theorem

We now turn to the proof of Theorem 1. Let Σ be a smoothly embedded oriented surface, dual to the class dH in ℂℙ^{2}. 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 [5]. We therefore assume that d is greater than 3. In this range we may suppose, without loss of generality, that Σ is not a sphere.

Regard Σ now as living in the connected sum X = ℂℙ^{2}#d^{2} ¯ℂℙ^{2}. Let S_{i} be a sphere in the i th copy of ¯ℂℙ^{2}, dual to the class −E, and let Σ̃ ⊂ X be the surface formed by taking an internal connected sum of Σ with all the S_{i}. This surface is dual to the class dH − E, where E = ∑ E_{i} as before. It has the same genus as Σ 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 Σ̃ has a product metric. Arrange that the metric on the factor Y ≅ S^{1} × Σ 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] × Y between X_{−} and X_{+}.

Lemma 10. For R sufficiently large, c_{1}(L) ⌣ [ω_{g_{R}}] is negative.

Proof. Let R_{i} be a sequence increasing to infinity, and normalize the forms ω_{i} = ω_{g_{R_{i}}} so that H ⌣ [ω_{i}] = 1. The forms ω_{i} then have ∥ω_{i}∥_{L^{2}} ≤ 1 and there is a subsequence converging on compact sets to an L^{2} form ω on the disjoint union of the two manifolds with cylindrical ends, X_{−}∪([ 0, ∞) × Y) and ((−∞, 0 ] × Y)∪X_{+}. (The convergence is understood by identifying a compact subset of the cylindricalend 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_{−} ∪ ([ 0, ∞) × Y) is empty, being the image of the compactly-supported cohomology in the ordinary cohomology. So [ω_{i}] · [Σ̃] goes to zero as i → ∞. Because of our normalization, we have

\begin{aligned}[ω_{i}] ⌣ c_{1}(L) & = [ω_{i}] · [Σ̃] − (d − 3)[ω_{i}] ⌣ H \\ & = [ω_{i}] · [Σ̃] − (d − 3). \end{aligned}

Since d is greater than 3, this quantity is eventually negative as claimed. □

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 ℝ × Y which is translation-invariant in a temporal gauge. Using Lemma 9 then, we obtain

c_{1}(L)[Σ̃] ≥ −(2g − 2).

The left-hand side is (3H−E)(dH−E), which is 3d−d^{2}. This inequality is therefore equivalent to 2g − 2 ≥ d^{2} − 3d, which is equivalent to the inequality we wished to prove. □

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Mathematical Institute, 24–29 St. Giles, Oxford OX1 3LB E-mail address: kronheim@maths.oxford.ac.uk

California Institute of Technology, Pasadena, CA 91125 E-mail address: mrowka@cco.caltech.edu