j. differential geometry 65 (2003) 469-511

BOUNDS ON GENUS AND GEOMETRIC INTERSECTIONS FROM CYLINDRICAL END

MODULI SPACES

SAŠO STRLE

Abstract

In this paper we present a way of computing a lower bound for the genus of any smooth representative of a homology class of positive self-intersection in a smooth four-manifold X with second positive Betti number b_{2}^{+}(X) = 1. We study the solutions of the Seiberg-Witten equations on the cylindrical end manifold which is the complement of the surface representing the class. The result can be formulated as a form of generalized adjunction inequality. The bounds obtained depend only on the rational homology type of the manifold, and include the Thom conjecture as a special case. We generalize this approach to derive lower bounds on the number of intersection points of n algebraically disjoint surfaces of positive self-intersection in manifolds with b_{2}^{+}(X) = n.

Introduction

Seiberg-Witten theory has proved very useful in the study of the minimal genus problem. After Kronheimer-Mrowka’s proof of the Thom conjecture [6], regarding the minimal genus problem in the complex projective plane, the following result (the so-called generalized Thom conjecture) was obtained by Morgan-Szabo-Taubes [13] (for classes of nonnegative self-intersection) and Ozsvath-Szabo [18] (the general case): any smooth symplectic curve in a closed symplectic four-manifold minimizes the genus in its homology class. Their proofs depend on results of Taubes [22] about Seiberg-Witten theory of symplectic manifolds, specifically the basic classes of such manifolds. In this paper we present a way of deriving genus bounds that does not depend on any special

Received 02/19/2002.

structure on the manifold. Rather than working over a closed manifold, we study Seiberg-Witten equations on an associated cylindrical end manifold; this approach is related to the work of Frøyshov [2]. The bounds obtained in this way depend only on the rational homology type of the manifold. Since the results are so general, the information about a possible symplectic structure on X is lost; in particular, the bounds are independent of (the sign of) the canonical class of X, whereas the bounds coming from the generalized Thom conjecture detect differences in canonical classes. An important advantage of our approach is that it can be used in manifolds with vanishing Seiberg-Witten invariants, in particular to study geometric intersections of surfaces.

Consider a divisible homology class dξ ∈ H_{2}(X; Z) of positive selfintersection in a smooth four-manifold X with b_{1}(X) = 0 and b_{2}^{+}(X) = 1. The divisibility d > 1 of the homology class is crucial (for technical reasons) while studying Seiberg-Witten equations on the cylindrical end manifold Z = X−Σ, where Σ is a smooth embedded surface representing dξ. The end of Z is modeled on a nontrivial circle bundle Y over Σ, and we work with Seiberg-Witten solutions on Z that exponentially decay to solutions on Y. This depends on description of the perturbed Seiberg-Witten moduli spaces on Y obtained by Mrowka-Ozsvath-Yu [15].

Even though the method requires us to consider a divisible class, the main result concerning genus bounds holds for primitive classes as well. The bound can be stated in the form of a generalized adjunction inequality.

Theorem A. Let X be a smooth closed oriented four-manifold with b_{1}(X) = 0 and b_{2}^{+}(X) = 1. If Σ ⊂ X is a smooth embedded surface of positive self-intersection, then

χ(Σ) + [Σ]^{2} ≤ |⟨c, [Σ]⟩| (1)

for any characteristic vector c ∈ H^{2}(X) that satisfies c^{2} > σ(X).

Based on this inequality it is straightforward to derive minimal genus formulae in (rational homology) CP^{2}, S^{2} × S^{2} and CP^{2}#\overline{CP}^{2}. In manifolds with rational homology of rational surfaces CP^{2}#n\overline{CP}^{2} with 2 ≤ n ≤ 9 the results are easiest to state for reduced classes, defined by Li-Li [8]. In particular, we prove that for any g > 0 there are only finitely many reduced classes of minimal genus g (see Proposition 14.2; also see [7]). We note that genuine rational surfaces mentioned above are ‘genus-minimal’ in the sense that minimal genus representatives in these manifolds have the smallest possible genus among all manifolds with the same rational homology type.

Above considerations generalize to manifolds X with b_{2}^{+}(X) = n in a way that allows us to study a collection of n surfaces in X. The counterpart of the adjunction inequality is the following result.

Theorem B. Let X be a smooth closed connected four-manifold with b_{1}(X) = 0 and b_{2}^{+}(X) = n > 1, and let Σ_{1}, …, Σ_{n} be disjoint embedded surfaces in X with positive self-intersections. If c ∈ H^{2}(X) is a characteristic vector satisfying

c^{2} > σ(X) and ⟨c, [Σ_{i}]⟩ ≥ 0 for all i,

then

χ(Σ_{i}) + [Σ_{i}]^{2} ≤ ⟨c, [Σ_{i}]⟩ (2)

holds for at least one i.

We use this to derive a lower bound on the number of intersection points of surfaces of low genus. For example, suppose that classes (p, q, 0, 0) and (0, 0, r, s) in H_{2}(S^{2} × S^{2}#S^{2} × S^{2}) are represented by spheres in the connected sum S^{2} × S^{2}#S^{2} × S^{2}. If p, q, r, s ≥ 2 and p + q ≥ r + s, then the number of intersection points of the two spheres is at least

pq + (r − 1)(s − 1).

In general, the lower bound on the number of intersection points obtained in this way is roughly by a factor of 2 better than the bounds obtained via the G-signature Theorem (see [4]). We also give an example where the bound on the number of intersection points is optimal.

This paper is divided in two parts. Part I is concerned with technical aspects of Seiberg-Witten moduli spaces over cylindrical end manifolds. The main results of this part are the dimension formula for the moduli space (Corollary 8.2) and compactness and regularity results of Section 9. In Part II we use results of Part I to derive genus bounds and bounds on the number of intersection points of surfaces. We first present a derivation of a genus bound in CP^{2} (which is equivalent to the Thom conjecture) and then proceed to the general case. This is described in Theorem 11.1, which can be rephrased as a generalized adjunction inequality stated above (see Section 12). After that we consider several examples in which one can derive explicit formulae for genus bounds and address the question of representability. In the last section of the paper we turn to geometric intersections of surfaces.

Acknowledgements. I would like to thank my advisor Daniel Ruberman for the numerous conversations and for all his support and guidance. I am grateful to Tom Mrowka and Jerry Levine for suggestions that helped in the preparation of this work. I would also like to thank Professor Jože Vrabec for all that he taught me and for his continued encouragement.

A part of this work was done while the author was a Liftoff mathematician with Clay Mathematics Institute. The author was partially supported by MSZS of the Republic of Slovenia Research Program No. 101-509.

Part I: Cylindrical end moduli spaces

1. The setup

Throughout X will denote a smooth closed connected oriented fourmanifold. If a smooth oriented surface Σ is embedded in X so that the image of its fundamental homology class [Σ] in H_{2}(X) is not a torsion class, we say that Σ represents this homology class. Denote by N ⊂ X a compact tubular neighborhood of Σ. It will be convenient to distinguish between ∂N, oriented as the boundary of N, and Y = −∂N, oriented as the ‘boundary’ of Z. More precisely, let Z_{0} be the closure of the complement of N in X; then Y = ∂Z_{0} and we think of Z = X − Σ as Z_{0} with a half-infinite cylinder attached, Z = Z_{0} ∪_{Y} [0, ∞) × Y. We refer to [0, ∞) × Y as the cylindrical end of Z and say that the end of Z is modeled on Y. The following proposition summarizes the relevant cohomological information about these spaces. Unless specified otherwise all the (co)homology groups have integer coefficients.

Proposition 1.1. Let X be a closed oriented four-manifold and let Σ ⊂ X be an embedded surface representing the class dξ, where ξ ∈ H_{2}(X) is a primitive class of nonzero self-intersection and d ≥ 1 is an integer. Denote by N a compact tubular neighborhood of Σ and by n = (dξ)^{2} the degree of the circle bundle ∂N → Σ. Then

H^{1}(∂N) ≅ H^{1}(Σ), H^{2}(∂N) ≅ H^{2}(Σ)/n[Σ]^{∗} ⊕ H^{1}(Σ),
H^{1}(Z) ≅ H^{1}(X), H^{2}(Z) ≅ H^{2}(X)/dα ⊕ F,

where [Σ]^{∗} denotes the fundamental cohomology class of Σ, α is the Poincaré dual of ξ and F is a subgroup of H^{1}(Σ). The restriction homomorphism H^{2}(X) → H^{2}(N) sends α to dξ^{2}[Σ]^{∗} and its image is a subgroup of index d. Moreover, the restriction homomorphism H^{2}(Z) → H^{2}(∂N) is injective on F.

Proof. The cohomology groups of ∂N follow easily from the Gysin exact sequence of the circle bundle S^{1} ↪ ∂N ⟶^{p} Σ with the Chern class c_{1} = n[Σ]^{∗}.

To determine the cohomology of Z, use the Poincaré duality and excision isomorphisms: H^{2}(Z) ≅ H_{2}(Z, ∂N) ≅ H_{2}(X, N). The last group can be computed using the exact sequence of the pair (X, N):

H_{3}(X) ⟶^{≅} H_{3}(X, N) → H_{2}(N) → H_{2}(X) → H_{2}(X, N) → H_{1}(N).

Since H_{1}(N) is a free abelian group, we have H_{2}(X, N) ≅ H_{2}(X)/dξ ⊕ F′ for some subgroup F′ of H_{1}(N). Similarly we obtain H^{1}(Z) ≅ H_{3}(X, N) ≅ H^{1}(X).

From ⟨α, [Σ]⟩ = ξ·dξ it follows that α ∈ H^{2}(X) restricts to dξ^{2}[Σ]^{∗} ∈ H^{2}(N). The remaining claims follow easily. q.e.d.

2. Seiberg-Witten solutions on a circle bundle

We describe the structure of the moduli spaces of solutions of certain perturbed Seiberg-Witten equations on a circle bundle p: Y → Σ of degree n ≠ 0 over an oriented smooth surface Σ, studied by Mrowka-Ozsvath-Yu [15]. The purpose of the perturbation is to make the equations behave as if the bundle Y were a product. This is achieved by choosing a ‘product’ connection as the background connection in T^{∗}Y in place of the Levi-Civita connection.

To define the background connection, choose a constant curvature metric g_{Σ} of volume 1 on Σ; denote by vol_{Σ} the corresponding volume form and let ω = p^{∗}(vol_{Σ}) be its pull-back to Y. The circle bundle Y admits a connection 1-form iφ: T Y → iR of constant curvature; observe that dφ = −2πnω, since Y has degree n. This connection determines a splitting T^{∗}Y = Rφ ⊕ H, where H ≅ p^{∗}T^{∗}Σ is the horizontal distribution. A metric on Y, compatible with this splitting, is given by g_{Y} = φ^{2} + p^{∗}g_{Σ}; the corresponding volume form is vol_{Y} = φ ∧ ω. Note that the radius of a fiber circle with respect to this metric is 1. The product connection on Y is defined by ∇^{Y} = d ⊕ p^{∗}∇^{Σ}, where ∇^{Σ} is the Levi-Civita connection of (Σ, g_{Σ}). Connection ∇^{Y} is compatible with the splitting and the metric. However, it is not torsion-free as Y → Σ is a nontrivial bundle.

The trivial Spin^{c} structure on Y is the one with the trivial bundle of spinors W^{Y}. By viewing W^{Y} as the pull-back of a spinor bundle over Σ, one can endow it with a ∇^{Y} -compatible spin connection that we denote by ∇^{Y} as well. The Clifford multiplication in W^{Y} is as follows: vectors in H act via the pull-back action, while φ acts by ±i on (W^{Y})^{±}, where the splitting W^{Y} = (W^{Y})^{+} ⊕ (W^{Y})^{−} is induced by the splitting of the spinor bundle over Σ.

Given a hermitian line bundle E → Y we say that the Spin^{c} structure on Y with the bundle of spinors W_{E}^{Y} = W^{Y} ⊗ E is determined by E. A unitary connection A in E induces a spin connection ∇^{Y} ⊗A in W^{Y} ⊗E; the Dirac operator of this connection is denoted by D_{A}. We need to understand the moduli space of solutions of the perturbed 3-dimensional Seiberg-Witten equations on Y (i.e., the equations defined using the above Dirac operator) in a given Spin^{c} structure. The space of reducible solutions in the Spin^{c} structure determined by E is nonempty only for torsion bundles E (i.e., the ones with torsion Chern class). If A_{0} is a smooth flat connection in E, then A_{0} + iα is a reducible solution if and only if α is a closed one-form on Y. This gives an identification between the space of reducible solutions and the space of closed one-forms on Y. The moduli space of reducible solutions, obtained by dividing the space of solutions by the gauge group action, is therefore identified with H^{1}(Y; R)/H^{1}(Y) ≅ H^{1}(Σ; S^{1}) via the choice of A_{0}.

Recall that the three-dimensional Seiberg-Witten equations are the equations for the critical points of the Chern-Simons-Dirac functional on R×Y (see [13]). This means that their linearization is self-adjoint, so for positive dimensional moduli spaces the linearization of the equations is not surjective. The appropriate notion of non-degeneracy of the moduli space is the following.

Definition 2.1. A component N of the moduli space of Seiberg-Witten solutions on Y is (Morse-Bott) nondegenerate, if the kernel of the linearization of the Seiberg-Witten equations at (A, Φ) is isomorphic to the tangent space to N at [A, Φ] for any point [A, Φ] ∈ N.

The following theorem, proved in [15], describes the moduli spaces of solutions to the perturbed Seiberg-Witten equations on Y for various Spin^{c} structures.

Theorem 2.2. Let Y be a circle bundle of degree n ≠ 0 over a surface Σ. The space of solutions in the Spin^{c} structure determined by E → Y is nonempty only if E is the pull-back of a line bundle F → Σ. Fix such an E and let c = c_{1}(F).

(a) The moduli space R(E) of reducible solutions is homeomorphic to the dual torus H^{1}(Σ; S^{1}). Moreover, if c ≢ 0 mod n or if Σ ≅ S^{2}, this space is nondegenerate. In particular, the above identification is a diffeomorphism.

(b) The irreducible components of the moduli space are parameterized by line bundles E_{0} → Σ satisfying

c_{1}(E_{0}) ≡ c mod n and 0 < |c_{1}(E_{0})| < g.

All of these components are compact and nondegenerate; they arise as the pull-backs of solutions to the vortex equations on Σ.

(c) Since the Chern class of the Spin^{c} structure is torsion, the Chern-Simons-Dirac functional descends to a real-valued function on the moduli space. If we normalize it so that it equals 0 on R(E), then its value on the component corresponding to a line bundle E_{0} is 8π^{2}(c_{1}(E_{0}))^{2}/n.

3. Seiberg-Witten solutions on a cylinder

The first step in understanding the structure of the space of Seiberg-Witten solutions on a manifold with a cylindrical end is to study the solutions on the cylindrical part. A standard approach which guarantees good limiting behavior of solutions at infinity is to consider only finite energy solutions (see [13] and [12]).

Let (A, Ψ) be a configuration on [0, ∞)×Y in a temporal gauge; denote by (A_{t}, Ψ_{t}) the path of configurations on Y obtained by restricting (A, Ψ) to the slices t × Y. Recall that the Seiberg-Witten equations on the cylinder take the form

\frac{∂}{∂t}(A_{t}, Ψ_{t}) = (∗_{Y} (q(Ψ_{t}) − F_{A_{t}}), D_{A_{t}}Ψ_{t}).

The energy of a configuration (A, Ψ) on [0, ∞) × Y in a temporal gauge is given by the square of the L^{2}-norm of the right-hand side in the above equation.

Any solution on Y, being a critical point of the above equation, gives rise to a static solution on the cylinder; such a solution clearly has finite energy. Moreover, any finite energy solution on the cylinder converges to a static solution exponentially fast. This result is the Seiberg-Witten analogue of the exponential decay results established by Morgan-Mrowka-Ruberman [12] in Donaldson’s theory. For our purposes, however, we do not need to know that all solutions are exponentially decaying to solutions on Y. That is, without referring to exponential decay results, we will consider only those configurations on the cylinder [0, ∞) × Y that decay exponentially to solutions on Y, for some appropriately chosen decay constant.

4. Seiberg-Witten equations on a cylindrical end manifold

Let Z_{0} denote a compact oriented 4-manifold with boundary, Y, a circle bundle of degree n ≠ 0 over a surface Σ. Choose a collar [−1, 0] × Y ⊂ Z_{0} and equip Z = Z_{0} ∪ [0, ∞) × Y with a cylindrical end metric that agrees with dt^{2} + g_{Y} on [−1, ∞) × Y, where g_{Y} is the metric on Y described in Section 2. As the background connection ∇^{Z} for the Dirac operator we use a metric compatible connection that agrees with the Levi-Civita connection on the complement of (−1, ∞)×Y and agrees with the pull-back of the connection ∇^{Y} on the cylinder [0, ∞) × Y.

Given a Spin^{c} structure on Z we denote the corresponding bundles of spinors by W = W^{+}⊕W^{−} and the determinant line by L = det(W^{+}). As the configuration space for the Seiberg-Witten equations on Z we choose the subset of uniformly exponentially decaying configurations. The restriction of an exponentially decaying configuration to the end [0, ∞)×Y differs from some static solution on the cylinder by a term that converges to zero exponentially fast along the cylinder (see Definition 5.1 for details). We will specify the rate of convergence in Proposition 6.2.

Suppose now that we are in the situation from Section 1: Z is the complement of a representative Σ of a multiple class dξ in a closed manifold X. In this case the following proposition shows that any Spin^{c} structure on Z which admits exponentially decaying solutions arises as the restriction of a Spin^{c} structure on X. We will use this to express the dimension of the moduli space of Seiberg-Witten solutions on Z in terms of the invariants of X.

Proposition 4.1. Let Σ ⊂ X be a smooth representative of the class dξ and let n = (dξ)^{2}; denote by N a compact tubular neighborhood of Σ and by Z = X − Σ the cylindrical end manifold with the end modeled on Y. Let S be a Spin^{c} structure on Z for which the space of exponentially decaying Seiberg-Witten solutions is nonempty.

(a) The restriction of S to Y is determined by a torsion line bundle and it extends to a Spin^{c} structure on the disk bundle N. We fix the extension W^{N} of the trivial Spin^{c} structure W^{Y} on Y to a Spin^{c} structure on N, which is uniquely determined by requiring that the Chern class of its determinant line is equal to n[Σ]^{∗}. Through this choice we obtain a canonical extension of the given Spin^{c} structure S on Z to a Spin^{c} structure on X.

(b) Any two Spin^{c} structures on X that induce the given Spin^{c} structure S on Z differ by a power of the line bundle on X with the Chern class dα, where α denotes the Poincaré dual of ξ.

Proof. The induced Spin^{c} structure on Y is determined by a line bundle E → Y satisfying L|_{Y} = E^{2}. Note that the existence of exponentially decaying Seiberg-Witten solutions on Z implies the existence of solutions on Y. Combining this with Theorem 2.2 we conclude that the line bundle E is torsion and hence the pull-back of a line bundle on Σ.

An extension of the trivial Spin^{c} structure W^{Y} on Y to a Spin^{c} structure on N is given as follows. Denote by N̂ → Σ the normal bundle of Σ in X, considered as a complex line bundle of degree n. As a complex manifold, the line bundle N̂ carries a canonical Spin^{c} structure with the bundles of spinors W^{+} = Λ^{0} ⊕ Λ^{0,2} and W^{−} = Λ^{0,1}. These bundles are determined up to isomorphism by their restrictions to the zero-section Σ ⊂ N̂. Note that W^{−}|_{Σ} ≅ N̂ ⊕ K_{Σ}^{−1}, where K_{Σ} is the canonical bundle of Σ. The determinant line is therefore isomorphic to N̂ ⊗ K_{Σ}^{−1}. Since the pull-back of N̂ is trivial over Y, we need to change the Spin^{c} structure by a square root of the pull-back of the canonical bundle of Σ.

A different extension of a Spin^{c} structure on Z to a Spin^{c} structure on X can be obtained by changing the Spin^{c} structure on N by a power of the pull-back of N̂ (since this operation preserves the Spin^{c} structure on Y). From c_{1}(N̂) = (dξ)^{2}[Σ]^{∗} and Proposition 1.1 it follows that the Chern class of the auxiliary bundle on X changes under this operation by a multiple of dα. To see that these are the only possibilities, consider two Spin^{c} structures on X that differ by a line bundle E. If they restrict to give the same Spin^{c} structure on Z, then E|_{Z} is trivial, hence c_{1}(E) lies in the kernel of the restriction homomorphism H^{2}(X) → H^{2}(Z). Recall that this kernel is generated by dα. q.e.d.

As in the case of Spin^{c} structures on Y we will say that a Spin^{c} structure on N is determined by a line bundle E → N (or by a line bundle E_{0} → Σ) if the bundle of spinors is of the form W^{N} ⊗ E (or W^{N} ⊗ p^{∗}E_{0}) .

Given a Spin^{c} structure on X, denote its determinant line by Det. With the above notation, we write ⟨c_{1}(Det), ξ⟩ = k + (2s + 1)dξ^{2} for some k ∈ {0, …, 2dξ^{2} − 1} and s ∈ Z. Note that k is the representative of the residue class

⟨c_{1}(Det), ξ⟩ + dξ^{2} mod 2dξ^{2}

in {0, …, 2dξ^{2} −1}; the purpose of the shift by dξ^{2} is to make k directly related to the induced Spin^{c} structure on Y. Indeed, the induced bundle of spinors over N is determined by a line bundle E_{0} → Σ with c_{1}(E_{0}) = e satisfying 2e = d(k + 2sdξ^{2}).

Possible values of k are constrained by the fact that c_{1}(Det) is a characteristic class; this motivates the following definition.

Definition 4.2. We call k ∈ {0, …, 2dξ^{2} −1} a characteristic number for (X, ξ, d), if there exists a Spin^{c} structure on X whose determinant line Det satisfies ⟨c_{1}(Det), ξ⟩ = k + (2s + 1)dξ^{2} for some s ∈ Z.

Clearly the parity of a characteristic number is uniquely determined. The following characterization of characteristic numbers is easily verified.

Lemma 4.3. Let X be a closed four-manifold, ξ ∈ H_{2}(X) a primitive class of positive self-intersection, and d ≥ 1 an integer. If ξ^{2} is even, then the set of characteristic numbers for (X, ξ, d) consists of all even numbers in {0, …, 2dξ^{2} − 1}; if ξ^{2} is odd, then the set of characteristic numbers consists of all numbers in {0, …, 2dξ^{2} − 1} with the parity opposite to that of d. In particular, kd is even in either case.

5. Configuration space over a cylindrical end manifold

In order to be able to consider the Seiberg-Witten equations on Z as an elliptic system of equations we need to choose appropriate function spaces in which to study the equations. Since we chose to work with configurations on Z that uniformly exponentially decay to configurations on Y, we can work with the weighted Sobolev spaces L_{r,δ}^{2} for some small δ > 0. We cannot choose δ = 0 because with this choice the operator on Y, associated to the linearization of the Seiberg-Witten equations on Z, has a nontrivial kernel and so the equations are not Fredholm (cf. [11]). Recall that the L_{δ}^{2} norm of a function f can be defined by ||f||_{δ}^{2} = ∫_{Z} |f|^{2}e^{δτ}, where τ: Z → R is a smooth function that is equal to −1 on the complement of (−1, ∞) × Y and agrees with the t coordinate on the cylinder [0, ∞) × Y; we further assume that τ depends only on t and is non-decreasing. Sobolev norms for r > 0 are defined analogously.

We are interested in the space of configurations on Z decaying to reducible solutions on Y. Fix for now a Spin^{c} structure on Z with nonempty space of exponentially decaying Seiberg-Witten solutions with reducible limits. For (A, Ψ) a smooth solution on Z denote its asymptotic value by (B_{∞}, 0). The space of all asymptotic values is an affine space whose underlying vector space is the space of imaginary-valued closed one-forms on Y. We let B be a smooth unitary connection in L (the determinant line of the Spin^{c} structure on Z) which agrees with the pull-back of B_{∞} on the cylinder [0, ∞) × Y. We use the connection B (resp. B_{∞}) to identify the space of unitary connections in L (resp. L|_{Y}) with the imaginary-valued 1-forms on Z (resp. Y).

Rather than working with the full configuration space, we restrict the possible asymptotic values of configurations (by fixing the gauge at infinity). Specifically, we replace the space of imaginary-valued closed one-forms on Y by the subspace H of imaginary-valued harmonic oneforms. Note that the subgroup of the gauge group G^{Y} that acts on H consists of harmonic gauge transformations and is therefore isomorphic to S^{1} × H^{1}(Y), where S^{1} corresponds to the constant gauge transformations. However, apart from the constant gauge transformations, the only harmonic gauge transformations on Y that extend to gauge transformations on Z are those that correspond to the classes in the image of H^{1}(Z) → H^{1}(Y). In our application of the cylindrical end moduli space to the genus problem we will assume that b_{1}(Z) = 0. Assuming the latter, the gauge group over Z for the restricted (i.e., gauge-fixed at infinity) configuration space consists of the gauge transformations that converge to the constant gauge transformations on Y. We will see later that the space of solutions on Z contains reducible configurations, so the subgroup of constant gauge transformations does not act freely on it. For this reason we consider the based moduli space, obtained by dividing the space of solutions by the action of the gauge group based at infinity. The group of constant gauge transformations still acts on the based moduli space and we will use this action to obtain our results.

Definition 5.1. The exponentially decaying configuration space on Z, corresponding to the reducible solutions on Y, is defined to be

H × L_{2,δ}^{2}(iΛ^{1}(Z) ⊕ W^{+}).

More precisely, the configuration associated to an element (h, α, Φ) is

(A, Ψ) = (B + α + τ̇h, Φ).

A gauge transformation σ ∈ L_{3,loc}^{2}(Z, S^{1}) belongs to the gauge group based at infinity, G_{∞}, if there exist a T > 0 and an f ∈ L_{3,δ}^{2}([T, ∞) × Y, iR) so that the restriction of σ to the cylinder [T, ∞) × Y is given by σ = exp(f).

6. Deformation complex

Recall that Spin^{c} structures on Z with nonempty exponentially decaying configuration space are induced from X. Moreover, we will consider only Spin^{c} structures on X for which the induced Spin^{c} structure on Y is nontrivial, unless Σ is a sphere; this way the moduli space of reducible solutions on Y is always nondegenerate (Theorem 2.2). Suppose (B_{∞}, 0) is the asymptotic value of a smooth solution (A, Ψ) on Z. We use the configuration space on Z as described in Definition 5.1. Since the configurations on Z converge to reducible solutions on Y, Seiberg-Witten equations on Z give rise to a map

SW: H ⊕ L_{2,δ}^{2}(iΛ^{1}(Z) ⊕ W^{+}) → L_{1,δ}^{2}(iΛ^{2,+}(Z) ⊕ W^{−}). (3)

This is well-defined because all the terms in the Seiberg-Witten map exponentially decay to 0 (recall that the multiplication L_{2,δ}^{2}⊗L_{2,δ}^{2} → L_{1,δ}^{2} is continuous). The deformation complex D_{(A,Ψ)} of the solution (A, Ψ), taking into account the action of G_{∞}, is

(4) 0 → L_{3,δ}^{2}(Z, iR) ⟶^{K_{(A,Ψ)}} H ⊕ L_{2,δ}^{2}(iΛ^{1}(Z) ⊕ W^{+}) →

⟶^{T_{(A,Ψ)}SW} L_{1,δ}^{2}(iΛ^{2,+}(Z) ⊕ W^{−}) → 0,

where K_{(A,Ψ)}(f) = (0, 2df, −fΨ) is the infinitesimal gauge group action and

T_{(A,Ψ)}SW(α̂, ψ) = (d^{+}α̂ − 2Q(Ψ, ψ), D_{A}ψ + α̂ · Ψ)

is the linearization of the Seiberg-Witten map at (A, Ψ). Here Q is the bilinear map associated to the quadratic map q in the Seiberg-Witten equations and α̂ = α+τ̇h is the one-form on Z corresponding to (h, α) ∈ H⊕L_{2,δ}^{2}(iΛ^{1}(Z)). The cohomology groups of this complex provide some local information about the based moduli space as described below. We first make the following observation.

Lemma 6.1. The zeroth cohomology group of the deformation complex is trivial.

Proof. If f ∈ L_{3,δ}^{2}(Z, iR) is in the kernel of K_{(A,Ψ)}, then df = 0. Thus f is constant and since it converges to 0 at infinity, it must be identically equal to zero. q.e.d.

The first cohomology group of the deformation complex (4) is called the Zariski tangent space of the moduli space and the second cohomology group is called the obstruction space. If (A, Ψ) is a regular point for the Seiberg-Witten map, then the obstruction space vanishes and the first cohomology of the complex is isomorphic to the (geometric) tangent space of the moduli space at [A, Ψ].

We will compute the index of the deformation complex D_{(A,Ψ)} via the index of the fiber complex F_{(A,Ψ)} associated to it; the latter is defined by the following exact sequence of complexes

0 → F_{(A,Ψ)} → D_{(A,Ψ)} → H → 0,

where H denotes the deformation complex of the asymptotic value (B_{∞}, 0) of (A, Ψ) and the morphism to H corresponds to taking limits at infinity. Here we identified the complex H with its only nonzero group (in dimension 1), namely the group of harmonic one-forms on Y. The fiber complex differs from the full deformation complex by a finite dimensional space H (of dimension 2g, where g is the genus of Σ), hence it suffices to compute the index of the fiber complex. The Fredholm properties of the fiber complex are determined by the asymptotic behavior of its ‘wrapped-up’ form [11], given by (5) below.

Proposition 6.2. There exists δ_{0} > 0 so that for any δ ∈ (0, δ_{0}], the Seiberg-Witten map (3), considered as a map of the spaces of L_{δ}^{2} sections, is Fredholm and its index is constant on the configuration space. Moreover, the asymptotic map of the linearization has trivial kernel.

Proof. Let (A, Ψ) be a configuration on Z with a reducible asymptotic value (B_{∞}, 0) on Y. The L_{δ}^{2} adjoint K_{δ}^{∗} of the infinitesimal gauge group action K_{(A,Ψ)} is defined with respect to the following inner products: for imaginary-valued forms α and β let

⟨α, β⟩_{δ} = ∫_{Z} α ∧ ∗β e^{δτ} ,

where ∗ is the complex anti-linear extension of the Hodge star-operator; for spinors ψ and ϕ let

⟨ψ, ϕ⟩_{δ} = 2Re ∫_{Z}⟨ψ(z), ϕ(z)⟩e^{δτ} vol_{Z}.

Then K_{δ}^{∗}(α, ψ) = 2e^{−δτ} d^{∗}e^{δτ} α + 2iIm⟨Ψ, ψ⟩; after dividing by 2 we obtain the ‘wrapped-up’ fiber complex

F_{δ}: L_{2,δ}^{2}(iΛ^{1}(Z) ⊕ W^{+}) → L_{1,δ}^{2}(iΛ^{0}(Z) ⊕ iΛ^{2,+}(Z) ⊕ W^{−}), (5)

F_{δ}(α, ψ) = (e^{−δτ} d^{∗}e^{δτ} α + i Im ⟨Ψ, ψ⟩, d^{+}α − 2Q(Ψ, ψ), D_{A}ψ + α · Ψ).

To analyze this map we conjugate it by the isometry T_{δ} = e^{−ετ}: L^{2} → L_{δ}^{2}, where ε = δ/2. This gives a map F between the spaces of L^{2} sections that sends (α, ψ) to

( \begin{matrix}d^{∗}α − ε⟨α, dτ⟩ + i Im ⟨Ψ, ψ⟩ \\ d^{+}α − ε(dτ ∧ α)^{+} − 2Q(Ψ, ψ) \\ D_{A}ψ − εdτ · ψ + α · Ψ \end{matrix} ) .

For the purpose of computing the asymptotic operator of F we only need to understand its form on the cylinder C = [0, ∞) × Y. Recall that W^{+} ≅ p_{2}^{∗}W_{E}^{Y} over C, where W_{E}^{Y} is the bundle of spinors on Y and p_{2} : [0, ∞)×Y → Y is the projection. Moreover, the Clifford multiplication by dt induces an isomorphism between W^{+} and W^{−}. The bundles of forms on the cylinder are given by Λ^{1}(C) = p_{2}^{∗}(Λ^{0}(Y) ⊕ Λ^{1}(Y)) and Λ^{2,+}(C) ≅ p_{2}^{∗}Λ^{1}(Y), where the last isomorphism follows from the fact that any self-dual two-form on the cylinder is of the form dt ∧ γ_{t} + ∗_{3}γ_{t} for some path γ_{t} of one-forms on Y. Writing α = ifdt + iβ, F (if, iβ, ψ) is given by

(−i(\frac{∂f}{∂t} −d_{3}^{∗}β+εf), i(\frac{∂β}{∂t} −d_{3}f+∗_{3}d_{3}β−εβ), \frac{∂ψ}{∂t} −D_{B_{∞}}ψ−εψ)+o(1).

Up to an obvious isomorphism, F is of the form \frac{∂}{∂t} −G+o(1), where the asymptotic operator G acts on the space of sections of Λ^{0}(Y) ⊕ Λ^{1}(Y) ⊕ W_{E}^{Y} via the matrix operator

[\begin{matrix}−ε & d_{3}^{∗} & 0 \\ d_{3} & − ∗_{3} d_{3} + ε & 0 \\ 0 & 0 & D_{B_{∞}} + ε \end{matrix}] .

Notice that G splits as the sum of (the perturbations of) the asymptotic operators corresponding to the anti-self-duality (ASD) operator and the Dirac operator on Y. By results of Lockhart and McOwen [11], the operator F (and hence F_{δ}) is Fredholm if the kernel of G is trivial. For the ASD part this follows from the computation of the spectrum of this operator (see the proof of Proposition 7.2): for ε = 0 the ASD asymptotic operator has nontrivial kernel, whereas for positive ε the kernel is trivial. For the Dirac part recall that by our choice of the Spin^{c} structure on X, the space of reducible solutions on Y in the induced Spin^{c} structure is nondegenerate, hence the kernel of D_{B_{∞}} is trivial. Moreover, the spectrum of D_{B_{∞}} depends only on the gauge equivalence class of B_{∞}; from compactness of the fundamental domain for the action of the gauge group on the space of flat connections on Y, invertibility of the operators, and the fact that D_{B_{∞}} has discrete spectrum, it follows that G has trivial kernel for all small enough positive ε.

Above we showed that the operator F (and hence F_{δ}), associated to an arbitrary configuration (A, Ψ), is Fredholm. To finish the proof we only need to show that the operator F_{δ} depends continuously on the configuration (A, Ψ). Suppose (A′, Ψ′) is another configuration; denote the difference (A′, Ψ′) − (A, Ψ) by (a, ϕ). Then the difference of the two linearizations is a bilinear map in ((a, ϕ), (α, ψ)). The required continuity now follows from the continuity of the Sobolev multiplication L_{2,δ}^{2} × L_{2,δ}^{2} → L_{1,δ}^{2}. q.e.d.

7. Index of the deformation complex

We use the Atiyah-Patodi-Singer index formula (see [1]) to compute the index of the fiber complex. This requires the operator to be independent of the t variable along the cylinder, which is not the case at a solution to the Seiberg-Witten equations. However, according to Proposition 6.2, the index can be computed using any configuration with a reducible limit, in particular one which agrees with the pull-back of a reducible configuration on Y along the cylinder. Recall that the Atiyah-Patodi-Singer index can be expressed in terms of the (extended) L^{2} solutions; since by Proposition 6.2 the asymptotic operator on L^{2} has no kernel, the Atiyah-Patodi-Singer index of the fiber complex agrees with its Fredholm index.

We first recall the relevant results from [1]. Let E_{0}, E_{1} → Z_{1} be hermitian vector bundles of the same fiber dimension over Z_{1} = {z ∈ Z | τ(z) ≤ 1} and let F: Γ(E_{0}) → Γ(E_{1}) be an elliptic operator. Assume that on the cylinder [0, 1] × Y ⊂ Z_{1} we have F = σ(\frac{∂}{∂t} − G), for some self-adjoint operator G: Γ(E) → Γ(E), where E = E_{0}|_{Y} and σ is an isomorphism between E_{0} and E_{1} over the cylinder. The new ingredient in the index formula (compared to the formula over a closed manifold) is a boundary correction term, which is a spectral function of G. More precisely, let η(s) = ∑_{λ≠0} sign(λ)|λ|^{−s}, where λ runs over the spectrum of

G, be the eta function of G. This series defines a holomorphic function in a half-plane Re(z) > z_{0} and extends to a meromorphic function on the entire plane; this extension has a finite value at 0. The correction term is defined in terms of η(0) and the dimension h of the kernel of G.

The Atiyah-Patodi-Singer domain of F, denoted by Γ(E_{0}, P), consists of all the sections of E_{0} whose restriction to the boundary 1 × Y of Z_{1} lies in the kernel of P. Here P = P_{≥0} : Γ(E) → Γ(E) denotes the spectral projection of G, corresponding to the nonnegative eigenvalues, i.e., the orthogonal projection onto the subspace spanned by the eigenvectors of G with nonnegative eigenvalues. Notice that with this choice of domain any solution of F = 0 extends to an L^{2} solution on Z.

Theorem 7.1. With the above notation, F: Γ(E_{0}, P) → Γ(E_{1}) has a finite index given by

ind_{AP S} F = ∫_{Z_{1}} k − \frac{h + η(0)}{2},

where k is a differential form on Z_{1} determined by F, called the index density of F. Moreover, ind_{AP S} F = h(E_{0}) − h(E_{1}) − h_{∞}(E_{1}), where h(E_{0}) is the dimension of the kernel of F on the space of L^{2} sections on Z, h(E_{1}) is the corresponding dimension for the adjoint F^{∗} of F, and h_{∞}(E_{1}) is the dimension of the space of asymptotic values of extended L^{2} solutions of F^{∗}.

To compute the index of the fiber complex we need the signature eta invariant of Y, which was computed by Komuro [5], and the eta invariant of the perturbed Dirac operator on Y (see Section 2), which was computed by Nicolaescu [17]; the latter paper also contains a derivation of the index formula.

Proposition 7.2. Let X be a closed four-manifold and Σ ⊂ X a smooth surface representing the homology class dξ, where d > 1 and ξ ∈ H_{2}(X) is a primitive class of positive self-intersection. Given a Spin^{c} structure on X with the determinant line Det, we write

⟨c_{1}(Det), ξ⟩ = k + (2s + 1)dξ^{2}

for some characteristic number k and some s ∈ Z. The index of the fiber complex (5), associated to the space of Seiberg-Witten solutions on the cylindrical end manifold Z = X − Σ, that along the end converge to a fixed reducible solution (B_{∞}, 0), is

\frac{1}{4} ∫_{Z} c_{1}(B)^{2} − \frac{σ(X)}{4} + \frac{(k − dξ^{2})^{2}}{4ξ^{2}} + 1 + b_{1}(X) − b_{2}^{+}(X) − 2g, (6)

where B is a unitary connection in Det|_{Z}, which agrees with the pullback of B_{∞} on the end [0, ∞) × Y.

Remark. By rewriting the above dimension formula, one can obtain the Frøyshov invariant (see [2]) of the circle bundle Y for a range of Spin^{c} structures.

Proof. We will compute the (real) index of the linearization of the Seiberg-Witten map at a configuration (B, 0) with asymptotic value (B_{∞}, 0). In this case the associated operator F (defined in the proof of Proposition 6.2) on the spaces of L^{2} sections takes the form

F (α, ψ) ↦ (d^{∗}α, d^{+}α, D_{B}ψ) − ε(⟨α, dτ⟩, (dτ ∧ α)^{+}, dτ · ψ).

Clearly F splits as the sum F_{0} ⊕ F_{1}, where F_{0} is a zeroth-order perturbation of the anti-self-duality operator A = d^{∗} ⊕ d^{+}, and F_{1} is a zeroth-order perturbation of the Dirac operator D_{B}. Hence we can split the index computation accordingly.

Index of the anti-self-dual part: For the purpose of invoking the Atiyah-Patodi-Singer index theorem we complexify the spaces of forms. The index density in the statement of Theorem 7.1 depends only on the principal symbol of the operator; for F_{0} it is therefore determined by A. The difference in the indices of F_{0} and A comes from the correction term in the index formula and can be described as the spectral flow of a family of associated asymptotic operators as made precise below.

Notice that A is (isomorphic to) the adjoint of the operator A_{+} from [1] and we will use the following fact from there. The Atiyah-Patodi-Singer index computation gives

\frac{1}{2}(σ(Z) + χ(Z)) = ∫_{Z} k − \frac{1}{2}η_{sign}(0),

where σ(Z) and χ(Z) are the signature and the Euler characteristic of Z respectively, k is the index density of A and η_{sign}(0) is its eta invariant. Therefore we obtain

ind_{AP S}(A) = − ∫_{Z} k − \frac{1}{2}(h − η_{sign}(0)) = −\frac{1}{2}(σ(Z) + χ(Z) + h),

where h is the dimension of the kernel of the asymptotic operator

G_{0} = [\begin{matrix}0 & d_{3}^{∗} \\ d_{3} & − ∗_{3} d_{3} \end{matrix}] : Ω^{0}(Y) ⊕ Ω^{1}(Y) → Ω^{0}(Y) ⊕ Ω^{1}(Y).

A pair (f, β) in the kernel of G_{0} satisfies d_{3}^{∗}β = 0 and d_{3}f = ∗_{3}d_{3}β = 0. Hence the kernel of G_{0} consists of harmonic forms and we have h = 1 + 2g. Using this, along with σ(X) = σ(Z) + 1, χ(X) = χ(Z) + 2 − 2g, and σ(X) + χ(X) = 2 − 2b_{1}(X) + 2b_{2}^{+}(X), we obtain σ(Z) + χ(Z) = −1 + 2b_{1}(X) + 2b_{2}^{+}(X) + 2g; thus ind_{AP S}(A) = b_{1}(X) − b_{2}^{+}(X) − 2g.

Notice that the asymptotic operator of F_{0} is of the form G_{0} + εE,

where E = [\begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix}]. It will be convenient to consider the family of

operators G_{u} = G_{0}+uεE for u ∈ [0, 1], connecting the limiting operators G_{0} and G_{1} of A and F_{0} respectively. The difference in the correction terms for G_{0} and G_{1} is equal to the spectral flow of the family G_{u}. More precisely, let η̂_{u} = \frac{1}{2}(h_{G_{u}} + η_{G_{u}}(0)) denote the reduced eta invariant of G_{u}. From the definition it is clear that η̂_{u} has a jump at u = a only if some eigenvalue λ(u) of G_{u} vanishes at a (but is not zero at least on one side of u = a). The case of interest to us is when any eigenvalue λ(u) behaves in one of the following ways: λ(u) = 0 for all u, λ(u) ≠ 0 for all u, or λ(u) = 0 iff u = 0 and this zero is transverse. Assume first that λ(u) is the only eigenvalue crossing 0 at u = 0. If λ′(0) > 0, then η̂_{1} = η̂_{0}, since at u = 0 the eigenvalue λ(0) contributes +1 to h and nothing to η, whereas for u > 0 it contributes +1 to η and nothing to h. Similar considerations in the case λ′(0) < 0 imply that η̂_{1} = η̂_{0} − 1, since in this case λ(u) contributes to h and η with the opposite signs. This clearly generalizes to a finite number of eigenvalues crossing 0 at u = 0.

To determine the difference of indices of A and F_{0} we therefore need to understand the behavior of the eigenvalues of the family G_{u}. A pair (f, β) ∈ Ω^{0}(Y) ⊕ Ω^{1}(Y) in the kernel of G_{u} satisfies

d_{3}^{∗}β = εuf, d_{3}f − ∗_{3}d_{3}β = −εuβ.

This implies d_{3}^{∗}d_{3}f = −u^{2}ε^{2}f, which has no solutions for u ≠ 0 since the Laplace operator d_{3}^{∗}d_{3} is positive definite. For u = 0 we computed above that the dimension of the kernel is 1 + 2g. The eigenvalue λ_{1}(u) = −εu, corresponding to the space of constant functions, has multiplicity 1 (at 0), whereas the multiplicity of λ_{2}(u) = εu, corresponding to the space of harmonic one-forms, is 2g. We conclude from the previous paragraph that η̂_{1} = η̂_{0} − 1, and hence the index of F_{0} is 1 + b_{1}(X) − b_{2}^{+}(X) − 2g.

Index of the Dirac part: Using similar considerations as above we see that the index of F_{1} is equal to the index of D_{B}; the reason for this is that all the asymptotic operators D_{B_{∞}} + uε are invertible (for u ∈ [0, 1]) by Theorem 2.2 and our choice of ε. The index density of D_{B} is \frac{1}{8}(c_{1}(B)^{2}−L(∇^{Z})), where L(∇^{Z}) = \frac{1}{3}p_{1}(∇^{Z}) is the Hirzebruch L-class associated to the background connection ∇^{Z} in Z; recall that ∇^{Z} agrees with the Levi-Civita connection ∇^{LC} on the complement of (−1, ∞)×Y and agrees with the pull-back of ∇^{Y} on [0, ∞) × Y. From Theorem 7.1 we get ind_{AP S}(D_{B}) = \frac{1}{8} ∫_{Z}(c_{1}(B)^{2} − L(∇^{Z})) − \frac{1}{2}η_{D_{B∞}} (0). The eta invariant of the Dirac operator on Y, coupled to the flat connection B_{∞}, was computed in [17] and is equal to

η_{D_{B∞}} (0) = −\frac{ad^{2}}{6} − \frac{k^{2}}{4a} + \frac{kd}{2},

where we used the fact that the radius of the fiber circles in Y is 1. The only other term we need to interpret is the integral of the L-class. If we were using the Levi-Civita connection as the background connection on Z, then we could use the fact that L(∇^{LC}) is the index density of the signature operator on Z. In particular, we have σ(Z) = ∫_{Z} L(∇^{LC}) − η_{sign}(0). The signature eta invariant η_{sign}(0) for Y was computed in [5] and is given by

η_{sign}(0) = 1 − \frac{ad^{2}}{3} + \frac{2ad^{2}}{3} (a^{2}d^{4} + 2g − 2).

In our case, however, there is another term coming from the difference in the L-classes of the two connections: we have

∫_{Z_{0}} L(∇^{Z}) = ∫_{Z_{−1}} L(∇^{LC}) + ∫_{[−1,0]×Y} L(∇^{Z}).

The last term in this expression can be computed explicitly; write ∇^{Z} = ∇^{∞} + f(t)α, where ∇^{∞} denotes the pull-back connection, α = ∇^{∞} − ∇^{LC} and f is a smooth non-decreasing function that maps [−1, 0] onto itself. The computation can be done with respect to a local orthonormal coframe (φ_{1}, φ_{2}, φ) on Y, where iφ is the connection of the circle bundle Y → Σ and (φ_{1}, φ_{2}) is the pull-back of a local coframe on Σ. This yields

∫_{[−1,0]×Y} L(∇^{Z}) = −\frac{2ad^{2}}{3} (a^{2}d^{4} + 2g − 2)

(see [17] for more details). Note that the index formula gives the complex index of the operator F_{1}. q.e.d.

8. Dimension of the cylindrical end moduli space

To express the formal dimension of the moduli space of Seiberg-Witten solutions on the cylindrical end manifold Z = X − Σ in terms of the data on the closed manifold X we need the following result.

Lemma 8.1. Let X be a closed four-manifold and let Σ ⊂ X be an embedded surface with self-intersection n ≠ 0. Denote by Z the complement X − Σ, thought of as a manifold with a cylindrical end [0, ∞) × Y. Given a Spin^{c} structure on X, let p = ⟨c_{1}(Det), [Σ]⟩ ∈ Z, where Det → X denotes the determinant line of the Spin^{c} structure. For any unitary connection B on L = Det |_{Z}, whose restriction to the cylinder [0, ∞) × Y agrees with the pull-back of a flat connection in Det |_{Y}, we have

∫_{Z} c_{1}(B)^{2} = c_{1}(Det)^{2} − \frac{p^{2}}{n} . (7)

Proof. Let Z_{1} = {z ∈ Z | τ(z) ≤ 1} and let Y be the oriented boundary of Z_{1}. We think of X as the union of Z_{1} and a compact tubular neighborhood N of Σ in X. Denote by [0, 1] × Y the oriented collar to the boundary Y of N. Suppose A is a connection in Det |_{N} that in a neighborhood of the boundary 0 × Y agrees with the pull-back of B_{∞}, the latter being the limit of B. Then A and B together define a (smooth) connection in Det and we have c_{1}(Det)^{2} = ∫_{Z} c_{1}(B)^{2} + ∫_{N} c_{1}(A)^{2}. We will evaluate the second integral. Combining Proposition 6.2 and the index formula (6), we see that, for the purpose of the computation, we can choose A to be any connection in Det |_{N} which in a neighborhood of Y agrees with the pull-back of some flat connection in Det|_{Y}. In what follows we use the notation from Section 2.

Let A_{1} be the pull-back of a constant curvature connection in Det |_{Σ} to Det |_{N}; then F_{A_{1}} = −2πipω. We let A = A_{1} + i\frac{p}{n}f(t)φ, where iφ is a constant curvature connection of the circle bundle Y → Σ and f: [0, 1] → [0, 1] is a smooth non-increasing function which is identically 1 in a neighborhood of 0 and identically 0 in a neighborhood of 1. Since the degree of Y is −n, we have dφ = 2πnω, so A is flat close to the boundary of N. A simple computation shows that F_{A} = 2πip(f(t) − 1)ω + i\frac{p}{n}f′(t)dt ∧ φ and hence

F_{A} ∧ F_{A} = −4π \frac{p^{2}}{n} (f(t) − 1)f′(t) dt ∧ vol_{Y} .

Using this along with vol (Y) = 2π and c_{1}(A) = \frac{i}{2π} F_{A} gives

∫_{Z} c_{1}(A)^{2} = 2\frac{p^{2}}{n} ∫_{0}^{1}(f(t) − 1)f′(t)dt = \frac{p^{2}}{n} .

q.e.d.

Now we can obtain a convenient formula for the (formal) dimension of the moduli space of Seiberg-Witten solutions on Z with reducible limits.

Corollary 8.2. Let X be a closed four-manifold with b_{1}(X) = 0 and b_{2}^{+}(X) = 1. Suppose Σ ⊂ X is a smooth surface representing the homology class dξ, where d > 1 and ξ ∈ H_{2}(X) is a primitive class of positive self-intersection. Let Z = X − Σ, thought of as a cylindrical end manifold. Given a Spin^{c} structure on X, let p = ⟨c_{1}(Det), ξ⟩, where Det → X denotes the determinant line of the Spin^{c} structure; we write p = k + (2s + 1)dξ^{2} for some characteristic number k and some s ∈ Z. The formal dimension of the based moduli space of Seiberg-Witten solutions on Z with reducible limits is given by

\frac{c_{1}(Det)^{2} − σ(X)}{4} + \frac{(k − dξ^{2})^{2} − p^{2}}{4ξ^{2}} . (8)

Moreover, this dimension depends only on the induced Spin^{c} structure on Z, i.e., it is independent of s.

Proof. Starting with the index of the fiber complex, given by (6), recall from the discussion preceding Proposition 6.2 that we need to add 2g to it (where g is the genus of Σ) to obtain the index of the full deformation complex. The formula now follows from the lemma above with n = (dξ)^{2} and p replaced by pd.

Considering c_{1}(Det) as a class in H^{2}(X; R), we write

c_{1}(Det) = \frac{p}{a}α + c

for some c ∈ H^{2}(X; R), where α is the Poincaré dual of ξ and a = ξ^{2}. It follows that c ∪ α = ⟨c, ξ⟩ = 0, hence c_{1}(Det)^{2} = p^{2}/a + c^{2}, which implies the following expression for the formal dimension:

\frac{c^{2} − σ(X)}{4} + \frac{(k − dξ^{2})^{2}}{4ξ^{2}} .

Note that since α and c are orthogonal and b_{2}^{+}(X) = 1, c^{2} cannot be positive.

Finally, consider another Spin^{c} structure on X which induces the same Spin^{c} structure on Z. Then its determinant line, denoted by Det′, satisfies c_{1}(Det′) = c_{1}(Det) + 2sdα for some s ∈ Z (Proposition 4.1). This shows that c′, defined analogously as c above, equals c; the last assertion of the corollary then follows from the above expression for the formal dimension. q.e.d.

9. Compactness and regularity of the cylindrical end moduli space

Let Σ be an embedded surface of positive self-intersection in a closed four-manifold X with b_{1}(X) = 0. Denote by Z the complement X − Σ, thought of as a manifold with a cylindrical end [0, ∞)×Y. We topologize the moduli space of exponentially decaying Seiberg-Witten solutions on Z by the weakest topology, containing the topology of uniform C^{k} convergence on compact subsets for some k ≥ 3, with respect to which the Chern-Simons-Dirac functional along the cylinder is continuous at infinity.

Proposition 9.1. Fix a Spin^{c} structure on X for which there are no irreducible Seiberg-Witten solutions on Y in the induced Spin^{c} structure. Then the moduli space of exponentially decaying Seiberg-Witten solutions on Z (in the induced Spin^{c} structure) with reducible limits is compact.

Proof. Given a sequence (A_{n}, Ψ_{n}) of Seiberg-Witten solutions on Z, observe that this sequence has a convergent subsequence on any submanifold Z_{t} ⊂ Z (after appropriate changes of gauge). This is essentially a consequence of [6, Lemma 4]. The uniform boundedness of |Ψ_{n}| on Z follows from the fact that the configurations converge to zero at infinity, and from a standard maximum principle argument [6, Lemma 2]. The only difference is that we are not using the Levi-Civita connection as the background connection. However, the two Dirac operators differ by Clifford multiplication by a one-form f(t)dt, where f is a smooth bounded function [15, Lemma 5.2.1], and it easily follows from this that the result holds for the perturbed Dirac operator as well.

By the diagonal argument we can find a subsequence of (A_{n}, Ψ_{n}) which, after appropriate changes of gauge, converges on all compact subsets of Z to some solution (A, Ψ) of the Seiberg-Witten equations on Z; we still denote this subsequence by (A_{n}, Ψ_{n}). Potential noncompactness therefore arises from the behavior of solutions on the end [0, ∞) × Y. In particular, the convergence of the sequence (A_{n}, Ψ_{n}) depends on the convergence of the sequence of its asymptotic values. Recall that the moduli space of asymptotic values is identified with the space H of imaginary valued harmonic one-forms on Y.

We first prove that the sequence of the asymptotic values of (A_{n}, Ψ_{n}) is bounded; in fact, the moduli space is contained in the fiber of the projection to the space of asymptotic values. Let (A, Ψ) and (B, Φ) be two solutions with reducible asymptotic values. Then α = A − B exponentially decays to a form α_{∞} ∈ H. To prove that α_{∞} = 0 it suffices to show that ∫_{K} α_{∞} = 0 for any embedded circle K ⊂ Y representing a homology class in the kernel of the morphism H_{1}(Y) → H_{1}(Z). Since such a K is the boundary of a surface S ⊂ Z, this is equivalent to ∫_{S}(F_{A} − F_{B}) = 0. As both F_{A} and F_{B} represent the same relative cohomology class on Z, the claim follows.

Suppose now that [(A_{n}, Ψ_{n})] does not converge to [(A, Ψ)] in the topology of the moduli space. This means that the Chern-Simons-Dirac functional has different values on the asymptotic configurations of (A_{n}, Ψ_{n}) and (A, Ψ). Then the asymptotic value of (A, Ψ) is irreducible (see [13, Proposition 8.5] for more details), which contradicts the assumption that the space of solutions on Y consists entirely of reducibles. q.e.d.

Now we turn to the question of regularity of the moduli space. We are particularly interested in the behavior of the Dirac operator at a reducible solution; a more general result regarding such Dirac operators on a closed manifold is proved in [21]. For technical reasons we choose to work with L_{k,δ}^{2} (k ≥ 4) configuration space.

Proposition 9.2. Let X be a closed oriented four-manifold with b_{1}(X) = 0 and b_{2}^{+}(X) = 1. Suppose that Σ ⊂ X is a smooth surface of positive self-intersection, and let Z = X − Σ, thought of as a cylindrical end manifold. Then for any small enough δ > 0 there exists a second category subset Ω of imaginary-valued L_{k−1,δ}^{2} self-dual two-forms on Z, such that for any ω ∈ Ω the following holds. For any exponentially decaying connection A in the determinant line L → Z satisfying F_{A}^{+} = ω, the Dirac operator D_{A} is either injective or surjective. Moreover, the irreducible part of the ω-perturbed moduli space of exponentially decaying Seiberg-Witten solutions on Z with reducible asymptotic values is a smooth orientable finite dimensional manifold.

Proof. Fix a smooth unitary connection A_{0} in the determinant line L → Z and let N be the manifold of all exponentially decaying configurations (A, Ψ) (with reducible limits) satisfying

T (A, Ψ) = (d^{∗}(A − A_{0}), D_{A}Ψ) = 0, Ψ ≠ 0.

To prove that N is a smooth manifold we need to verify that the differential

D_{(A,Ψ)}T (α, ψ) = (d^{∗}α, D_{A}ψ + α · Ψ)

of T at (A, Ψ) is onto. The adjoint of this operator is

(f, χ) ↦ (df + i⟨ · Ψ, χ⟩ − δf dτ, D_{A}χ + δdτ · χ),

where we used the usual L_{δ}^{2} inner product on the space of imaginary valued one-forms and the real part of the hermitian L_{δ}^{2} inner product on spinors. Expression i⟨ ·Ψ, χ⟩ denotes the imaginary valued one-form characterized by

⟨α, i⟨ · Ψ, χ⟩⟩ = Re ⟨α · Ψ, χ⟩,

for all imaginary valued one-forms α.

Rather than proving directly that the kernel of the adjoint is trivial, we will replace the adjoint by the operator of the same kind with δ = 0. The injectivity of thus obtained operator D implies the injectivity of the adjoint for δ > 0 small enough.

Suppose that D(f, χ) = 0. Computing with respect to a local orthonormal coframe {φ_{1}, …, φ_{4}} on Z we obtain

id^{∗}df = Re ⟨iD_{A}Ψ, χ⟩ + ⟨iΨ, D_{A}^{∗}χ⟩

− Re ⟨i ∑_{j}(∇_{φ_{j}}φ_{j} + (∗d ∗ φ_{j})φ_{j}) · Ψ, χ⟩ .

The first two terms in this expression vanish as Ψ and χ are harmonic. Moreover, the one-form inside the last term vanishes for Levi-Civita connection. The background connection we are using differs from the Levi-Civita connection on the cylinder [0, ∞) × Y by a multiple of

( \begin{matrix}0 & φ_{3} & φ_{2} & 0 \\ −φ_{3} & 0 & −φ_{1} & 0 \\ −φ_{2} & φ_{1} & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} )

(see [15, Lemma 5.2.1]), where we took (φ_{1}, φ_{2}) to be the pull-back of an orthonormal coframe on Σ and iφ_{3} to be the connection one-form of the circle bundle Y. So the last term in the above expression vanishes as well. This implies that f is constant, and since f exponentially decays to zero, f = 0. Finally, since ⟨α·Ψ, χ⟩ = 0 for any α, where Ψ and χ are harmonic, it follows from unique continuation property for harmonic spinors that χ = 0 (see [14, Lemma 6.2.1]). This proves that N is smooth. In particular, at any point (A, Ψ) ∈ N,

im D_{A} + {α · Ψ | d^{∗}α = 0} = L_{3,δ}^{2}(W^{−}).

Let Ω_{0} be the set of regular values of the map N → L_{3,δ}^{2}(iΛ^{2,+}(Z)), (A, Ψ) ↦ F_{A}^{+}. Given ω ∈ Ω_{0}, let (A, Ψ) ∈ N be a point satisfying F_{A}^{+} = ω. Now the differential d^{+} from the tangent space to N at (A, Ψ) to the space of imaginary self-dual two-forms is onto by the choice of ω. Further, from Hodge decomposition of L_{k,δ}^{2} forms on Z (see [10]; note that we chose δ so that the Laplace operator is Fredholm), and the assumption H^{1}(Z) = 0, it follows that the space of co-closed one-forms maps isomorphically onto the space of self-dual two-forms. Thus for any co-closed one-form α there exists a spinor ψ so that D_{A}ψ + α · Ψ = 0. Combining this with the observation at the end of the previous paragraph proves that D_{A} is onto.

The proof of regularity of the irreducible part proceeds as for a closed manifold (see [14]); we get a set Ω_{1} of regular perturbation parameters and let Ω = Ω_{0} ∩ Ω_{1}. q.e.d.

Part II: Genus bounds

10. An example: genus bounds in CP^{2}

Before stating and proving the main theorem (in the next section), we will demonstrate the argument in the simplest possible case, for X = CP^{2}. Let ξ = [CP^{1}] be the standard generator of H_{2}(X); then ξ^{2} = 1. We fix d > 1 and consider a smooth genus g representative Σ of the class dξ. A Spin^{c} structure on X is uniquely determined by p = ⟨c_{1}(Det), ξ⟩. As before we write p = k+d for some characteristic number k. We will see in the proof of Theorem 11.1 that it suffices to consider k ∈ {0, …, d}. Recall from the discussion preceding Definition 4.2 that the line bundle, determining the induced Spin^{c} structure on Y, is the pull-back of a line bundle E_{0} → Σ with c_{1}(E_{0}) = kd/2. According to part (b) of Theorem 2.2, irreducible solutions in the given Spin^{c} structure on Y exist only if

g > \frac{kd}{2}. (9)

The general formula (8) for the formal dimension of the based moduli space of Seiberg-Witten solutions with reducible limits on Z in the case we are considering becomes

\frac{(k − d)^{2} − 1}{4}. (10)

Notice that this number is even and by 1 greater than the expected dimension of the moduli space.

Using inequality (9) and the dimension formula (10) we obtain a lower bound on the genus g of Σ based on the following observation (explained in the proof of Theorem 11.1): if the moduli space of Seiberg-Witten solutions with reducible limits on Z is compact and positive dimensional, this leads to a contradiction. In other words, using Proposition 9.1, if the moduli space is positive dimensional, then (9) must hold.

The dimension (10) of the based moduli space is positive for k < d−1 in the range of k’s considered. Since d and k have different parity, we conclude (for d ≥ 3) that

g > \frac{(d − 3)d}{2}.

Since the classes d[CP^{1}] for d = 1, 2 are represented by spheres, this inequality is equivalent to the Thom conjecture, which was first established by Kronheimer and Mrowka [6].

Theorem 10.1. Let Σ ⊂ CP^{2} represent the class d[CP^{1}] for some d ≥ 1. Then the genus g of Σ satisfies

g ≥ \frac{(d − 1)(d − 2)}{2}. (11)

Moreover, this lower bound is attained by a smooth holomorphic curve representing this homology class.

Remark.

(a) Note that analogous genus bound holds for classes d[CP^{1}] with d ≤ −1, where d gets replaced by |d| in (11). This observation is true in general, since we can always replace ξ by −ξ.

(b) From above computations we see that genus bound (11) holds in any X which is a rational homology CP^{2}: by possibly changing the orientation we can assume that X is positive definite and let ξ be a generator of H_{2}(X). However, it is not true in general that this bound is the best possible for any rational homology CP^{2}. As an example, consider Mumford surface [16], which is an algebraic surface of general type with the canonical class 3. According to the generalized symplectic Thom conjecture (see [13]) the minimal genus in the class of multiplicity d > 0 equals

\frac{(d + 1)(d + 2)}{2}.

11. The main theorem

In the previous section we saw how the moduli space of Seiberg-Witten solutions on the complement of an embedded surface, representing a given homology class in CP^{2}, can be used to derive a lower bound for the genus of any smooth representative of this class. Below we generalize this result to a bigger class of four-manifolds with b_{2}^{+} = 1. The bound is not explicit, but it can be effectively computed in many specific cases. The idea of the proof is analogous to Kronheimer’s proof of Donaldson’s Theorem on definite intersection forms of closed manifolds (see [21]).

Theorem 11.1. Let X be a smooth closed oriented four-manifold with b_{1}(X) = 0 and b_{2}^{+}(X) = 1. Suppose Σ ⊂ X is a smooth surface representing homology class dξ, where d > 1 and ξ ∈ H_{2}(X) is a primitive class of positive self-intersection. Let K be the set of all characteristic numbers k ∈ {0, …, dξ^{2}} for (X, ξ, d) which satisfy the following condition: there exists a Spin^{c} structure on X such that

c_{1}(Det)^{2} > σ(X) + 4kd, (12)

and ⟨c_{1}(Det), ξ⟩ = k+dξ^{2}, where Det is the determinant line of the Spin^{c} structure. Suppose that K is not empty and let k_{0} be the maximum of K. Then the genus g of Σ satisfies

g > \frac{k_{0}d}{2}. (13)

Proof. We will use our standard notation Z = X − Σ for the cylindrical end manifold and Y for the boundary of a tubular neighborhood of Σ (oriented as the ‘boundary’ of Z). Choose a regular perturbation ω ∈ Ω (see Proposition 9.2); then the irreducible part M^{∗} of the perturbed moduli space M is smooth. Notice that (12) is equivalent to the dimension of the (perturbed) based moduli space M̃ being positive (by Corollary 8.2); since this dimension is even, this also implies that the dimension of M is positive. With this remark, the statement of the theorem is equivalent to the following claim, which we prove below: if the moduli space of Seiberg-Witten solutions on Z is positive dimensional for a given Spin^{c} structure, then it is not compact.

Suppose contrary to the statement of the theorem that for some k ∈ K, kd/2 ≥ g. Fix a Spin^{c} structure on X for which (12) holds for k. According to part (b) of Theorem 2.2 there are no irreducible solutions on Y in the induced Spin^{c} structure and therefore the moduli space M is compact (see Proposition 9.1).

Next we show that M contains a unique reducible point [A, 0]. Since ω ∈ Ω, d^{+} is a surjection from the space of extended one-forms to the space of self-dual two-forms. Hence the equation F_{A}^{+} = ω has a solution. Suppose now that (A′, 0) is another solution and write A′ = A + iα for some one-form α on Z; clearly d^{+}α = 0 and therefore dα = 0. By the choice of gauge the asymptotic value h of α is harmonic. Since the class of α is trivial in H^{1}(Z; R), h represents the trivial class in H^{1}(Y; R), hence h = 0. Thus there exists a function f on Z, exponentially decaying to 0 at infinity, such that α = 2df, i.e., (A, 0) and (A′, 0) are gauge equivalent.

To finish the proof of the claim, we need to understand the structure of M at the reducible point [A, 0]. Recall that the index of the ASD part of the linearization at (A, 0) is zero. Since its kernel also vanishes (by an argument similar to the one in the previous paragraph), the (Zariski) tangent space and the obstruction space at (A, 0) correspond to the kernel and cokernel of D_{A} respectively. As we assumed that the index is positive, Proposition 9.2 implies that the cokernel of the Dirac operator vanishes, so the based moduli space M̃ is smooth. The action of the group S^{1} of constant gauge transformations on the kernel of D_{A} is by complex multiplication, hence a closed neighborhood V of [A, 0] in M is a cone on some projective space CP^{n}. Let N be the smooth compact submanifold of M with the boundary CP^{n}, obtained as the closure of the complement of V. Denote by c the Chern form of the S^{1} bundle Ñ → N, where Ñ ⊂ M̃ is the preimage of N. Note that the induced S^{1} bundle over the boundary CP^{n} is the tautological bundle. So ∫_{CP_{n}} c^{n} = ∫_{N} d(c^{n}) = 0 is a contradiction.

We remark that for k = 0, the condition g > 0 that we obtain from the argument above is consistent with the assumption that Σ is a sphere (needed for non-degeneracy of solutions on Y).

Finally, we check that it suffices to consider characteristic numbers ≤ ad, i.e., that for ad < k < 2ad we do not get any new restrictions on the genus (here a = ξ^{2}). Suppose that for some k in (ad, 2ad) condition (12) holds for some Spin^{c} structure on X. Then we claim that there exists a Spin^{c} structure on X with the characteristic number k′ = 2ad − k, for which (12) holds as well, hence k′ ∈ K. To see this first change the given Spin^{c} structure by the line bundle E on X with c_{1}(E) = −dα, where α is the Poincaré dual of ξ. The characteristic number of the inverse of thus obtained Spin^{c} structure is k′. The expression for the dimension of the moduli space is unaffected by these changes of the Spin^{c} structure. For the first change this follows from Corollary 8.2. For the second note that c_{1}(−Det) = −c_{1}(Det), hence the class c, defined by ⟨c_{1}(Det), ξ⟩ = \frac{p}{a}α+c, changes sign. The claim now follows from the alternative form of the dimension formula in the proof of Corollary 8.2. Since k′ ∈ K, we have seen above that g > k′d/2 = |k − 2ad|d/2; however, by part (b) of Theorem 2.2, this implies that there exist irreducible Seiberg-Witten solutions on Y in the Spin^{c} structure induced by the one given on X and hence the moduli space need not be compact. q.e.d.

Remark. We note that the leading term in the genus bound for a divisible class dξ, obtained from the above theorem, equals (dξ)^{2}/2, which is by a factor of 2 better than the bounds obtained via the Gsignature Theorem (cf. Rohlin [19]).

A special case of interest occurs when the class ξ ∈ H_{2}(X) is characteristic, that is, its Poincaré dual is a characteristic class.

Corollary 11.2. With notation as in Theorem 11.1, assume that H_{1}(X) = 0, the signature of X is negative, and that ξ is characteristic. Then the genus g of any smooth surface Σ representing dξ satisfies

g > \binom{d}{2}ξ^{2}.

Proof. By Furuta’s 10/8 Theorem [3] (in fact by a Theorem of Donaldson), X is odd. Consider the Spin^{c} structure on X characterized by c_{1}(Det) = (2d − 1)α, where α is the Poincaré dual of ξ. From ⟨c_{1}(Det), ξ⟩ = (2d − 1)a = k + ad we get k = a(d − 1), where a = ξ^{2}. Since the class c (defined in the proof of Corollary 8.2) is equal to 0 in this case, (12) is equivalent to (k − ad)^{2} > aσ(X), which is clearly true for k = ad − a and σ(X) < 0. This implies g > a(d − 1)d/2. q.e.d.

12. Geometric constructions

Let X be a smooth four-manifold. For a class ξ ∈ H_{2}(X) denote by g_{ξ}(d) the minimal genus of a smooth representative of dξ; we write g_{ξ} for g_{ξ}(1). In this section we will show, using some simple geometric constructions, that asymptotically g_{ξ}(d) does not grow faster than (dξ)^{2}/2. Combining this with genus bounds from Theorem 11.1 we conclude that (dξ)^{2}/2 describes the dominant term in g_{ξ}(d) (as d → ∞) in a manifold X with b_{2}^{+}(X) = 1.

Proposition 12.1. Let X be a smooth four-manifold and let ξ ∈ H_{2}(X) be a class of positive self-intersection. Then

g_{ξ}(d) ≤ \frac{(dξ)^{2}}{2} − (\frac{ξ^{2}}{2} + 1 − g_{ξ})d + 1 (14)

for any d > 1. Moreover, there exists a smooth representative of dξ with the genus given by the right-hand side of the above inequality.

Proof. Let Σ ⊂ X be a smooth embedded surface of genus g_{ξ} representing ξ and let Σ′ denote Σ with the interiors of a := ξ^{2} disjoint disks removed. Think of the normal bundle ν_{Σ} of Σ in X as being obtained from the product bundle over Σ′ by adding a degree 1 bundle over a 2-disk for each puncture. To construct d copies Σ_{i} of Σ in general position, we choose d distinct parallel copies Σ′_{i} of Σ′. Over each 2-disk we cap-off Σ′_{i} by adding a disk in such a way that any two disks intersect transversely in a single point and any intersection point is common to two disks only. It is clear from the construction that the total number of intersection points between the surfaces Σ_{i} thus obtained equals a\binom{d}{2}. A neighborhood of each intersection point looks like a pair of transversely intersecting disks. Removing these and replacing them by annuli gives Σ(d). Since the surfaces, obtained from Σ_{i}, i = 1, …, d by removing small (disjoint) disks around the intersection points are disjoint, we need d − 1 annuli to make a connected surface; each of the remaining annuli increases the genus by 1. q.e.d.

As an immediate consequence of the above inequality we obtain the following bound on the genus of a representative of a primitive class.

Corollary 12.2. Let X be a smooth four-manifold and ξ ∈ H_{2}(X) a class of positive self-intersection. If g_{ξ}(d) > (dξ^{2} − Δ_{d})d/2 for some d > 1, then g_{ξ} > (ξ^{2} − Δ_{d})/2.

Assume now that X is a smooth closed oriented four-manifold with b_{1}(X) = 0 and b_{2}^{+}(X) = 1. If ξ ∈ H_{2}(X) is a primitive class of positive self-intersection a, then for any integer d > 1 we know from Theorem 11.1 that g_{ξ}(d) > (ad − Δ_{d})d/2 for some Δ_{d}. For example, if the signature of X is negative and ξ is characteristic, we can take Δ_{d} = a (see Corollary 11.2). The previous corollary then implies that a characteristic class in X is not represented by an embedded sphere.

In general we obtain the following consequence of Theorem 11.1. This result is equivalent to the generalized adjunction inequality of Theorem A.

Corollary 12.3. Let X be a smooth closed oriented four-manifold with b_{1}(X) = 0 and b_{2}^{+}(X) = 1. For any (primitive) class ξ ∈ H_{2}(X) of positive self-intersection there exists Δ ≥ 0 so that

g_{ξ}(d) > \frac{(dξ^{2} − Δ)d}{2}

for all d ≥ 1.

Proof. We first check that the set K in the statement of Theorem 11.1 is nonempty for d large enough. Choose a Spin^{c} structure on X with c_{1}(Det) = 2dα − γ, where γ is a characteristic vector satisfying γ^{2} > σ(X) and γ ∪ α ≥ 0. Such a characteristic vector clearly exists – starting with any characteristic vector we can get one satisfying these conditions by adding to it a large enough multiple of α. Then k = ad − ⟨γ, ξ⟩ belongs to [0, ad] for large enough d. We need to check that (12) also holds:

c_{1}(Det)^{2} − σ(X) − 4kd = γ^{2} − σ(X) > 0.

Fix some d for which K is nonempty and a characteristic number k ∈ K for (X, ξ, d); let Δ := ad − k. Denote by c_{1} = c_{1}(Det) the Chern class of the Spin^{c} structure that satisfies (12) with k and d, and let c_{1}(Det′) = c_{1} + 2nα for some integer n. Then Δ′ = a(d + n) − k′ = Δ and it follows from the alternative form of the dimension formula in the proof of Corollary 8.2 that the Spin^{c} structure with determinant Det′ satisfies (12) with k′ and d+n in place of k and d. This implies that the genus bound in the statement of the corollary holds for all multiplicities d + n for which k′ ≥ 0; it clearly holds for the rest. q.e.d.

13. Manifolds with signature zero

In this section X denotes a smooth closed oriented four-manifold with b_{1}(X) = 0, b_{2}^{+}(X) = 1 and signature σ(X) = 0. Up to isomorphism, there are only two possible intersection forms such a manifold can have, distinguished by the parity. The even intersection form is

given by H = [\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}] and is realized for example by S^{2} × S^{2}; the odd

intersection form is given by E = [\begin{matrix}1 & 0 \\ 0 & −1 \end{matrix}] and is realized for example

by CP^{2}#\overline{CP}^{2}. Genus bounds that follow from Theorem A depend only on the intersection pairing of the manifold, so we need only consider two cases.

Proposition 13.1. Suppose the intersection pairing of X is isomorphic to H; let {ξ_{1}, ξ_{2}} be a basis of H_{2}(X) modulo the torsion subgroup with respect to which the intersection pairing is given by H. Then any class ξ ∈ H_{2}(X), whose image in H_{2}(X; R) is given by pξ_{1} + qξ_{2} with pq ≠ 0, satisfies

g_{ξ} ≥ (|p| − 1)(|q| − 1).

Proof. After possibly changing the orientation of X we may assume that the self-intersection 2pq of ξ is positive; then we can further assume p, q > 0. Denote by ξ_{i}^{∗} ∈ H^{2}(X; R) the Hom-dual of ξ_{i} and consider a characteristic vector c satisfying c = 2ξ_{1}^{∗} + 2ξ_{2}^{∗} in H^{2}(X; R). Since c^{2} > 0, the claimed bound follows from adjunction inequality (1). q.e.d.

The following result for S^{2} × S^{2} and the corresponding result regarding classes in CP^{2}#\overline{CP}^{2} (see below) was proved independently by Ruberman [20] and by Li-Li [8].

Corollary 13.2. With notation as in the previous proposition, assume further that ξ_{1} and ξ_{2} are represented by spheres which intersect transversely in one point. Then

g_{ξ} = \begin{cases}(|p| − 1)(|q| − 1); & pq ≠ 0 \\ 0; & pq = 0. \end{cases}

Proof. Let Σ_{i} be a representative of ξ_{i} as in the statement. Since the self-intersection of ξ_{i} is zero, any class with pq = 0 is represented by a sphere. Suppose now that pq ≠ 0; we may assume p, q > 0. To construct a representative of ξ with genus (p − 1)(q − 1), take p disjoint copies of Σ_{1} and q disjoint copies of Σ_{2}, so that any copy of Σ_{1} intersects any copy of Σ_{2} in exactly one point. Resolving the intersection points gives the required representative. q.e.d.

Proposition 13.3. Suppose the intersection pairing of X is isomorphic to E; let {ξ_{1}, ξ_{2}} be a basis of H_{2}(X) modulo the torsion subgroup with respect to which the intersection pairing is given by E. Then any class ξ ∈ H_{2}(X) of positive self-intersection satisfies

g_{ξ} > \frac{p^{2} − q^{2} − 3|p| + |q|}{2},

where pξ_{1} + qξ_{2} is the image of ξ in H_{2}(X; R).

Proof. We may assume (by possibly changing the sign of ξ_{i}) that p > q ≥ 0. Let c be a characteristic class whose real image is 3ξ_{1}^{∗} − ξ_{2}^{∗}, where ξ_{i}^{∗} ∈ H^{2}(X; R) denotes the Hom-dual of ξ_{i}. As c^{2} > 0, the adjunction inequality implies the claimed genus bound, except for q = 0 and p ≤ 2. In the latter cases the claimed bound states g_{ξ} ≥ 0, which is the best possible bound since in CP^{2}#\overline{CP}^{2} these classes are represented by spheres. q.e.d.

Corollary 13.4. With notation as in the previous proposition, assume further that ξ_{i} are represented by disjoint spheres. Then

g_{ξ} = \begin{cases}\frac{p^{2} − q^{2} − 3|p| + |q|}{2} + 1; & |p| > |q| \\ \frac{q^{2} − p^{2} − 3|q| + |p|}{2} + 1; & |q| > |p| \\ 0; & |p| = |q|. \end{cases}

Proof. Let Σ_{i} be a representative of ξ_{i} as in the statement. Note that any class of the form (±1, ±1) is represented by a sphere of selfintersection 0. Hence any class (p, q) with |p| = |q| is represented by a sphere.

Suppose that |p| > |q| (the remaining case is analogous). We may assume p > q > 0 for the purpose of construction; if q = 0, the situation is as in CP^{2}. To construct a representative of ξ with the stated genus, decompose ξ as (p, q) = q(1, 1) + (p − q)(1, 0). Represent q(1, 1) by q disjoint spheres, and (p − q)(1, 0) by a surface Σ of genus (p − q − 1)(p − q − 2)/2 which intersects each of the spheres in p − q points. Finally resolve the intersection points. q.e.d.

14. Manifolds with negative signature

Let X be a smooth closed oriented four-manifold with b_{1}(X) = 0, b_{2}^{+}(X) = 1 and signature σ(X) = 1 − n with n ≥ 2. We will assume that the intersection form of X is odd. This is always the case for n ≤ 8, since any such form is odd. Without the restriction on n, the assumption holds for manifolds without 2-torsion according to Furuta’s 10/8 Theorem [3].

Fix a primitive class ξ ∈ H_{2}(X) of positive self-intersection and choose a basis {ξ_{0}, …, ξ_{n}} of H_{2}(X) (modulo the torsion) with respect to which the intersection form is given by ⟨1⟩ ⊕ n⟨−1⟩, and ξ = (p, q_{1}, …, q_{n}) with p > 0 and q_{i} ≥ q_{i+1} ≥ 0; then ξ^{2} = p^{2} − ∑ q_{i}^{2}. Denote by m = m_{ξ} the number of nonzero q_{i}’s.

It turns out that the genus bounds obtained from adjunction inequality (1) with c the ‘canonical class’ (3, −1, …, −1) are only optimal for reduced classes. The notion of a reduced class, used by Li-Li [8] to study genus bounds in rational surfaces CP^{2}#n\overline{CP}^{2} for n ≤ 9, extends naturally to manifolds we are considering.

Definition 14.1. A class ξ ∈ H_{2}(X) as above is called reduced with respect to the basis {ξ_{0}, …, ξ_{n}} provided m_{ξ} ≤ 9 and p ≥ q_{1} + q_{2} + q_{3}, where q_{3} = 0 if n = 2.

It is proved in [8, Lemma 4.1] that in rational surfaces with n ≤ 9 any class of positive self-intersection can be mapped to a reduced class (with respect to the standard basis) by an orientation preserving diffeomorphism. The argument there also proves that any class ξ ∈ H_{2}(X) with m_{ξ} ≤ 9 is reduced with respect to some basis as above.

Proposition 14.2. With above notation, suppose that 2 ≤ m ≤ 9 and ξ is reduced. Then

g_{ξ}(d) > \frac{(dξ^{2} − 3p + ∑ q_{i})d}{2}.

Moreover, g_{ξ}(d) > 0 unless d = 1, m = 2 and ξ = (p, p − 1, 1) for some p > 1. Excluding the latter classes, given any g > 0 there is only a finite number of reduced classes with minimal genus no greater than g.

Remark. The last statement gives an affirmative answer to a conjecture of Li and Li [8]. In fact, B.H. Li proved [7] that the above lower bound is sharp in rational surfaces X = CP^{2}#n\overline{CP}^{2} for n ≤ 9.

Proof. Let c be a characteristic vector whose real image with respect to the Hom-dual basis is (3, −1, …, −1); clearly c^{2} > σ(X). Let Δ = ⟨c, ξ⟩ = 3p − ∑ q_{i}, where the sum, as all other sums over i, runs from 1 to m. Since ξ is reduced, Δ ≥ 0.

Suppose first that m ≥ 3. Since ξ is reduced,

p − \frac{3}{2} ≥ (q_{1} − \frac{1}{2}) + (q_{2} − \frac{1}{2}) + (q_{3} − \frac{1}{2}) .

Using this along with q_{i} ≥ q_{i+1}, we obtain

ξ^{2} − Δ ≥ (9 − m)(q_{3}^{2} − q_{3}) ≥ 0,

thus g_{ξ}(d) ≥ 1 for any d ≥ 1. Note in general that to establish the last claim of this proposition, it suffices to show that an upper bound on minimal genus implies an upper bound on q_{1}; this is enough since increasing the value of p (while keeping q_{i}’s fixed) increases ξ^{2} −Δ. The argument is simple if m ≤ 8, as then ξ^{2} − Δ > (q_{1} − 1/2)(q_{2} − 1/2). The last inequality implies that an upper bound on minimal genus yields an upper bound on q_{1}. For m = 9 the first inequality of this paragraph gives

ξ^{2} − Δ ≥ ∑_{i=4}^{7}(q_{1} − q_{i}) (q_{i} − \frac{1}{2}) + ∑_{i=8}^{9}(q_{2} − q_{i}) (q_{i} − \frac{1}{2});

we need to consider several cases. If q_{1} > q_{7}, a bound on minimal genus implies a bound on q_{1}. Same holds if q_{1} = q_{7}, but q_{1} > q_{9}. Finally, if q_{1} = q_{9} = q,

ξ^{2} − Δ = (p − 3q)(p + 3q − 3),

but positive square condition implies p > 3q; again it follows that there is only a finite number of such vectors whose minimal genus is at most g.

If m = 2, then p ≥ q_{1} + q_{2} implies

ξ^{2} − Δ + 2 ≥ 2(q_{1} − 1)(q_{2} − 1),

which is strictly positive unless q_{2} = 1. Note also that for q_{2} > 1 there are only finitely many classes (p, q_{1}, q_{2}) with minimal genus at most g. For q_{2} = 1 we get

ξ^{2} − Δ + 2 = (p − q_{1} − 1)(p + q_{1} − 2);

this equals 0 only if p = q_{1} + 1, and for p ≥ q_{1} + 2 there are only finitely many classes (p, q_{1}, 1) with minimal genus no greater than g. q.e.d.

One can verify that the bound in the above proposition is the best possible bound obtainable from Theorem A. However, this also follows from work of Li-Li [8] and [9], where they prove that the minimal genus bound for a reduced class ξ = (p, q_{1}, …, q_{m}) of positive self-intersection in a rational surface CP^{2}#n\overline{CP}^{2} with n ≤ 9 is given by

g_{ξ}(d) = \frac{(dξ^{2} − 3p + ∑ q_{i})d}{2} + 1.

In the next proposition we give a construction of a minimal genus representative in a special case.

Proposition 14.3. Let X, ξ_{i}, ξ = (p, q_{0}, …, q_{m}) and m ≤ 9 be as above, and let \overline{m} be the largest value of i for which q_{i} > 2. Assume further that ∑\overline{m} ; i=1 q_{i} ≤ p and that for i = 0, …, m the classes ξ_{i} are represented by disjoint spheres. Then ξ has a representative of genus

g_{ξ} = (ξ^{2} − 3p + ∑ q_{i}) /2 + 1.

Proof. Assume first that \overline{m} = m and let q := ∑\overline{m} ; i=1 q_{i}. Then ξ can be decomposed as (p−q)ξ_{0} +∑ q_{i}(ξ_{0} +ξ_{i}). By assumption ξ_{0} and ξ_{0} +ξ_{i} (for i = 1, …, m) can be represented by spheres Σ_{i} (i = 0, …, m) any two of which intersect transversely in one point. Moreover, the spheres Σ_{i} for i ≥ 1 have self-intersection zero. To construct a representative for ξ, take p − q copies of Σ_{0} and q_{i} disjoint copies of Σ_{i}, so that the whole collection of spheres is in general position and any two spheres that intersect have exactly one point in common. Note that the total number of intersection points of these p spheres is

\binom{p − q}{2} + q(p − q) + ∑_{i<j} q_{i}q_{j},

so after resolving the intersection points we obtain a minimal genus representative.

If m > \overline{m}, for any q_{i} = 2 take two spheres representing ξ_{i} that intersect transversely in one point. Then connect one of the two spheres representing ξ_{i} to a sphere Σ_{j} for some j ≤ \overline{m}, obtaining a surface Σ. Now cancel the −1 intersection point with one of the +1 intersection points of Σ_{j}. To this end choose a curve γ ⊂ Σ connecting the two intersection points and replace the complements of small disks around the intersection points (cut out from the other surfaces, not Σ) by a tube which is the restriction to γ of the normal circle bundle of Σ. Resolving the remaining intersection points again gives a minimal genus representative. Finally, if q_{i} is 1, connect the corresponding sphere to the surface constructed before. q.e.d.

15. Geometric intersections of surfaces

Let X be a smooth closed connected four-manifold. We say that a collection of classes ξ_{1}, …, ξ_{n} ∈ H_{2}(X) is algebraically disjoint if ξ_{i} · ξ_{j} = 0 for all pairs i ≠ j. Classes in an algebraically disjoint collection can clearly be represented by disjoint surfaces – starting with any choice of representatives in general position, we can eliminate a pair of ±1 intersection points between two surfaces by adding a one-handle to one of them. An important question is whether the classes can be represented by disjoint surfaces of low genus. It turns out that the minimal genus representatives of the classes intersect in general and we will derive a lower bound for the number of pairs of ±1 intersection points.

Proof of Theorem B. We assume for the purpose of the proof that Σ_{i} is representing a divisible class d_{i}ξ_{i} for some d_{i} > 1. The result for d_{i} = 1 then follows as in Corollary 12.2. Consider Z = X − ∪_{i=1}^{n}Σ_{i} as a cylindrical end manifold with cylindrical ends [0, ∞) × Y_{i}, where Y_{i} is the boundary of a tubular neighborhood of Σ_{i} (with the opposite orientation). Note that Z is negative semi-definite, so we can adapt the argument that we used to derive genus bounds to this context.

We work with exponentially decaying Seiberg-Witten configuration space on Z with reducible asymptotic values; the background connection on Z agrees on the end [0, ∞)×Y_{i} with the pull-back of the product connection on Y_{i} (see the proof of Proposition 7.2). By gauge fixing at infinity we can assume that the asymptotic values differ by imaginaryvalued harmonic one-forms on each end. We use gauge group based at infinity on the first end [0, ∞) × Y_{1}. To compute the index of the deformation complex, we follow the proof of Proposition 7.2. The quotient of the deformation complex by the fiber complex is

0 → ⊕_{i=2}^{n}R → ⊕_{i=1}^{n}H(Y_{i}) → 0 → 0,

where H(Y_{i}) denotes the space of imaginary-valued one-forms on Y_{i}. Hence the index of the deformation complex is the index of the fiber complex plus 2 ∑ g_{i} − (n − 1), where g_{i} is the genus of Σ_{i}. The computation of the fiber index splits in the anti self-dual part and the Dirac part. Note for the first that each end contributes 1 to the spectral flow; it follows that the fiber index of the ASD part equals −2 ∑ g_{i} + n − 1. The index of the deformation complex is thus given by

\frac{1}{4}(c_{1}(Det) − σ(X)) − ∑_{i=1}^{n} k_{i}d_{i}

(see also Lemma 8.1), where Det is the determinant line of a Spin^{c} structure on X satisfying

k_{i} = ⟨c_{1}(Det), ξ_{i}⟩ − d_{i}ξ_{i}^{2} ∈ [0, d_{i}ξ_{i}^{2}]

for all i.

Next we argue as in the proof of Theorem 11.1. The based moduli space of Seiberg-Witten solutions on Z is smooth (after choosing an appropriate perturbation) and contains a unique reducible point. From the structure of the moduli space close to the reducible point we conclude that if the moduli space is positive dimensional, then it is not compact. Therefore g_{i} > k_{i}d_{i}/2 for at least one i. Finally, setting c_{1}(Det) = ∑ 2d_{i}α_{i} − c, where α_{i} is the Poincaré dual of ξ_{i}, we see that the formal dimension of the (based) moduli space is positive if and only if c^{2} > σ(X) (see the proof of Corollary 12.3). The condition at the end of the previous paragraph then becomes ⟨c, [Σ_{i}]⟩ ∈ [0, [Σ_{i}]^{2}], while the genus bound can be expressed as χ(Σ_{i}) + [Σ_{i}]^{2} ≤ ⟨c, [Σ_{i}]⟩. q.e.d.

In what follows we will restrict our attention to manifolds with b_{2}^{+}(X) = 2 in order to keep the discussion simple.

Theorem 15.1. Let X be a smooth closed connected four-manifold with b_{1}(X) = 0 and b_{2}^{+}(X) = 2, and let Σ_{1}, Σ_{2} be embedded surfaces in general position, representing algebraically disjoint classes of positive self-intersection. Suppose that a characteristic vector c ∈ H^{2}(X) satisfies

c^{2} > σ(X), ⟨c, [Σ_{i}]⟩ ≥ 0 for i = 1, 2,
and χ(Σ_{i}) + [Σ_{i}]^{2} > ⟨c, [Σ_{i}]⟩ for i = 1, 2.

Then

g(Σ_{1}) + g(Σ_{2}) + N ≥ \frac{[Σ_{1}]^{2} + [Σ_{2}]^{2} − ⟨c, [Σ_{1}] + [Σ_{2}]⟩}{2} + 1,

where N denotes the number of pairs of ±1 intersection points between Σ_{1} and Σ_{2}.

Proof. If condition (2) fails for both surfaces, then by Theorem B the surfaces are not disjoint. To construct disjoint representatives, we trade pairs of ±1 intersection points for one-handles – this way elimination of a pair of intersection points increases the genus of one of the surfaces by 1. We add the maximal possible number of handles to Σ_{1}, so that the resulting surface still does not satisfy (2), and add the rest of the handles to Σ_{2}. Since the sum of the genera of thus constructed disjoint surfaces equals g(Σ_{1}) + g(Σ_{2}) + N, the claimed inequality follows from Theorem B. q.e.d.

We compare this to bounds obtained using G-signature Theorem. As is the case for genus bounds, the result we obtained is roughly by a factor of 2 better. Specifically, we state the following consequence of a Theorem of Gilmer [4].

Proposition 15.2 (Gilmer [4]). Let X be a smooth closed connected four-manifold with b_{1}(X) = 0 and b_{2}^{+}(X) = 2, and let Σ_{1}, Σ_{2} be embedded surfaces in general position, representing algebraically disjoint classes of positive self-intersection. If Σ_{1} and Σ_{2} are not disjoint and [Σ_{1}] + [Σ_{2}] is divisible by 2, then

g(Σ_{1}) + g(Σ_{2}) + N ≥ \frac{[Σ_{1}]^{2} + [Σ_{2}]^{2}}{4} − 1,

where N denotes the number of pairs of ±1 intersection points between Σ_{1} and Σ_{2}.

15.1 Examples

(1) Let X = (S^{2} × S^{2})#(S^{2} × S^{2}) and let ξ_{1} = (p, q, 0, 0) and ξ_{2} = (0, 0, r, s) be classes of positive self-intersection, expressed with respect to the standard basis for H_{2}(X). We may assume that p, q, r, s > 0. If ξ_{i} is primitive, it is represented by an embedded sphere in X, according to a Theorem of Wall [23]; however, for p, q, r, s ≥ 2 it is not represented by a sphere in its summand. Let Σ_{1} and Σ_{2} be smooth representatives of ξ_{1} and ξ_{2} in general position. Denote by g_{i} the genus of Σ_{i} and by N the number of pairs of ±1 intersection points between Σ_{1} and Σ_{2}. Using characteristic vectors c = (2, 2, 0, 0) and c = (0, 0, 2, 2) in Theorem 15.1 gives the following lower bounds for g_{1} + g_{2} + N:

\begin{cases}(p − 1)(q − 1) + rs; & p, q ≥ 2, g_{1} < (p − 1)(q − 1) and g_{2} ≤ rs \\ pq + (r − 1)(s − 1); & r, s ≥ 2, g_{1} ≤ pq and g_{2} < (r − 1)(s − 1). \end{cases}

In particular, if ξ_{1} and ξ_{2} with p, q, r, s ≥ 2 are represented by spheres Σ_{1} and Σ_{2} in X, then

N ≥ max{(p − 1)(q − 1) + rs, pq + (r − 1)(s − 1)}.

(2) Consider now X = CP^{2}#CP^{2} and let ξ_{1} = (p, q) and ξ_{2} = (q, −p) for some p, q > 0, expressed with respect to the standard basis for H_{2}(X). Note that ξ_{i} has a smooth representative Σ_{i} of genus

g = \frac{p^{2} + q^{2} − 3(p + q)}{2} + 2,

obtained as the connected sum of minimal genus representatives for classes of divisibility p and q in CP^{2}. Let Σ_{1} and Σ_{2} be genus g representatives of ξ_{1} and ξ_{2} in general position, and let N be the number of pairs of ±1 intersection points. Using characteristic vectors c = (3, −1), c = (3, 1) and c = (1, −3) in Theorem 15.1, we obtain

N ≥ \begin{cases}p + 2q − 3; & p ≥ 2 \\ 2p + q − 3; & q ≥ 2 and p ≤ 3q \\ p + 4q − 3; & q ≥ 2 and p ≥ 3q. \end{cases}

In particular, for q = 1 we have N ≥ p − 1 and this bound is sharp, which can be seen as follows. Decompose (p, 1) = (p − 1)(1, 0) + (1, 1) and (1, −p) = (1, −1) − (p − 1)(0, 1). Since the classes (1, 1) and (1, −1) can be represented by disjoint embedded spheres, N = p − 1.

(3) As the last example consider X = 2CP^{2}#2\overline{CP}^{2}, and let ξ_{1} = (p, 0, q, 0) for some p > q > 0 and ξ_{2} = (0, r, 0, s) for some r > s > 0. These classes are represented by spheres in X (according to the Theorem of Wall mentioned above), but not in their copy of CP^{2}#\overline{CP}^{2}, unless p = q + 1 and r = s + 1. By choosing representatives Σ_{i} for ξ_{i} of small genus and in general position, we get the following bound based on Theorem 15.1 (using c = (3, 1, −1, −1) and c = (1, 3, −1, −1)):

g_{1} + g_{2} + N ≥ \begin{cases}\frac{p^{2} − q^{2} − 3p + q}{2} + \frac{r^{2} − s^{2} − r + s}{2} + 1; & p ≥ q + 2 \\ \frac{p^{2} − q^{2} − p + q}{2} + \frac{r^{2} − s^{2} − 3r + s}{2} + 1; & r ≥ s + 2. \end{cases}

Small genus here means that for a formula to hold, g_{1} has to be no greater than the first summand and g_{2} has to be no greater than the second summand.

Remark. It is an interesting question whether the bounds obtained in the above examples are optimal.

References

[1] M.F. Atiyah, V.K. Patodi & I.M. Singer, Spectral asymmetry and Riemannian geometry: I, Math. Proc. Camb. Phil. Soc. 77 (1975) 43–69, MR 0397797 (53 #1655a), Zbl 0297.58008.

[2] K. Frøyshov, The Seiberg-Witten equations and four-manifolds with boundary, Math. Res. Lett. 3(3) (1996) 373–390, MR 1397685 (97i:57037), Zbl 0872.57024.

[3] M. Furuta, Monopole equation and the \frac{11}{8} -conjecture, Math. Res. Lett. 8(3) (2001) 279–291, MR 1839478 (2003e:57042), Zbl 0984.57011.

[4] P. Gilmer, Configurations of surfaces in 4-manifolds, Trans. Amer. Math. Soc. 264 (1981) 353–380, MR 0603768 (83h:57027), Zbl 0509.57023.

[5] M. Komuro, On Atiyah-Patodi-Singer η-invariant for S^{1}-bundles over Riemann surfaces, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 30(3) (1984) 525–548, MR 0731516 (85a:58091), Zbl 0534.58036.

[6] P.B. Kronheimer & T.S. Mrowka, The genus of embedded surfaces in the projective plane, Math. Res. Lett. 1(6) (1994) 797–808, MR 1306022 (96a:57073), Zbl 0851.57023.

[7] B.-H. Li, Representing nonnegative homology classes of CP^{2}#n\overline{CP}^{2} by minimal genus smooth embeddings, Trans. Amer. Math. Soc. 352(9) (2000) 4155–4169, MR 1637082 (2000m:57052), Zbl 0947.57034.

[8] B.-H. Li & T.-J. Li, Minimal genus smooth embeddings in S^{2}×S^{2} and CP^{2}#n\overline{CP}^{2} with n ≤ 8, Topology 37(3) (1998) 575–594, MR 1604887 (99b:57059), Zbl 0899.57022.

[9] B.-H. Li & T.-J. Li, Symplectic genus, minimal genus and diffeomorphisms, Asian J. Math. 6(1) (2002) 123–144, MR 1902650 (2003c:57027), Zbl 1008.57024.

[10] R. Lockhart, Fredholm, Hodge and Liouville theorems on noncompact manifolds, Trans. Amer. Math. Soc. 301(1) (1987) 1–35, MR 0879560 (88g:58190), Zbl 0623.58019.

[11] R.B. Lockart & R.C. McOwen, Elliptic differential operators on noncompact manifolds, Annali d. Scuola Norm. Sup. de Pisa, 12 (1985) 409–448, MR 0837256 (87k:58266), Zbl 0615.58048.

[12] J. Morgan, T. Mrowka & D. Ruberman, The L^{2}-moduli space and a vanishing theorem for Donaldson polynomial invariants, Monographs in Geometry and Topology, Vol. 2, International Press, 1994, MR 1287851 (95h:57039), Zbl 0830.58005.

[13] J. Morgan, Z. Szabo & C. Taubes, A product formula for the Seiberg–Witten invariants and the generalized Thom conjecture, J. Differential Geo. 44 (1996) 706–788, MR 1438191 (97m:57052), Zbl 0974.53063.

[14] J.W. Morgan, The Seiberg-Witten equations and applications to the topology of smooth four-manifolds, Mathematical Notes, Vol. 44, Princeton University Press, Princeton, NJ, 1996, MR 1367507 (97d:57042), Zbl 0846.57001.

[15] T. Mrowka, P. Ozsváth & B. Yu, Seiberg-Witten monopoles on Seifert fibered spaces, Comm. Anal. Geom. 5(4) (1997) 685–791, MR 1611061 (98m:58017), Zbl 0933.57030.

[16] D. Mumford, An algebraic surface with K ample, (K^{2}) = 9, p_{g} = q = 0, Amer. J. Math. 101(1) (1979) 233–244, MR 0527834 (80j:14032), Zbl 0433.14021.

[17] L.I. Nicolaescu, Eta invariants of Dirac operators on circle bundles over Riemann surfaces and virtual dimensions of finite energy Seiberg-Witten moduli spaces, Israel J. Math. 114 (1999) 61–123, MR 1738674 (2001h:58032), Zbl 0998.58022.

[18] P. Ozsváth & Z. Szabó, The symplectic Thom conjecture, Ann. of Math. (2) 151(1) (2000) 93–124, MR 1745017 (2001a:57049), Zbl 0967.53052.

[19] V.A. Rohlin, Two-dimensional submanifolds of four-dimensional manifolds, Functional Anal. and Appl. 5 (1971) 39–48, MR 0298684 (45 #7733), Zbl 0268.57019.

[20] D. Ruberman, The minimal genus of an embedded surface of nonnegative square in a rational surface, Turkish J. Math. 20 (1996) 129–135, MR 1392668 (97k:57036), Zbl 0870.57025.

[21] D. Salamon, Spin geometry and Seiberg-Witten invariants, Preliminary version.

[22] C.H. Taubes, The Seiberg-Witten invariants and symplectic forms, Math. Res. Lett. 1(6) (1994) 809–822, MR 1306023 (95j:57039), Zbl 0853.57019.

[23] C.T.C. Wall, Diffeomorphisms of 4-manifolds, Jour. Lond. Math. Soc. 39 (1964) 131–140, MR 0163323 (29 #626), Zbl 0121.18101.

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