j. differential geometry 55 (2000) 385-440

HIGHER TYPE ADJUNCTION INEQUALITIES IN SEIBERG-WITTEN THEORY

PETER OZSVÁTH & ZOLTÁN SZABÓ

Abstract

In this paper, we derive new adjunction inequalities for embedded surfaces with non-negative self-intersection number in four-manifolds. These formulas are proved by using relations between Seiberg-Witten invariants which are induced from embedded surfaces. To prove these relations, we develop the relevant parts of a Floer theory for four-manifolds which bound circlebundles over Riemann surfaces.

1. Introduction

In this paper, we prove certain adjunction inequalities, which give relations between the Seiberg-Witten invariants of a four-manifold X and the genus of embedded surfaces in X. These results are generalizations of results from [12], [21], [24], see also [13].

The investigations center on a construction of an appropriate Seiberg-Witten-Floer functor for manifolds which bound circle bundles Y over Riemann surfaces (with sufficiently large Euler number), which relies on the calculations of [22]. Special cases of this theory were studied in [24], where the authors used similar techniques to prove the symplectic Thom conjecture. That problem requires an analysis of those Spin_{ℂ} structures over Y for which the Seiberg-Witten moduli space contains only reducible solutions, which simplifies the corresponding Floer homology. In this paper, we work out the theory in the other, more complicated cases. We will give more applications of these techniques in [23].

Received July 13, 1998, and, in revised form, February 2, 1999. The first author was partially supported by NSF grant number DMS 9304580, and the second author was partially supported by NSF grant number DMS-9704359, a Sloan Foundation Fellowship, and a Packard Fellowship.

Before stating the results, we set up some notation. Let X be a closed, connected, smooth four-manifold equipped with an orientation for which b_{2}^{+}(X) > 0 (where b_{2}^{+}(X) is the dimension of a maximal positive-definite linear subspace H^{+}(X; ℝ) of the intersection pairing on H^{2}(X; ℝ)) and an orientation for H^{1}(X; ℝ)⊕H^{+}(X; ℝ). Given such a four-manifold, together with a Spin_{ℂ} structure 𝔰, the Seiberg-Witten invariants (see [31], [19], [26]) form an integer-valued function

SW_{X,𝔰}: 𝔸(X) ⟶ ℤ,

where 𝔸(X) denotes the graded algebra obtained by tensoring the exterior algebra on H_{1}(X) (graded so that H_{1}(X) has grading one) with the polynomial algebra ℤ[U] on a single two-dimensional generator. The invariants are constructed via intersection theory on the moduli space M_{X}(𝔰) of solutions (A, Φ) modulo gauge to the Seiberg-Witten equations in 𝔰:

(1)  ρ(TrF_{A}^{+}) = i{Φ, Φ}_{0} − ρ(iη)
(2)  D̸_{A}Φ = 0,

where Φ is a section of W^{+}, A is a spin-connection in the spinor bundle W^{+} of 𝔰, D̸_{A} denotes the associated Dirac operator, ρ denotes Clifford multiplication, η is some fixed self-dual two-form, and {Φ, Φ}_{0} is the usual quadratic map (see [31]). Note that the invariants are zero on homogeneous elements whose degree is not d(𝔰), where

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

denotes the formal dimension of the moduli space M_{X}(𝔰). When b_{2}^{+}(X) > 1, SW_{X,𝔰} is a diffeomorphism invariant of the four-manifold; when b_{2}^{+}(X) = 1, the invariants depend on a chamber structure (see [19], [24]). There are two distinguished chambers corresponding to the two components of K(X) = {ω ∈ H^{2}(X; ℝ) − 0 | ω^{2} ≥ 0}. Given a component K_{0} of K(X), the corresponding invariant (still denoted SW_{X,𝔰}) is calculated using the moduli space of solutions to the Seiberg-Witten equations perturbed by any generic self-dual two-form η, provided that the sign of −2πc_{1}(𝔰)·ω_{g} +∫_{X} η∧ω_{g} agrees with the sign of γ ·ω_{g}, where γ is any class in K_{0}, and ω_{g} ≠ 0 is a harmonic (with respect to the metric g), self-dual two-form over X. Note that SW_{X,𝔰} is a diffeomorphism invariant of X (and the component K_{0}).

Those Spin_{ℂ} structures 𝔰 for which the invariant SW_{X,𝔰} is non-trivial are called basic classes.

Our results are easiest to state when b_{1}(X) = 0, where we have the following.

Theorem 1.1. Let X be a smooth, closed, connected, oriented fourmanifold with b_{2}^{+}(X) > 0 and b_{1}(X) = 0, and let Σ ⊂ X be a smoothlyembedded surface with genus g(Σ) > 0 representing a non-torsion homology class with self-intersection number [Σ] · [Σ] ≥ 0. If b_{2}^{+}(X) > 1, then we have the following adjunction inequality

|⟨c_{1}(𝔰), [Σ]⟩| + [Σ] · [Σ] + 2d(𝔰) ≤ 2g(Σ) − 2,

for each basic class 𝔰 ∈ Spin_{ℂ}(X). Furthermore, when b_{2}^{+}(X) = 1, for each basic class 𝔰 of X for the component of K(X) which contains PD[Σ] with

−⟨c_{1}(𝔰), [Σ]⟩ + [Σ] · [Σ] ≥ 0,

we have an inequality

−⟨c_{1}(𝔰), [Σ]⟩ + [Σ] · [Σ] + 2d(𝔰) ≤ 2g(Σ) − 2.

Remark 1.2. The above theorem should be seen as a refinement of the adjunction inequality proved by Kronheimer-Mrowka and Morgan-Szabó-Taubes (see [12], [21], [3]). Analogous results for immersed spheres were obtained by Fintushel and Stern, see [7].

In fact, Theorem 1.1 follows from a more general version. To state this, note first that an inclusion i: Σ ⟶ X induces a map

i_{∗}: 𝔸(Σ) ⟶ 𝔸(X).

Theorem 1.3. Let X be a smooth, closed, connected, oriented fourmanifold with b_{2}^{+}(X) > 0. Let Σ ⊂ X be a surface with genus g(Σ) > 0 representing a non-torsion homology class with self-intersection number [Σ] · [Σ] ≥ 0. Let ℓ be an integer so that there is a symplectic basis {A_{j}, B_{j}}_{j=1}^{g} for H_{1}(Σ) so that i_{∗}(A_{j}) = 0 in H_{1}(X; ℝ) for i = 1, …, ℓ. Fix any element a ∈ 𝔸(X), and let and b ∈ 𝔸(Σ) be an element whose degree satisfies d(b) ≤ ℓ. If b_{2}^{+}(X) > 1 then for each Spin_{ℂ} structure 𝔰 so that SW_{X,𝔰}(a · i_{∗}(b)) is non-zero, we have

|⟨c_{1}(𝔰), [Σ]⟩| + [Σ] · [Σ] + 2d(b) ≤ 2g(Σ) − 2.

Furthermore, when b_{2}^{+}(X) = 1 then for each Spin_{ℂ} structure 𝔰 of X with

−⟨c_{1}(𝔰), [Σ]⟩ + [Σ] · [Σ] ≥ 0,

for which SW_{X,𝔰}(a·i_{∗}(b)) is non-zero, when calculated in the component of K(X) containing PD[Σ], we have an inequality

−⟨c_{1}(𝔰), [Σ]⟩ + [Σ] · [Σ] + 2d(b) ≤ 2g(Σ) − 2. (3)

The Adjunction Inequality (3) does not hold without homological restrictions on X, as we can see by looking at the ruled surface X = S^{2} × Σ. In general, one can obtain only a weaker inequality (losing the factor of 2 on the dimension d(b)), as follows.

Theorem 1.4. Let X be a smooth, closed, connected, oriented fourmanifold with b_{2}^{+}(X) > 1. Let Σ ⊂ X be a surface with genus g(Σ) > 0 representing a non-torsion homology class with self-intersection number [Σ] · [Σ] ≥ 0. Let a ∈ 𝔸(X) and b ∈ 𝔸(Σ). If b_{2}^{+}(X) > 1 and if SW_{X,𝔰}(a · i_{∗}(b)) is non-zero for some b ∈ 𝔸(Σ) of degree d(b), then we have

|⟨c_{1}(𝔰), [Σ]⟩| + [Σ] · [Σ] + d(b) ≤ 2g(Σ) − 2.

If b_{2}^{+}(X) = 1 and 𝔰 is a Spin_{ℂ} structure with

−⟨c_{1}(𝔰), [Σ]⟩ + [Σ] · [Σ] ≥ 0,

for which SW_{X,𝔰}(a·i_{∗}(b)) is non-zero, when calculated in the component of K(X) containing PD[Σ], then we have

−⟨c_{1}(𝔰), [Σ]⟩ + [Σ] · [Σ] + d(b) ≤ 2g(Σ) − 2. (4)

Remark 1.5. Adjunction inequalities for surfaces of positive square in Donaldson’s theory were first obtained in the influential paper of Kronheimer and Mrowka (see [13]). These inequalities were strengthened under similar, but more restrictive, hypotheses in their preprint [14]; see also [6]. The conjectured relationship between the Donaldson and Seiberg-Witten invariants gives a correspondence between the adjunction inequalities arising in these two theories. For more on this correspondence, see [27], [31], [18], [5], [25], and [8].

We illustrate Theorem 1.4 with the following application:

Corollary 1.6. Let T ⊂ S^{1} × S^{3} be an embedded torus for which the restriction of the one-dimensional cohomology from H^{1}(S^{1} × S^{3}) is non-trivial. Let (X, ω) be a symplectic four-manifold and Σ ⊂ X be a symplectic submanifold with non-negative self-intersection. Then, in the connected sum X#(S^{1} × S^{3}), the internal connected sum Σ#T ⊂ X#(S^{1} × S^{3}) minimizes genus among all embedded surfaces Σ′ ⊂ X#(S^{1} × S^{3}) which are homologous to Σ#T and for which the one-dimensional cohomology from S^{1} × S^{3} restricts non-trivially to Σ′.

Theorem 1.3 follows from a relation which holds for embedded surfaces with arbitrary self-intersection number. This relation can be viewed as a generalization of the relation appearing in [24]. Once again, we begin by stating the case when b_{1}(X) = 0, in the interest of exposition.

Theorem 1.7. Let X be a smooth, closed, connected, oriented fourmanifold with b_{1}(X) = 0, and let Σ ⊂ X be a smoothly embedded surface with genus g(Σ) > 0. Then, for each Spin_{ℂ} structure 𝔰 with

−⟨c_{1}(𝔰), [Σ]⟩ + [Σ] · [Σ] ≥ 0

and

−⟨c_{1}(𝔰), [Σ]⟩ + [Σ] · [Σ] + 2d(𝔰) > 2g(Σ) − 2,

we have

SW_{X,𝔰}(U^{d}) = SW_{X,𝔰−PD[Σ]}(U^{d′}),

where d and d′ denote the dimensions of 𝔰 and 𝔰 − PD[Σ] respectively. In the case where b_{2}^{+}(X) = 1, both invariants are to be calculated in the same component of K(X).

More generally, we have the following.

Theorem 1.8. Let X be a smooth, closed, connected, oriented fourmanifold with b_{2}^{+}(X) > 0. Let Σ ⊂ X be a surface with genus g(Σ) > 0. Let ℓ be an integer so that there is a symplectic basis {A_{j}, B_{j}}_{j=1}^{g} for H_{1}(Σ) so that i_{∗}(A_{j}) = 0 in H_{1}(X; ℝ) for i = 1, …, ℓ. For each Spin_{ℂ} structure 𝔰 with

−⟨c_{1}(𝔰), [Σ]⟩ + [Σ] · [Σ] ≥ 0

and each b ∈ 𝔸(Σ) of degree d(b) ≤ ℓ with

−⟨c_{1}(𝔰), [Σ]⟩ + [Σ] · [Σ] + 2d(b) > 2g(Σ) − 2, (5)

there is an element b′ ∈ 𝔸(Σ) with d(b′) ≥ d(b) so that for any a ∈ 𝔸(X), we have

SW_{X,𝔰}(a · i_{∗}(b)) = SW_{X,𝔰−PD[Σ]}(a · i_{∗}(b′)). (6)

Furthermore, if b = U^{d/2}, then b′ − U^{d′/2} lies in the ideal generated by H_{1}(Σ) in 𝔸(Σ). Once again, in the case where b_{2}^{+}(X) = 1, both invariants are to be calculated in the same component of K(X).

Theorem 1.3 is a simple consequence of Theorem 1.8, as the following proof shows.

Theorem 1.8 ⇒ Theorem 1.3. Suppose Theorem 1.3 were false; i.e., suppose there were X, Σ, 𝔰, a, and b which satisfy the hypotheses of the theorem, but which violate Adjunction Inequality (3). We can assume without loss of generality that

−⟨c_{1}(𝔰), [Σ]⟩ + [Σ] · [Σ] ≥ 0,

by reversing the orientation of Σ if necessary (when b_{2}^{+}(X) > 1). Thus, Theorem 1.8 applies. Let b′ be the element which satisfies Relation (6), so we have that SW_{X,𝔰−PD[Σ]}(a · i_{∗}(b′)) ≠ 0. Since b and b′ are homogeneous elements with the same degree modulo two, and d(b′) ≥ d(b), it follows that we can find elements a′ ∈ 𝔸(X) and b′′ ∈ 𝔸(Σ) with d(b′′) = d(b), and SW_{X,𝔰−PD[Σ]}(a′ · i_{∗}(b′′)) ≠ 0. Now, since [Σ] · [Σ] ≥ 0 and d(b′′) = d(b), we see that Σ also violates the adjunction inequality for 𝔰 − PD[Σ], a′ ∈ 𝔸(X), and b′′ ∈ 𝔸(Σ). Proceeding in this way, we see that 𝔰 − nPD[Σ] is a Seiberg-Witten basic class for all n ≥ 0. If b_{2}^{+}(X) > 1, then there are only finitely many basic classes of X, so since Σ is not a torsion class, we get a contradiction, proving Theorem 1.3 in this case.

The above argument works also when b_{2}^{+}(X) = 1, since there are still only finitely many basic classes of the form 𝔰−nPD[Σ] with n ≥ 0 in the chamber corresponding to PD[Σ]. We see this as follows. Fix a metric g on X and a generic self-dual two-form η. Clearly, if 𝔰 is fixed and n is sufficiently large, the sign of PD[Σ]·ω_{g} agrees with the sign of −2πc_{1}(𝔰− nPD[Σ])·ω_{g}+∫ η∧ω_{g}; i.e., for all large n, the η-perturbed moduli spaces for 𝔰 − nPD[Σ] can be used calculate the invariant in the component which contains PD[Σ]. But the usual compactness argument shows that all but finitely many of these moduli spaces are empty. Again, we have the contradiction completing the proof of Theorem 1.3. q.e.d.

By blowing up, Theorem 1.8 is reduced to the case where the selfintersection number of Σ is sufficiently negative. The theorem is then proved by expressing the Seiberg-Witten invariants of a four-manifold with such an embedded surface Σ in terms of relative invariants, which take values in a Seiberg-Witten-Floer homology associated to non-trivial circle bundles over Σ. In the presence of the topological hypotheses on the inclusion of H_{1}(Σ) in H_{1}(X), the above relation then follows from properties of this Floer homology.

The outline of this paper is as follows. In Section 2, we give examples of the adjunction formulae. Our examples include four-manifolds with b_{2}^{+}(X) = 1, and also examples where both b_{2}^{+}(X) > 1 and b_{1}(X) > 0, and a proof of Corollary 1.6. In Section 3, we show how Theorem 1.8 can be deduced from properties of a product formula, which relates the Seiberg-Witten invariants of a four-manifold containing an embedded surface with sufficiently negative self-intersection number with certain relative invariants associated to X − Σ. For completeness, we also show how a modified version of Theorem 1.8 implies Theorem 1.4. In Section 4 we review the gauge theory for circle bundles over Riemann surfaces as developed in [22]. There is one Spin_{ℂ} structure in which the moduli space of reducibles has singularities (to which we return in a later section). In Section 5, we prove the product formula introduced in Section 3, assuming technical facts about the moduli spaces over N, the tubular neighborhood of Σ. In Section 6, we define an invariant with irreducible boundary values and use properties of this relative invariant to analyze the terms appearing in the product formula, completing the proof of Theorem 1.8. In Section 7, we prove the technical facts about the moduli spaces over N which were used in earlier sections. In Section 8, we show how to extend the results of Sections 4 and 7 to deal with the remaining Spin_{ℂ} structure. Finally, in Section 9, which should be viewed as an appendix, we discuss representatives for the cohomology classes used throughout the paper.

Acknowledgements. The authors wish to thank Vicente Muñoz for his very helpful comments on an early version of this paper. They also wish to thank the referee for providing useful suggestions.

2. Examples

We give examples of embedded surfaces Σ ⊂ X with b_{2}^{+}(X) > 1 and b_{1}(X) > 0, where the inequality in Theorem 1.3 is in fact an equality. (It is an open problem whether manifolds with b_{2}^{+}(X) > 1 and b_{1}(X) = 0 can admit basic classes of non-zero dimension.) But first, we need to find such four-manifolds X which admit basic classes of non-zero dimension.

To construct these examples, we use the following construction.

Definition 2.1. Let X be smooth four-manifold and let S ⊂ X be an embedded two-sphere with zero self-intersection number. Let X′ denote the manifold obtained as surgery on S; i.e.,

X′ = (X − nd(S)) ∪_{ϕ} S^{1} × D^{3},

where nd(S) is an open tubular neighborhood of S and

ϕ: ∂(X − nd(S)) ⟶ S^{1} × S^{2}

is a orientation-reversing diffeomorphism. Let C ⊂ X′ denote the closed curve which is the core of the added S^{1} × D^{3}. Note that there is a diffeomorphism X − S ≅ X′ − C.

Of course, X′ = X#(S^{1} × S^{3}) is an example of this construction.

Proposition 2.2. Let X be a closed, smooth, oriented four-manifold with b_{2}^{+}(X) > 0, and let S ⊂ X be a homologically trivial embedded twosphere. For each Spin_{ℂ} structure 𝔰 on X, there is a unique induced Spin_{ℂ} structure 𝔰′ on X′ with the property that

𝔰|_{X−S} = 𝔰′|_{X′−C}.

Then, d(𝔰′) = d(𝔰) + 1; and for all a ∈ 𝔸(X)

SW_{X′,𝔰′}(a · μ(C)) = SW_{X,𝔰}(a),

for some homology orientation on X′. (When b_{2}^{+}(X) = 1, both invariants are to be calculated in the same chamber.)

Proof. The dimension statement is straightforward.

To prove the relation, we pull X apart along S^{1} ×S^{2} = ∂ nd(S), and study the corresponding moduli spaces (see Section 5 for more discussion on such matters). Let X_{0} denote the complement X − S, given a cylindrical-end metric modeled on the product metric [0, ∞) × S^{1} × S^{2}, where S^{2} is given its standard, round metric. Note that this metric can be extended over both S^{1} × D^{3} and D^{2} × S^{2} to give metrics with non-negative scalar curvature. Consequently, the moduli spaces of solutions over S^{1} × S^{2}, S^{1} × D^{3}, and D^{2} × S^{2} consist entirely of smooth reducibles (i.e., the moduli spaces are identified with S^{1}, S^{1}, and a point respectively).

Let M_{X_{0}}(𝔰_{0}) denote the moduli space of finite energy solutions to the Seiberg-Witten equations over X_{0} in the Spin_{ℂ} structure 𝔰_{0} = 𝔰|_{X_{0}}. Thus, we can think of the boundary map as a map

ρ: M_{X_{0}}(𝔰_{0}) ⟶ S^{1}.

Gluing theory gives a diffeomorphism for all sufficiently large T > 0:

M_{X(T)}(𝔰) ≅ ρ^{−1}(x_{0}),

where X(T) denotes the metric on X with neck-length T and x_{0} ∈ S^{1} corresponds to the unique reducible on S^{1}×S^{2} which extends to D^{2}×S^{2}. Consequently,

SW_{X,𝔰}(a) = ⟨M_{X_{0}}(𝔰_{0}), μ(a) ∪ μ(C)⟩, (7)

since μ(C) is represented by the holonomy class around C (see Proposition 9.1).

Similarly, gluing gives a diffeomorphism of

M_{X_{0}}(𝔰_{0}) ≅ M_{X′(T)}(𝔰′),

and consequently

SW_{X′,𝔰′}(a · C) = ⟨M_{X_{0}}(𝔰_{0}), μ(a · C)⟩. (8)

Together, Equations (7) and (8) prove the proposition. q.e.d.

With this proposition in hand, we can provide a proof of Corollary 1.6:

Proof of Corollary 1.6. According to Taubes’ non-vanishing theorem (see [29]), the canonical Spin_{ℂ} structure 𝔰_{0} of the symplectic manifold has SW_{(X,𝔰_{0})} = ±1; thus, according to Proposition 2.2, if 𝔰′ is any Spin_{ℂ} structure over X′ = X#(S^{1} ×S^{3}) whose restriction to the X side agrees with 𝔰_{0}, and b ∈ H_{1}(X′; ℤ) is a non-zero homology class coming from the S^{1} × S^{3} factor, then SW_{(X′,𝔰′)}(b) ≠ 0. Thus, in view of the usual adjunction formula for the symplectic submanifold

−⟨c_{1}(𝔰_{0}), [Σ]⟩ + Σ · Σ = 2g(Σ) − 2,

Inequality (4) gives that g(Σ′) > g(Σ), as required.

Note that in the case where b_{2}^{+}(X) = 1, the additional hypothesis −⟨c_{1}(𝔰′), [Σ′]⟩ + [Σ′] · [Σ′] > 0 follows immediately from the adjunction formula except, of course, when g(Σ) = 0, in which case the result is vacuously true. q.e.d.

2.1 Examples of Theorem 1.3 with b_{2}^{+}(X) > 1

We can use Proposition 2.2 to construct examples where Theorem 1.3 is sharp, as well.

Fix natural numbers n, k, and m with 2k ≥ n > 1, and let X be the four-manifold E(n)#m(S^{3} × S^{1}), where E(n) is a simply-connected elliptic surface with no multiple fibers and with geometric genus n − 1. Let Σ_{0} ⊂ E(n) denote a symplectic submanifold representing the homology class S + kF, where S and F denote the homology classes of a section and a fiber respectively of the elliptic fibration. Let T_{i} ⊂ X denote a fiber in the elliptic fibration of the i^{th} summand S^{3} × S^{1}. Let Σ ⊂ X denote the internal connected sum of Σ_{0}#F_{1}# … #F_{m}. Note that g(Σ) = k + m and Σ · Σ = 2k − n ≥ 0. Let 𝔰 be the Spin_{ℂ} structure over X induced from the canonical Spin_{ℂ} structure on E(n), and let b = 𝔸(Σ) be the product B_{1} · . . . · B_{m} where B_{i} ∈ H_{1}(X) generates H_{1} of the i^{th} copy of S^{1} × S^{3}. Note that d(b) = m and Σ has a symplectic basis {A_{i}, B_{i}}_{i=1}^{k+m} for which A_{1}, …, A_{m} are homologically trivial in X. By Proposition 2.2,

SW_{X,𝔰}(B_{1} · . . . · B_{m}) = 1,

so the data X, b, Σ, 𝔰 satisfy the hypotheses of Theorem 1.3. In fact, we see that

Σ · Σ + ⟨c_{1}(𝔰), [Σ]⟩ + 2d(b) = 2g(Σ) − 2,

which shows that the inequality of the theorem is sharp, for all choices of g(Σ) > 0, Σ · Σ ≥ 0, and d(b).

2.2 Ruled surfaces: the homological hypotheses on H_{1}(Σ)

By looking at ruled surfaces, we give examples where Inequality (4) is sharp, and hence that some homological hypotheses are necessary for the stronger inequality (which appears in Theorem 1.3) to hold.

As mentioned before, one cannot hope for the adjunction inequality of Theorem 1.3 to be valid without additional topological hypotheses on the inclusion of Σ in X. Indeed, fix n ≥ 0 and g > 0, and let X be the two-sphere bundle over a surface Σ of genus g, associated to the circle bundle with Euler number n. In particular, X contains an embedded copy of Σ with Σ·Σ = n. In the chamber corresponding to PD[Σ], there is a zero-dimensional basic class 𝔰_{0} with c_{1}(𝔰_{0}) = −K_{X}, where K_{X} is the canonical class of X viewed as Kähler manifold. Moreover, letting F be the class of the two-sphere fiber in X, we see that the moduli space associated to 𝔰_{0} + dPD[F] is identified with Sym^{d}(Σ), and U is the symmetric product of the volume form of Σ (see Proposition 6.10 for a related discussion). Thus, if 𝔰 = 𝔰_{0} + dPD[F], then SW_{𝔰}(U^{d}) ≠ 0, and

⟨c_{1}(𝔰), [Σ]⟩ = 2d.

Clearly, Adjunction Inequality (4) is sharp for all values of k, d, n, and g provided that −n ≤ k, where k = ⟨c_{1}(𝔰), [Σ]⟩, 2d = d(b), n = Σ · Σ, and g = g(Σ). (This construction, strictly speaking, only gives us even values of d(b). For odd values, one can attach an S^{1}×S^{3}.) In particular, we see that some homological criterion on the embedding of Σ ⊂ X is necessary for the stronger Inequality (3) to hold.

3. From product formulas to relations

The aim of this section is to outline the proof of Theorem 1.8. By employing the blowup formula in a manner analogous to [24], we reduce to the case where the self-intersection number of Σ is very negative (Proposition 3.1). The invariants in this latter case are studied via a product formula, which we state (and prove in Section 5), whose terms are then related with other Seiberg-Witten invariants of X. In the end of the section, we discuss the modifications which are needed to prove Theorem 1.4.

We reduce Theorem 1.8 to the following special case.

Proposition 3.1. Theorem 1.8 holds, under the additional hypothesis that

0 ≤ −⟨c_{1}(𝔰), [Σ]⟩ + [Σ] · [Σ] ≤ 2g(Σ) − 2 and −[Σ] · [Σ] > 2g − 2.

The reduction involves the following basic result of Fintushel and Stern.

Theorem 3.2. (Blowup Formula: [7] and [26]). Let X be a smooth, closed four-manifold, and let X̂ = X#\overline{ℂℙ}^{2} denote its blow-up, with exceptional class E ∈ H^{2}(X̂; ℤ). If b_{2}^{+}(X) > 1, then for each Spin_{ℂ} structure 𝔰̂ on X̂ with d(𝔰̂) ≥ 0, and each a ∈ 𝔸(X) ≅ 𝔸(X̂), we have

SW_{X̂,𝔰̂}(a) = SW_{X,𝔰}(U^{m}a),

where 𝔰 is the Spin_{ℂ} structure induced on X obtained by restricting 𝔰̂, and 2m = d(𝔰) − d(𝔰̂). If b_{2}^{+}(X) = 1, there is a one-to-one correspondence between components of Ω^{+}(X) and Ω^{+}(X̂), and the above relation holds provided both invariants are calculated in chambers associated to corresponding components.

Before showing how to reduce Theorem 1.8 to the special case, we point out that another special case of Theorem 1.8 was already proved in Theorem 1.3 [24]. More specifically, the following was shown:

Theorem 3.3. [24] Let X be a smooth, closed, connected, oriented four-manifold with b_{2}^{+}(X) > 0. Let Σ ⊂ X be a surface with genus g(Σ) > 0 and negative self-intersection. For each Spin_{ℂ} structure 𝔰 with

−⟨c_{1}(𝔰), [Σ]⟩ + [Σ] · [Σ] > 2g(Σ) − 2,

there is an element b′ ∈ 𝔸(Σ) so that that for any a ∈ 𝔸(X), we have

SW_{X,𝔰}(a) = SW_{X,𝔰−PD[Σ]}(a · i_{∗}(b′)).

Furthermore, b′ − U^{d′/2} lies in the ideal generated by H_{1}(Σ) in 𝔸(Σ).

Remark 3.4. In the language of Theorem 1.8, this case corresponds to ℓ = 0 and b = 1.

Proposition 3.1 ⇒ Theorem 1.8. Let g = g(Σ), fix an integer m with

m > [Σ] · [Σ] + 2g − 2,

let X̂ = X#m\overline{ℂℙ}^{2}, and let Σ̂ be the “proper transform” of Σ, the embedded surface obtained by internal connected sum of Σ with the m exceptional spheres in the \overline{ℂℙ}^{2} summands; i.e.,

PD[Σ̂] = PD[Σ] − E_{1} − ⋯ − E_{m}.

Finally, let 𝔰̂ denote the Spin_{ℂ} structure on X̂ which agrees with 𝔰 in the complement of the exceptional spheres, whose Chern class satisfies

c_{1}(𝔰̂) = c_{1}(𝔰) − E_{1} − ⋯ − E_{m}.

It is easy to check that:

−[Σ̂] · [Σ̂] = m − [Σ] · [Σ] > 2g − 2,
−⟨c_{1}(𝔰̂), [Σ̂]⟩ + [Σ̂] · [Σ̂] = −⟨c_{1}(𝔰), [Σ]⟩ + [Σ] · [Σ];
d(𝔰) = d(𝔰̂).

Now, if

−⟨c_{1}(𝔰), [Σ]⟩ + [Σ] · [Σ] ≥ 2g,

the hypotheses of Theorem 3.3 are satisfied; and otherwise, the hypotheses of Proposition 3.1 are. In either case, for each b ∈ 𝔸(Σ) of degree d(b) ≤ ℓ, we can find b′ ∈ 𝔸(Σ) with

SW_{X̂,𝔰̂}(a · i_{∗}(b)) = SW_{X̂,𝔰̂−PD[Σ̂]}(a · i_{∗}(b′)). (9)

According to the blow-up formula,

SW_{X̂,𝔰̂}(a · i_{∗}(b)) = SW_{X,𝔰}(a · i_{∗}(b)); (10)

and, since 𝔰̂ − PD[Σ̂] agrees with 𝔰 − PD[Σ] away from the exceptional spheres and

c_{1}(𝔰̂ − PD[Σ̂]) = c_{1}(𝔰 − PD[Σ]) − E_{1} − ⋯ − E_{m},

we see from another application of the blowup formula that

SW_{X̂,𝔰̂−PD[Σ̂]}(a · i_{∗}(b′)) = SW_{X,𝔰−PD[Σ]}(a · i_{∗}(b′)). (11)

Theorem 1.8 then follows by combining Equations (9), (10) and (11). q.e.d.

We now turn to the special case considered in Proposition 3.1. We will study the Seiberg-Witten invariant of X by decomposing it into two pieces

X = N ∪_{Y} (X − N),

where Y a circle bundle over Σ (as in the proposition), and N is the associated disk bundle. Following [22], the moduli space of Seiberg-Witten monopoles over Y decomposes into an irreducible and a reducible component. (Actually, there is one Spin_{ℂ} structure over Y, where it is necessary to perturb the equations for this decomposition to occur; this perturbation is studied Section 8.) Correspondingly, we construct relative invariants of X − Σ, denoted SW_{𝔰}^{irr} and SW_{𝔰}^{red}, arising from the L^{2} moduli spaces on X − Σ with irreducible and reducible boundary values. In Section 5 (see Lemma 5.6, and the discussion following it), we prove the following:

Proposition 3.5. Suppose

0 ≤ −⟨c_{1}(𝔰), [Σ]⟩ + [Σ] · [Σ] ≤ 2g(Σ) − 2 and −[Σ] · [Σ] > 2g − 2.

Then,

SW_{X,𝔰} = SW_{𝔰}^{irr} + SW_{𝔰}^{red}.

We can interpret the latter invariant in terms of the closed manifold as follows.

Definition 3.6. Let Σ be a surface of genus g, and let {A_{i}, B_{i}}_{i=1}^{g} be a standard symplectic basis for H_{1}(Σ; ℤ). For j = 0, …, g, let ξ_{j}(Σ) ∈ 𝔸(Σ) be the degree 2j component of

∏_{i=1}^{g} (1 + U + A_{i} · B_{i}) ∈ 𝔸(Σ);

i.e., ξ_{0} = 1, ξ_{1}(Σ) = gU + ∑ A_{i} · B_{i}, …, ξ_{g}(Σ) = ∏_{i=1}^{g} (U + A_{i} · B_{i}).

Proposition 3.7. Suppose

0 ≤ −⟨c_{1}(𝔰), [Σ]⟩ + [Σ] · [Σ] ≤ 2g(Σ) − 2 and −[Σ] · [Σ] > 2g − 2.

Then, letting

e = g − 1 + \frac{⟨c_{1}(𝔰), [Σ]⟩ − [Σ] · [Σ]}{2},

we have that

SW_{𝔰}^{red}(a) = SW_{X,𝔰−PD[Σ]}(a · ξ_{g−1−e}(Σ))

for all a ∈ 𝔸(X).

Furthermore, under the homological condition of Theorem 1.8, we will express SW_{𝔰}^{irr} in terms of SW_{X,𝔰−PD[Σ]}, as follows.

Proposition 3.8. Suppose

0 ≤ −⟨c_{1}(𝔰), [Σ]⟩ + [Σ] · [Σ] ≤ 2g(Σ) − 2 and −[Σ] · [Σ] > 2g − 2,

and let ℓ be an integer so that there is a symplectic basis {A_{i}, B_{i}}_{i=1}^{g} for H_{1}(Σ) so that i_{∗}(A_{i}) = 0 in H_{1}(X; ℝ) for i = 1, …, ℓ. Then, for each b ∈ 𝔸(Σ) of degree e < d(b) ≤ ℓ, there is an element b′′ ∈ 𝔸(Σ) so that

SW_{𝔰}^{irr}(a · i_{∗}(b)) = SW_{𝔰−PD[Σ]}(a · i_{∗}(b′′)).

Furthermore, b_{2} lies in the ideal generated by H_{1}(Σ) in 𝔸(Σ).

The proof of Proposition 3.8 is given in the end of Section 6.

Proposition 3.1 follows immediately from Propositions 3.5–3.8. In the proof of these latter propositions, we will construct a natural Seiberg-Witten-Floer functor for four-manifolds which bound Y.

Before proceeding, we pause to tie up one more loose end: Theorem 1.4. That result can be reduced to a relation which replaces Theorem 1.8, using the same argument given in the proof of Theorem 1.3. The relevant relation in this case is:

Theorem 3.9. Let X be a smooth, closed, connected, oriented fourmanifold with b_{2}^{+}(X) > 0. Let Σ ⊂ X be a surface with genus g(Σ) > 0. For each Spin_{ℂ} structure 𝔰 with

−⟨c_{1}(𝔰), [Σ]⟩ + [Σ] · [Σ] ≥ 0

and each b ∈ 𝔸(Σ) of degree d(b) with

−⟨c_{1}(𝔰), [Σ]⟩ + [Σ] · [Σ] + d(b) > 2g(Σ) − 2, (12)

there is an element b′ ∈ 𝔸(Σ) with d(b′) ≥ d(b) so that for any a ∈ 𝔸(X), we have

SW_{X,𝔰}(a · i_{∗}(b)) = SW_{X,𝔰−PD[Σ]}(a · i_{∗}(b′)). (13)

Furthermore, if b = U^{d}, then b′−U^{d′} lies in the ideal generated by H_{1}(Σ) in 𝔸(Σ).

Once again, via the blowup formula, this relation can be reduced to the case where the self-intersection number Σ is very negative; i.e.,

0 ≤ −⟨c_{1}(𝔰), [Σ]⟩ + [Σ] · [Σ] ≤ 2g(Σ) − 2 and −[Σ] · [Σ] > 2g − 2.

(compare Proposition 3.1). Like Proposition 3.1, this special case also follows from the product formula in Proposition 3.5, the relation in Proposition 3.7, together with the following analogue of Proposition 3.8 (whose proof is also given in the end of Section 6):

Proposition 3.10. Suppose

0 ≤ −⟨c_{1}(𝔰), [Σ]⟩ + [Σ] · [Σ] ≤ 2g(Σ) − 2 and −[Σ] · [Σ] > 2g − 2.

Then, for each b ∈ 𝔸(Σ) of degree 2e < d(b), there is an b′′ ∈ 𝔸(Σ) so that

SW_{𝔰}^{irr}(a · i_{∗}(b)) = SW_{𝔰−PD[Σ]}(a · i_{∗}(b′′)).

Furthermore, b′′ lies in the ideal generated by H_{1}(Σ).

4. Gauge theory on ℝ × Y

The Seiberg-Witten moduli spaces over Y and ℝ × Y were studied for Seifert fibered three-manifolds Y in [22]. We summarize these results here, for Y a circle-bundle over a Riemann surface Σ with g(Σ) > 0 and Euler number −n, where n > 2g − 2.

Y admits a canonical Spin_{ℂ} structure whose bundle of spinors is ℂ ⊕ π^{∗}(K_{Σ}^{−1}), which we use to identify the Spin_{ℂ} structures on Y with H^{2}(Y; ℤ) ≅ ℤ^{2g} ⊕ ℤ/nℤ.

Let N_{Y} (𝔱) denote the moduli space of solutions to the Seiberg-Witten equations over Y in the Spin_{ℂ} structure 𝔱. Here, we use the metric g_{Y} and SO(3)-connection over T Y of [22]. Given a pair of components C_{1}, C_{2} in N_{Y} (𝔱), let M(C_{1}, C_{2}) denote the moduli space of solutions [A, Φ] to the Seiberg-Witten equations on ℝ × Y for which

lim_{t↦−∞}[A, Φ]|_{{t}×Y} ∈ C_{1}, and lim_{t↦∞}[A, Φ]|_{{t}×Y} ∈ C_{2}.

This moduli space admits a translation action by ℝ. Let M̂(C_{1}, C_{2}) denote the quotient of M(C_{1}, C_{2}) by the translation action.

In general, these spaces admit a Morse-theoretic interpretation. If c_{1}(𝔱) is a torsion class, there is a real-valued functional

CSD: B(Y, 𝔱) ⟶ ℝ

defined over the configuration space B(Y, 𝔱) of pairs (B, Ψ) of spinconnections B in 𝔱 and spinors Ψ modulo gauge (cf. Equation (28) below; when e = g − 1, we will use the perturbation given in Equation (32)). The critical manifolds are the moduli spaces N (Y; 𝔱). When c_{1}(𝔱) is not torsion, the functional is circle-valued. The Seiberg-Witten equations on ℝ×Y are the upward gradient-flow equations for this functional. In keeping with this interpretation, we call M̂(C_{1}, C_{2}) the space of unparameterized flows from C_{1} to C_{2}.

Theorem 4.1. [22] Let Y be a circle-bundle over a Riemann surface with genus g > 0 and Euler number −n < 2 − 2g. The moduli space N_{Y} (𝔱) is empty unless 𝔱 corresponds to a torsion class in H^{2}(Y; ℤ). So, suppose 𝔱 corresponds to e ∈ ℤ/nℤ ⊂ H^{2}(Y; ℤ).

(1) If 0 ≤ e < g − 1 then N_{Y} (𝔱) contains two components, a reducible one J, identified with the Jacobian torus H^{1}(Σ; ℝ/ℤ), and a smooth irreducible component C diffeomorphic to Sym^{e}(Σ).

Both of these components are non-degenerate in the sense of Morse-Bott. There is an inequality CSD(J) > CSD(C), so the space M̂(J , C) is empty. The space M̂(C, J) is smooth of expected dimension 2e; indeed it is diffeomorphic to Sym^{e}(Σ).

(2) If g − 1 < e ≤ 2g − 2, the Seiberg-Witten moduli spaces over both Y and ℝ × Y in this Spin_{ℂ} structure are naturally identified with the corresponding moduli spaces in the Spin_{ℂ} structure 2g − 2 − e, which we just described.

(3) For all other e ≠ g − 1, N_{Y} (𝔱) contains only reducibles. Furthermore, it is smoothly identified with the Jacobian torus.

In the Spin_{ℂ} structure corresponding to g − 1 ∈ ℤ/nℤ, the unperturbed Seiberg-Witten equations used in Theorem 4.1 are inconvenient, since the corresponding reducible manifold is not smooth in the sense of Morse-Bott. To overcome this difficulty, when working in this Spin_{ℂ} structure, we use a perturbation of the equations where the theory resembles the case where 0 ≤ e < g − 1 (and, in particular, the reducibles are smooth). A thorough discussion of the perturbation is given in Section 8.

5. The product formula

In this section, we define two quantities, SW^{irr} and SW^{red}, and prove that the Seiberg-Witten invariant decomposes into a sum of these (Propostion 3.5). Furthermore, we express SW^{red} in terms of another Seiberg-Witten invariant of X (Proposition 3.7).

Decompose X as

X = N ∪_{Y} X_{0},

where Y is unit circle bundle over Σ with Euler number −n, with n > 2g − 2. N is a tubular neighborhood of the surface Σ (which is diffeomorphic to the disk bundle associated to Y), and X_{0} is the complement in X of the interior of N. Fix metrics g_{X_{0}}, g_{N}, and g_{Y} for which g_{X_{0}} and g_{N} are isometric to

dt^{2} + g_{Y}^{2}

in a collar neighborhood of their boundaries (where t is a normal coordinate to the boundary). Let X(T) denote the Riemannian manifold which is diffeomorphic to X and whose metric g_{T} is obtained from the description

X(T) = N ∪_{∂N={−T }×Y} [−T, T] × Y ∪_{{T }×Y =−∂X_{0}} X_{0};

i.e., g_{T} |_{N} = g_{N}, g_{T} |_{[−T,T]×Y} = dt^{2}+g_{Y}^{2} , and g_{T} |_{X_{0}} = g_{X_{0}}. Our goal here is to provide, for all sufficiently large T, a description of the moduli space M_{X(T)}(𝔰) on X(T) in terms of the moduli spaces for Y, N_{Y} (𝔰|_{Y}), and the finite-energy, cylindrical-end moduli spaces associated to X_{0} and N, denoted M_{X_{0}}(𝔰|_{X_{0}}), and M_{N}(𝔰|_{N}) respectively. In this context, finite energy means that the total variation of the Chern-Simons-Dirac functional over the infinite cylinder is bounded. Henceforth, X_{0} and N will denote the cylindrical-end manifolds obtained by attaching [0, ∞)× Y (with appropriate orientations) to the corresponding subsets of X.

In the case where b_{2}^{+}(X) = 1, we choose the perturbing form η to be compactly supported in X_{0} in such a way that

−2πc_{1}(𝔰) · ω_{∞} + ∫_{X_{0}} η ∧ ω_{∞}

has the same sign as γ · ω_{∞}, where γ is a compactly supported representative for a class in the chosen component K_{0} ⊂ K(X), and ω_{∞} is a self-dual harmonic two-form over X_{0} with ∫_{X_{0}} ω_{∞} ∧ ω_{∞} = 1. Note that such a γ and ω_{∞} can be found since Σ · Σ < 0, forcing b_{2}^{+}(X_{0}) = 1 (see [1]). Now, the moduli spaces of the η-perturbed Seiberg-Witten equations over X(T) calculate the invariant in the chosen chamber for all sufficiently large T.

We collect useful facts about the moduli spaces M_{N}(𝔰|_{N}), most of which we defer to Section 7 (see also [24]), but first we introduce some notation. The map

Spin_{ℂ}(N) → ℤ

given by

𝔰 ↦ ⟨c_{1}(𝔰), [Σ]⟩

induces a one-to-one correspondence between Spin_{ℂ} structures and integers which are congruent to n modulo 2. Note that the Spin_{ℂ} structure over Y 𝔰|_{Y} corresponds to the mod n reduction of

e = g − 1 + \frac{⟨c_{1}(𝔰), [Σ]⟩ + n}{2}

appearing in Theorem 4.1.

By taking limits at the end of the tube, one can define maps

ρ: M_{N}(𝔰) ⟶ N_{Y} (𝔰|_{Y}) and ρ: M_{X_{0}}(𝔰) ⟶ N_{Y} (𝔰|_{Y})

(see [20]). If C is a connected manifold of N_{Y} (𝔰|_{Y}), then M_{N}(𝔰, C) and M_{X_{0}}(𝔰, C) denotes the pre-image of C under ρ. Throughout the following discussion, we will use the perturbation discussed in Section 8 over X_{0}, N, and Y, when 𝔰|_{Y} corresponds to e = g − 1 (in the notation of Section 4); i.e., in this case, N_{Y} (𝔰|_{Y}), M_{N}(𝔰) and M_{X_{0}}(𝔰|_{X_{0}}) will denote the perturbed versions of these moduli spaces, with perturbation parameter u in the range 0 < u < 2, in the notation of Section 8. (We will show in Section 8 that this is an allowable perturbation to use when b_{2}^{+}(X) = 1; i.e., we are computing the Seiberg-Witten invariants in the correct chamber.) When they are clear from the context, we leave the Spin_{ℂ} structures out of the notation. Note that on the cylinders, the analogous boundary value maps factor through the unparameterized spaces, defining

ρ_{J} : M̂(C, J) ⟶ J and ρ_{C}: M̂(C, J) ⟶ C,

where J and C are the critical manifolds of Theorem 4.1.

Proposition 5.1. Suppose that −n − 2g + 2 ≤ ⟨c_{1}(𝔰), [Σ]⟩ ≤ −n, and let

e = g − 1 + \frac{⟨c_{1}(𝔰), [Σ]⟩ + n}{2},

Then according to Theorem 4.1, and Theorem 8.1 when e = g − 1, N_{Y} (𝔰|_{Y}) has two components, J and C, where C is diffeomorphic to Sym^{e}(Σ). Furthermore, the expected dimensions of the moduli spaces over N and X_{0} are given by:

(14)  e-dimM_{N}(J) = 2e + 1
(15)  e-dimM_{N}(C) = 2e
(16)  e-dimM_{X_{0}}(J) = 2d + 2g − 2e − 2
(17)  e-dimM_{X_{0}}(C) = 2d,

where d = d(𝔰) and g = g(Σ). Moreover, M_{N}^{∗}(J), M_{N}(C), M_{X_{0}}(J), and M_{X_{0}}(C) are transversally cut out by the Seiberg-Witten equations (in particular, they are manifolds of the expected dimension).

Proof. This is a combination of Proposition 7.9 and 7.10 when ⟨c_{1}(𝔰), [Σ]⟩ ≠ n, and Proposition 8.3 in the remaining case. q.e.d.

When studying the deformation theory of reducibles inside M_{N}(J), the L^{2} kernel and the cokernel of the Dirac operator (on the cylindricalend version of N) play a central role. These spaces can be concretely understood, thanks to the holomorphic interpretation of the Dirac operator (see also [24]).

Proposition 5.2. Suppose that −n − 2g + 2 ≤ ⟨c_{1}(𝔰), [Σ]⟩ ≤ −n, then there is a natural correspondence between reducibles [(A, 0)] ∈ M_{N}(J) with holomorphic line bundles E of degree e over Σ which identifies

KerD̸_{A} = H^{0}(Σ, E) and CokerD̸_{A} = H^{1}(Σ, E).

Proof. This follows from Theorem 7.4 and Proposition 7.5 (see also the proof of Theorem 8.1 in the perturbed case). q.e.d.

The above proposition allows us to understand an important class of reducibles.

Definition 5.3. The jumping locus Θ ⊂ M_{N}(J) is the locus of reducible solutions [(A, 0)] ∈ M_{N}(J) for which KerD̸_{A} is non-trivial.

Corollary 5.4. Suppose that −n − 2g + 2 ≤ ⟨c_{1}(𝔰), [Σ]⟩ ≤ −n, then the jumping locus Θ ⊂ J = M_{N}^{red}(J) is the image of a smooth map Sym^{e}(Σ) ⟶ J.

Proof. According to Proposition 5.2, the space Θ ⊂ J is identified with the space of degree e line bundles over Σ with non-trivial H^{0}. The forgetful map Sym^{e}(Σ) ⟶ J which takes a degree e divisor, thought of as a complex line bundle with section, to the underlying complex line bundle gives the surjection to this locus. q.e.d.

We will also need to understand those Spin_{ℂ} structures 𝔰 ∈ Spin_{ℂ}(N) for which −n < ⟨c_{1}(𝔰), [Σ]⟩ ≤ n.

Proposition 5.5. If

−n < ⟨c_{1}(𝔰), [Σ]⟩ ≤ n,

then the moduli space M_{N}(J) contains only reducibles. Moreover, the space of reducibles is smoothly identified with the Jacobian torus J (i.e., the kernel and the cokernel of the Dirac operator coupled to any reducible vanishes). Furthermore, M_{N}(C) is empty.

Proof. When |⟨c_{1}(𝔰), [Σ]⟩| < n, this is proved in Section 7, where it appears as Proposition 7.6. The remaining case is covered by Proposition 8.2. q.e.d.

With these preliminaries in place, we turn to the Seiberg-Witten invariants of X, by investigating the moduli spaces over X(T). Specifically, choose some a ∈ 𝔸(X) of degree d(𝔰), and indeed choose representatives for the corresponding homology classes which are compactly supported in X_{0}. Let V (a) denote the corresponding representatives for μ(a) in the configuration spaces for X_{0} and X(T) as appropriate (see Section 9 for a discussion of such representatives). Recall that SW_{X,𝔰}(a) is the number of points in M_{X(T)}(𝔰) ∩ V (a), counted with appropriate sign.

Lemma 5.6. Suppose that −n − 2g + 2 ≤ ⟨c_{1}(𝔰), [Σ]⟩ ≤ −n, then for each ϵ > 0, there is a T_{0} > 0 so that for all T ≥ 2T_{0} the restriction of [(A, Φ)] ∈ M_{X(T)}(𝔰) ∩ V (a) to any slice {t} × Y with t ∈ [−T_{0}, T_{0}] lies within ϵ (in the C^{∞} topology) from either J or C. Accordingly, if ϵ is sufficiently small, then [(A, Φ)] satisfies exactly one of the following two conditions:

(H-1) [(A, Φ)]|_{N} is C^{∞} close to smooth reducible and [(A, Φ)]|_{X_{0}} is C^{∞} close to (the restriction to X_{0}) of a configuration in M_{X_{0}}(J) ∩ V (a).

(H-2) [(A, Φ)]|_{N} is C^{∞} close to a configuration in M_{N}(C), and [(A, Φ)]|_{X_{0}} is C^{∞} close to a configuration in the cut-down moduli space M_{X_{0}}(C) ∩ V (a).

Proof. This is a dimension-counting argument. Suppose we have a sequence [A_{i}, Φ_{i}] ∈ M_{X(T_{i})}(𝔰) ∩ V (a), for some increasing, unbounded sequence {T_{i}}_{i=1}^{∞} of real numbers. By local compactness, there is a subsequence which converges in C_{loc}^{∞} to a pair of configurations (A_{N}, Φ_{N}) and (A_{X_{0}}, Φ_{X_{0}}) over N and X_{0} respectively. By the usual compactness arguments (see [12]), the total variation of the Chern-Simons-Dirac functional of (A_{i}, Φ_{i}) over the cylinder [−T_{i}, T_{i}]×Y remains globally bounded (independent of i), so (A_{N}, Φ_{N}) and (A_{X_{0}}, Φ_{X_{0}}) both have finite energy.

First, we prove that either Hypothesis (H-1) or (H-2) is satisfied. There are a priori four cases, according to which critical manifolds ρ[A_{X_{0}}, Φ_{X_{0}}] and ρ[A_{N}, Φ_{N}] lie in.

(P-1) The case where ρ[A_{N}, Φ_{N}] ∈ J while ρ(A_{X_{0}}, Φ_{X_{0}}) ∈ C is excluded because CSD(C) > CSD(J).

(P-2) The case where ρ[A_{N}, Φ_{N}] ∈ C while ρ(A_{X_{0}}, Φ_{X_{0}}) ∈ J is excluded by a dimension count, as follows. In this case, we see that ρ[A_{X_{0}}, Φ_{X_{0}}] ∈ ρ_{J} (M(C, J)) ∩ ρ(M_{X_{0}}(J) ∩ V (a)). But

ρ_{J} (M(C, J)) = ρ_{J} (M̂(C, J)),

so

e-dim(ρ_{J} (M(C, J)) ∩ ρ(M_{X_{0}}(J) ∩ V (a)) = −2.

It follows from Theorems 4.1 and 8.1 that M(C, J) is smooth of the expected dimension, so from the usual transversality results, the above intersection is generically empty.

(P-3) Suppose that ρ(A_{N}, Φ_{N}) ∈ J and ρ(A_{X_{0}}, Φ_{X_{0}}) ∈ J. Then we see that

ρ[A_{N}, Φ_{N}] = ρ[A_{X_{0}}, Φ_{X_{0}}] ∈ ρ(M_{N}(J)) ∩ ρ(M_{X_{0}}(J) ∩ V (a));

but, according to Proposition 5.1

e-dimρ(M_{N}^{∗}(J)) ∩ ρ(M_{X_{0}}(J) ∩ V (a))
= e-dimM_{N}(J) + M_{X_{0}}(J) − 2d − 2g
= −1,

which is generically empty. Thus, it follows that [A_{N}, Φ_{N}] must be reducible. Moreover, according to Corollary 5.4,

e-dimρ(Θ) ∩ ρ(M_{X_{0}}(J) ∩ V (a)) = 2e + e-dimM_{X_{0}}(J) − 2d − 2g
= −2,

which is also generically empty. Hence, [A_{N}, Φ_{N}] and [A_{X_{0}}, Φ_{X_{0}}] satisfy Hypotheses (H-1).

(P-4) If ρ[A_{N}, Φ_{N}] and ρ[A_{X_{0}}, Φ_{X_{0}}] both lie in C, then the Hypotheses (H-2) are satisfied.

The assertion at the beginning of the proposition follows easily. q.e.d.

The above proposition says that we can partition the points in the cut-down moduli space (which is an oriented, zero-dimensional manifold) for sufficiently large T into two disjoint sets, the subsets of configurations which satisfy (H-1) and (H-2) respectively. Thus, if we let SW_{𝔰}^{red}(a) and SW_{𝔰}^{irr}(a) be the signed number of points satisfying (H-1) and (H-2) respectively, then

SW_{X,𝔰}(a) = SW_{𝔰}^{red}(a) + SW_{𝔰}^{irr}(a). (18)

As we shall see, gluing theory allows us to compute both of these quantities in terms of cylindrical-end moduli spaces. So, in the next step, we study these cylindrical-end moduli spaces.

Lemma 5.7. For all Spin_{ℂ} structures 𝔰 on X the corresponding moduli spaces M_{N}(C), M_{X_{0}}(J), and M_{X_{0}}(C) ∩ V (a) are all compact manifolds.

Proof. The compactness of M_{X_{0}}(J) and M_{N}(C) follows from the usual compactness arguments, together with the facts that the Chern-Simons-Dirac functional is real-valued, CSD(J) > CSD(C), and there are no other critical manifolds. Compactness of M_{X_{0}}(C) ∩ V (a) follows from this, together with a straightforward dimension count (see the discussion above in the proof of Lemma 5.6, part (P-2)). q.e.d.

Compactness of M_{X_{0}}(J) allows us to define a relative invariant with reducible boundary values. We pause to discuss some relevant properties of this invariant.

Definition 5.8. Let 𝔰_{0} be a Spin_{ℂ} structure on X_{0} which extends over X. Since the moduli space M_{X_{0},𝔰_{0}}(J) is compact, there is a relative Seiberg-Witten invariant

SW_{(X_{0},𝔰_{0},J)}: 𝔸(X_{0}) ⟶ ℤ,

defined by the pairing SW_{(X_{0},𝔰_{0},J)}(a) = ⟨[M_{X_{0},𝔰_{0}}(J)], μ(a)⟩.

This relative invariant is related to an absolute invariant, according to the following.

Proposition 5.9. If 𝔰 satisfies −n < ⟨c_{1}(𝔰), [Σ]⟩ ≤ n, then for all a ∈ 𝔸(X),

SW_{X,𝔰}(a) = SW_{(X_{0},𝔰_{0},J)}(a),

where 𝔰_{0} = 𝔰|_{X_{0}}.

Proof. Recall that M(𝔰|_{N}) consists entirely of reducibles all of which are smooth, according to Proposition 5.5; thus, gluing theory identifies the moduli spaces M_{X(T)}(𝔰) for large T with M_{X_{0},𝔰_{0}}(J). (See also [24], where this result appears as Proposition 2.7.) q.e.d.

We now return to the discussion of SW^{red} and SW^{irr}. Although the definitions of both terms implicitly use T, we show now that if T is sufficiently large, then the terms can be computed from absolute invariants (and hence are independent of the parameter).

Proposition 5.10. Suppose that 𝔰 satisfies

−n − 2g + 2 ≤ ⟨c_{1}(𝔰), [Σ]⟩ ≤ −n,

where Σ has self-intersection number −n, and let

e = g − 1 + \frac{⟨c_{1}(𝔰), [Σ]⟩ + n}{2}.

Then, for all sufficiently large T,

SW_{𝔰}^{red}(a) = SW_{X,𝔰−PD[Σ]}(a · ξ_{g−1−e}(Σ)),

where ξ_{g−1−e}(Σ) ∈ 𝔸(Σ) is the element defined in Definition 3.6.

Proof. The moduli space M_{N}^{red}(J) − Θ comes equipped with an obstruction bundle Ξ ⟶ M_{N}^{red}(J) − Θ, defined by Ξ_{[(A,0)]} = CokerD̸_{A}. (whose K-theory class canonically extends over all of M_{N}^{red}(J)). The dimension count in Lemma 5.6 guarantees that each solution in M_{X_{0}}(J)∩ V (a) extends uniquely to a smooth reducible over N. Thus, gluing theory gives that

SW_{𝔰}^{red}(a) = ⟨[M_{X_{0}}(J) ∩ V (a)], e(L ⊗ ρ^{∗}(Ξ))⟩
= ⟨[M_{X_{0}}(J) ∩ V (a)], c_{g−1−e}(L ⊗ ρ^{∗}(Ξ))⟩
= ⟨[M_{X_{0}}(J)], μ(a) ∪ c_{g−1−e}(L ⊗ ρ^{∗}(Ξ))⟩,

where e(L⊗ρ^{∗}(Ξ)) denotes the Euler class of the bundle over M_{N}^{red}(J)− Θ. According to the Riemann-Roch formula, dim(Ξ) = 2g − 2 − 2e so the Euler class agrees with the top Chern class c_{g−1−e}, which in turn extends over all of J. Using the index theorem for families, together with the holomorphic interpretation of the obstruction bundle Ξ given in Proposition 5.2, it is a straightforward computation that the total Chern class of Ξ is

∏_{i=1}^{g}(1 + μ(A_{i})μ(B_{i}))

(see also [24] Proposition 2.6); thus,

c_{g−1−e}(L ⊗ ρ^{∗}(Ξ)) = ξ_{g−1−e}(Σ).

Putting all this together, we have that

SW_{𝔰}^{red}(a) = SW_{(X_{0},𝔰_{0},J)}(a · ξ_{g−1−e}(Σ)), (19)

where 𝔰_{0} = 𝔰|_{X_{0}}. Since n − 2g + 2 ≤ ⟨c_{1}(𝔰 − PD[Σ]), [Σ]⟩ ≤ n and 𝔰 − PD[Σ]|_{X_{0}} = 𝔰_{0}, the proposition then follows from Proposition 5.9.

q.e.d.

Proposition 5.11. For sufficiently large T,

SW_{𝔰}^{irr}(a) = #M_{X_{0}}(C) ∩ V (a).

Proof. Gluing shows that

SW_{𝔰}^{irr}(a) = (#M_{X_{0}}(C) ∩ V (a))( deg(ρ: M_{N}(C) → C)).

According to Propositions 7.9 and 8.3, ρ: M_{N}(C) → C either has degree +1, or M_{N}(C) is empty. The latter case would force SW_{𝔰}^{irr}(a) ≡ 0 (for the given genus and self-intersection number).

To rule out this latter case, we need only look at an example where the irreducible term is non-zero. Let X be a ruled surface X over Σ associated to the line bundle with Euler number −n. Let Σ ⊂ X denote the section with self-intersection number −n, and fix any 0 ≤ e ≤ g − 1. Let 𝔰 denote the Spin_{ℂ} structure over X given by 𝔰 = 𝔰_{0} + ePD[F], where 𝔰_{0} is the canonical Spin_{ℂ} structure on X associated to the Kähler structure, and F denotes a fiber in the ruling. It is easy to see that SW_{X,𝔰−PD[Σ]} ≡ 0, as the corresponding space of divisors is empty (see Proposition 7.5). Moreover, we know that SW_{X,𝔰} ≢ 0 (compare the example in Section 2.2). Thus, in light of Equation (18) and Proposition 5.10, we have examples where SW^{irr} ≢ 0, forcing the degree to be non-zero. q.e.d.

We will give the seemingly ad hoc quantity #M_{X_{0}}(C)∩V (a) a more intrinsic formulation in Section 6. With the help of this formulation, we can then prove a vanishing result for this term under suitable algebrotopological hypotheses on the embedding of Σ ⊂ X (Proposition 3.8).

6. Relative invariants

Let X_{0} be a smooth, oriented manifold-with-boundary with b_{2}^{+}(X_{0}) > 0, whose boundary is identified with ∂X_{0} = −Y, a circle bundle over a Riemann surface Σ of genus g > 0 with Euler number −n, where n > 2g − 2.

In Section 5, we studied the moduli space M_{X_{0}}(C), and used it to define a relative invariant

SW_{𝔰}^{irr}: 𝔸(X_{0}) ⟶ ℤ,

by cutting down the moduli space M_{X_{0}}(C) by submanifolds representing μ(a) which are induced from compactly supported representatives for homology in X_{0} (see Proposition 5.11). When a ∈ 𝔸(Y), there are alternate representatives which are supported “at infinity.” The advantage of these representatives is that the corresponding relative invariant inherits relations arising from the cohomology ring of C. In view of the non-compactness of M_{X_{0}}(C), the two types of representatives do not necessarily give rise to the same invariant. However, the difference can be explicitly computed in terms of other Seiberg-Witten invariants. In this section, we recast this discussion in a more algebraic setting, defining an invariant

SW_{(X_{0},C)} : 𝔸(X_{0}) ⊗ H^{∗}(C) ⟶ ℤ.

which simultaneously captures both types of representatives; in particular,

SW_{𝔰}^{irr}(a) = SW_{(X_{0},C)}(a ⊗ 1).

Proposition 3.8 then follows from properties of this invariant.

A subtlety arises in the definition of SW_{(X_{0},C)}, since the moduli space M_{X_{0}}(C) is not compact. However, we have the following weak compactness theorem.

Definition 6.1. A sequence of configurations {[A_{i}, Φ_{i}]}_{i=1}^{∞} is said to converge weakly to a configuration

[B, Ψ] × [A, Φ] ∈ M̂(C, J) ×_{J} M_{X_{0}}(J)

if [A_{i}, Φ_{i}] converges to [A, Φ] in C_{loc}^{∞}, and there is an increasing, unbounded sequence of real numbers {T_{i}}_{i=1}^{∞} with T_{i} > i, so that the translates of {[A_{i}, Φ_{i}]|_{[0,2T_{i}]×Y} }_{i=1}^{∞}, viewed as a sequence of configurations on [−T_{i}, T_{i}]×Y, converge in C_{loc}^{∞} to a configuration which is equivalent (under translations) to [B, Ψ].

Proposition 6.2. Weak convergence gives the space

\overline{M}_{X_{0}}(C) = M_{X_{0}}(C) ∐ M̂(C, J) ×_{J} M_{X_{0}}(J)

the structure of a compact Hausdorff space.

Proof. This a standard argument from Morse-Floer theory. A general discussion of compactness results for the anti-self-duality equation can be found in [20] (see especially Theorem 6.3.3 of [20]); so we sketch the argument here only briefly.

A sequence [A_{i}, Φ_{i}] ∈ M_{X_{0}}(C) converges in C_{loc}^{∞}, after passing to a subsequence, to some solution [A, Φ] to the Seiberg-Witten equations on X_{0}. Since each of the [A_{i}, Φ_{i}] have finite energy, so does [A, Φ]; thus, it has a boundary value. If ρ[A, Φ] ∈ C, then the length-energy estimates of L. Simon [28] can be used to show that the convergence is C^{∞} as in [20].

If, on the other hand, ρ[A, Φ] ∉ C, it must be the case that ρ[A, Φ] ∈ J. Now, let T_{i} ∈ ℝ be the number so that

CSD[A_{i}, Φ_{i}]_{\{T_{i}\}×Y} = \frac{CSD(J) + CSD(C)}{2}.

Clearly, T_{i} ↦ ∞. After passing to a subsequence, we can find a configuration [B, Ψ] so that the sequence [A_{i}, Φ_{i}]|_{[0,2T_{i}]×Y}, viewed as a sequence of configurations over [−T_{i}, T_{i}], converges in C_{loc}^{∞} to [B, Ψ]. In fact, [B, Ψ] must solve the Seiberg-Witten equations and it must have finite energy, so [B, Ψ] ∈ M(C, J). The usual length-energy estimates then guarantee that the boundary values match up. q.e.d.

The topological space \overline{M}_{X_{0}}(C) defined in Proposition 6.2 is called the compactified moduli space. The following result follows immediately from its definition.

Proposition 6.3. The inclusion maps

i: M_{X_{0}}(C) ⟶ B^{∗}(X_{0} − (0, ∞) × Y)

and

i ∘ Π_{2} : M̂(C, J) ×_{J} M_{X_{0}}(J) ⟶ B^{∗}(X_{0} − (0, ∞) × Y)

fit together to give a continuous map

\overline{i}: \overline{M}_{X_{0}}(C) ⟶ B^{∗}(X_{0} − (0, ∞) × Y),

where B^{∗} denotes the irreducible configurations.

Similarly, we can extend the restriction map over the compactified moduli space, as follows.

Proposition 6.4. The restriction maps

ρ_{C}: M_{X_{0}}(C) ⟶ C

and

ρ_{C} ∘ Π_{1} : M̂(C, J) ×_{J} M_{X_{0}}(J) ⟶ C

fit together to give a continuous map

\overline{ρ}_{C}: \overline{M}_{X_{0}}(C) ⟶ C.

Proof. If a sequence [A_{n}, Φ_{n}] ∈ M_{X_{0}}(C) converges to an ideal point

[B, Ψ] × [A, Φ] ∈ M̂(C, J) ×_{J} M_{X_{0}}(J),

then there is a divergent sequence {T_{n}}_{n=1}^{∞} of real numbers so that

lim_{n↦∞} τ_{n}^{∗}[A_{n}, Φ_{n}]|_{[T_{n},∞)×Y} = [B, Ψ]|_{[0,∞)×Y} ,

where

τ_{n}: [0, ∞) × Y ⟶ [T_{n}, ∞) × Y

is the map induced by translation by T_{n} on the first coordinate. Since each path has finite energy, continuity of the restriction maps (see [20]) guarantees that

lim_{n↦∞} ρ[A_{n}, Φ_{n}]|_{{t}×Y} = lim_{n↦∞} ρτ_{n}^{∗}[A_{n}, Φ_{n}] = ρ[B, Ψ].

q.e.d.

Gluing gives this space more structure.

Proposition 6.5. Gluing endows \overline{M}_{X_{0}}(C) with the structure of a manifold. The space of ideal solutions

M̂(C, J) ×_{J} M_{X_{0}}(J)

has the structure of a smooth submanifold of codimension two. In particular, a fundamental class for M_{X_{0}}(C) gives rise to a unique fundamental class for \overline{M}_{X_{0}}(C).

Proof. Gluing describes the end of M(X, C) as a fibered product

(M̂^{0}(C, J) ×_{J} M_{X_{0}}^{0}(J) × (0, ∞))/S^{1},

where the superscript denotes based versions of the moduli spaces. This gives the space of ideal solutions a disk-bundle neighborhood in \overline{M}_{X_{0}}(C). q.e.d.

In light of the above result, we can define the relative Seiberg-Witten invariant SW_{(X_{0},C)}, as follows.

Definition 6.6. The relative Seiberg-Witten invariant

SW_{(X_{0},C)} : 𝔸(X_{0}) ⊗ H^{∗}(C) ⟶ ℤ

is defined by

SW_{(X_{0},C)}(a ⊗ ω) = ⟨[\overline{M}_{X_{0}}(C)], \overline{i}^{∗}(μ(a)) ∪ \overline{ρ}_{C}^{∗}(ω)⟩.

We now spell out the strategy for proving Proposition 3.8. First, it is shown that for b ∈ 𝔸(Y), SW_{(X_{0},C)}(a·b⊗ω) can be expressed in terms of SW_{(X_{0},C)}(a ⊗ b · ω) and SW_{(X_{0},J)} (Lemma 6.7 and Proposition 6.9). Here, b · ω denotes the action of 𝔸(Y) on H^{∗}(C) induced from the inclusion of C in B^{∗}(Y). (Note the cohomology classes over \overline{M}_{X_{0}}(C) induced from 𝔸(Y) through the action on H^{∗}(C), and pulled back via \overline{ρ}, correspond to divisor representatives over X_{0} which are supported “at infinity.”) Then, it is shown that SW_{(X_{0},C)}(a ⊗ b · ω) vanishes, when b has sufficiently high degree. This follows from algebraic considerations, according to which b · ω = b′ · ω, where b′ ∈ 𝔸(Y) lies in the ideal generated by the cycles in Y which bound in X_{0} (Proposition 6.12). It is then easy to see that SW (a ⊗ b′ · ω) vanishes (Corollary 6.13).

Now, we express the “commutator” SW (a ⊗ b · ω) − SW (a · b ⊗ ω). First note that if b is induced from H_{1}(Y), the commutator vanishes, as follows.

Lemma 6.7. Let [γ] ∈ H_{1}(Y), then for all a ∈ 𝔸(X) and ω ∈ H^{∗}(C),

SW_{(X_{0},C)}(a ⊗ μ[γ] · ω) = SW_{(X_{0},C)}(a · μ[γ] ⊗ ω).

Proof. We must show that \overline{ρ}_{C}^{∗}(μ[γ]) is homologous to \overline{i}^{∗}(μ[γ]). It suffices to verify this over the subset M(X_{0}, C) ⊂ \overline{M}_{X_{0}}(C), since the complement has codimension two, and the classes in question are onedimensional. Over the subset, now, the claim is easy to verify. On M_{X_{0}}(C), ρ^{∗}(μ[γ]) is represented by (Hol_{γ} ∘ ρ_{C})^{∗}(dθ), the holonomy around a representative of γ “at infinity” (see Proposition 9.1); while i^{∗}(μ[γ]) is represented by Hol_{γ_{0}}^{∗}(dθ), where γ_{0} = 0 × γ ⊂ 0 × Y ⊂ [0, ∞) × Y ⊂ X. Now, the cylinder [0, ∞) × γ provides a homotopy between Hol_{γ} ∘ ρ_{C} and Hol_{γ_{0}}. q.e.d.

It remains to see how the point class commutes. For this class, we can express the commutator in terms of SW_{(X,J)} and another Seiberg-Witten invariant, defined below.

Definition 6.8. There is a Seiberg-Witten invariant of the tube

ŜW_{(C,J)} : H^{∗}(C) ⟶ H^{∗}(J) ⊂ 𝔸(X_{0})

which raises degree by dimension 2g − dim M̂(C, J) = 2g − dim C, defined by

ŜW_{(C,J)}(ω) = (P_{2})_{∗}((ρ_{J} × Id)^{∗}PD[Δ] ∪ (ρ_{C} ∘ P_{1})^{∗}ω),

where P_{1} and P_{2} are the projection maps

P_{1} : M̂(C, J) × J ⟶ M̂(C, J) and P_{2} : M̂(C, J) × J ⟶ J ,

and PD[Δ] denotes the Poincaré dual of the diagonal Δ ⊂ J ×J. Thus, ŜW_{(C,J)} satisfies:

⟨M̂(C, J) ×_{J} M_{X_{0}}(J), \overline{i}^{∗}(μ(a)) ∪ \overline{ρ}_{C}^{∗}(ω)⟩ = SW_{(X_{0},J)}(a · ŜW_{(C,J)}(ω)).

We can now calculate the commutator, which involves comparing the cohomology classes \overline{ρ}_{C}^{∗}μ(y) and \overline{i}^{∗}μ(x) over \overline{M}_{X_{0}}(C), where y is a point in Y and x is a point in X_{0}.

Proposition 6.9. Choose points x ∈ X_{0} and y ∈ Y. In \overline{M}_{X_{0}}(C), we have

\overline{ρ}_{C}^{∗}(μ(y)) − \overline{i}^{∗}(μ(x)) = PD[M̂(C, J) ×_{J} M_{X_{0}}(J)].

Consequently, there is a relation between Seiberg-Witten invariants:

SW_{(X_{0},C)}(a ⊗ μ(y) · ω) − SW_{(X_{0},C)}(a · μ(x) ⊗ ω)

= SW_{(X_{0},J)}(a · ŜW_{(C,J)}(ω)).

Proof. Clearly, the difference \overline{ρ}_{C}^{∗}(μ(y)) − \overline{i}^{∗}(μ(x)) is the first Chern class of the circle bundle Hom_{S^{1}}(L_{x}, L_{y}). Here, L_{z} denotes the moduli space based at z; see Section 9. To prove the proposition, we must verify that this bundle admits a section σ in the complement of

M̂(C, J) ×_{J} M_{X_{0}}(J) ⊂ \overline{M}_{X_{0}}(C)

(i.e., over M_{X_{0}}(C) ⊂ \overline{M}_{X_{0}}(C)) and that, with respect to a trivialization of the circle bundle over a disk transverse to the submanifold, the restriction of the section to the boundary induces a map from the circle to the circle which has degree one.

The section σ is induced by parallel transport, as follows. Let γ be a half-infinite arc formed by joining [0, ∞) × y to any arc which connects x to 0 × y. Over the point [A, Φ] ∈ M_{X_{0}}(C), parallel transport via A along γ induces a homomorphism in Hom_{S^{1}}(L_{x}, L_{y}).

We now verify that the trivialization induces a degree one map around circles transverse to the submanifold. For any point in the submanifold

[A_{1}, Φ_{1}] × [A_{2}, Φ_{2}] ∈ M̂(C, J) ×_{J} M_{X_{0}}(J),

fix fibers

[A_{1}, Φ_{1}, λ_{1}] ∈ L_{x}|_{[A_{1},Φ_{1}]} and [A_{2}, Φ_{2}, λ_{2}] ∈ L_{y}|_{[A_{2},Φ_{2}]}.

These choices induce a trivialization of Hom_{S^{1}}(L_{x}, L_{y}) over a disk in \overline{M}_{X_{0}}(C) transverse to [A_{1}, Φ_{1}]×[A_{2}, Φ_{2}] (obtained by varying the gluing and translation parameters). Calculating the desired degree amounts to seeing how the holonomy along γ varies as the gluing parameter is rotated. But holonomy along any path which crosses the gluing region once varies as a degree one function of the gluing parameter. q.e.d.

We can understand the action of 𝔸(Y) on H^{∗}(C) explicitly, under the identification C ≅ Sym^{k}(Σ).

Before describing this, we begin with a few preliminaries about the homology of symmetric products of Σ (for an extensive discussion of this topic, see [17]). Recall that Sym^{k}(Σ) can be viewed as the quotient of the k-fold Cartesian product Σ^{×k} by the action of the symmetric group on k letters. We denote the quotient map by

q: Σ^{×k} ⟶ Sym^{k}(Σ).

According to elementary properties of the transfer homomorphism,

q_{∗}: H_{∗}(Σ^{×k}) ⟶ H_{∗}(Sym^{k}(Σ))

is surjective. Dually, we have a map

q^{∗}: H^{∗}(Sym^{k}(Σ)) ⟶ H^{∗}(Σ^{×k})

which identifies H^{∗}(Sym^{k}(Σ)) with the elements of H^{∗}(Σ^{×k}) ≅ H^{∗}(Σ)^{⊗k} which are invariant under the symmetric group action. In particular, by summing over the action, we obtain a map

Sym^{k}: H^{∗}(Σ) ⟶ H^{∗}(Sym^{k}(Σ)).

Thus, if we fix any collection of points {p_{2}, …, p_{k}} ⊂ Σ, given ω ∈ H^{∗}(Σ), Sym^{k}(Σ) is the class characterized by the property that

⟨Sym^{k}(ω), q_{∗}(Z × p_{2} × ⋯ × p_{k})⟩ = ⟨ω, Z⟩,

for any cycle Z ⊂ Σ. Equivalently, given a cycle Z ∈ H_{∗}(Σ), Sym^{k}(PD[Z]) is Poincaré dual to the cycle q(Z × Σ × ⋯ × Σ). The above discussion works over rational coefficients (which suffices for our purposes), but in fact it works over ℤ as well, since H_{∗}(Sym^{k}(Σ)) has no torsion (see [17]).

Proposition 6.10. Under the identification C ≅ Sym^{k}(Σ), the canonical map

𝔸(Y) ⟶ H^{∗}(Sym^{k}(Σ)),

induced from the inclusion of Sym^{k}(Σ) = C ⟶ B^{∗}(Y), takes μ(y) for y ∈ H_{∗}(Y) to the cohomology class Sym^{k}(PD[π_{∗}(y)]) ∈ H^{2−∗}(Sym^{k}(Σ)), where π: Y → Σ is the projection map.

Proof. We can reduce to a corresponding statement for configurations over Σ, as follows. Let E be a line bundle over Σ, so that W ≅ π^{∗}(E ⊗ (ℂ ⊕ K_{Σ})). Then, pull-back induces a map

π^{∗}: B(Σ, E) = A(E) × Γ(E)/Map(Σ, S^{1}) ⟶ B(Y, W),

to the configurations where the fiber-wise holonomy of the connection is constant, and the section is covariantly constant around each fiber. The identification between the critical manifolds and the symmetric powers C ≅ Sym^{k}(Σ) described in [22] is obtained by proving that C lies in the image of this pull-back map, and indeed that it lies in the pull-back of the vortex moduli space, which, according to [11] (see also [2]), is in turn identified with the space of divisors, by looking at the zeroset of the section. The key points we need presently are that C lies in π^{∗}(B(Σ, E)), and that configurations are the pull-backs of configurations [A, Φ] ∈ B(Σ, E), where Φ is \overline{∂}_{A}-holomorphic section.

Over B(Σ, E), there is a universal line bundle L(Σ), defined in the usual manner. Note that

L(Y)|_{π^{∗}(B(Σ))×Y} ≅ π^{∗}(L(Σ)),

so μ(y)|_{π^{∗}(B(Σ))} for y ∈ H_{∗}(Y) agrees with μ([π_{∗}(y)]), where the former μ-map is induced from L(Y), and the latter from L(Σ). We have thus reduced the proof of the proposition to a statement purely over Σ; so for the duration of the proof, L will refer to L(Σ), B will refer to B(Σ, E), and all μ-maps will be calculated over Σ.

To facilitate the proof over Σ, we pause for a discussion about the canonical section σ of the universal line bundle L, which takes the configuration [A, Φ] × {x} ∈ B × Σ to the based configuration [A, Φ, Φ(x)]. This section has the property that, under the canonical identification of L|_{[A,Φ]×Σ} ≅ E (where E is the bundle over Σ with Chern number k), the restriction σ|_{[A,Φ]×Σ} is identified with the section Φ of E. In particular, if Φ ≢ 0 is a holomorphic section, then σ|_{[A,Φ]×Σ} has at most k zeros; moreover, if it has k zeros, then each is transverse.

Now, if y ∈ Σ is a point (i.e., a generator of H_{0}(Σ; ℤ)), then by definition, μ(y) is the element of H^{2}(C) whose pairing against any homology class [S] ∈ H_{2}(C) is given by

⟨μ(y), [S]⟩ = ⟨c_{1}(L_{y}), [S]⟩.

where, as usual, L_{y} denotes the restriction of L to B(Σ, E) × {y}. Choose points {p_{2}, …, p_{k}} ⊂ Σ which are distinct from y. Recall that H_{2}(Sym^{k}(Σ)) is generated by the surface q(Σ × p_{2} × ⋯ × p_{k}) (where p_{2}, …, p_{k} are points on Σ), and the tori of the form q(C_{1} × C_{2} × p_{3} × ⋯ × p_{k}), where C_{1}, C_{2} ⊂ Σ are closed curves in Σ, which we can choose to miss y. The canonical section σ restricted to a torus of the form q(C_{1} × C_{2} × p_{3} × ⋯ × p_{k}) × {y} clearly vanishes nowhere (as all k of the zeros have been constrained to lie in the set C_{1} ∪ C_{2} ∪ {p_{3}, …, p_{k}} which does not include the point y); thus,

⟨c_{1}(L_{y}), [q(C_{1} × C_{2} × ⋯ × p_{k})]⟩ = 0.

Over q(Σ × p_{2}, ⋯ × p_{k}) the canonical section vanishes at the single point q(y×p_{2}×⋯×p_{k}). We verify transversality of this zero, as follows. View σ as a section over Σ × Σ = q(Σ × p_{2} × ⋯ × p_{k}) × Σ; we know that σ(y, y) ≡ 0, and that Dσ_{(y,y)} induces an isomorphism from 0 ⊕ T_{y}Σ to E_{y} (i.e., that the zero of σ|_{{y}×Σ} at y is transverse). Differentiating the equation that σ(y, y) ≡ 0, we see that

Dσ_{(y,y)}(0, v) = −Dσ_{(y,y)}(v, 0).

Thus, the section σ|_{Σ×{y}} of L_{y}|_{[Σ]} has a single, transverse zero, which shows that

⟨c_{1}(L), [q(Σ × p_{2} × ⋯ × p_{k})]⟩ = ±1.

Moreover, the sign is positive since the section is holomorphic.

Hence, we have proved the result when y ∈ H_{0}(Σ). Proving the result for classes coming from H_{1}(Σ) amounts to proving that, if C_{1} and C_{2} closed curves in Σ which meet transversally, then

⟨c_{1}(L), q(C_{1} × p_{2} × ⋯ × p_{n}) × C_{2}⟩ = −#C_{1} ∩ C_{2} = #C_{2} ∩ C_{1}.

Note first that the zeros of the canonical section σ, restricted to q(C_{1} ×p_{2} ×⋯×p_{k})×C_{2} are the points C_{1} ∩C_{2} (a zero of σ corresponds to a point where the section Φ vanishes at some point of C_{2}, but the zeros of Φ lie in C_{1} ∪ {p_{2}, …, p_{k}}, and {p_{2}, …, p_{k}} ∩ C_{2} is empty). We must now consider the local contribution of each zero (and check transversality).

Consider the map C_{1} × C_{2} : S^{1} × S^{1} ⟶ Sym^{k}(Σ) × Σ defined by C_{1} × C_{2}(s, t) = q(C_{1}(s) × p_{2} × ⋯ × p_{k}) × C_{2}(t). Suppose for notational simplicity that C_{1}(0) = C_{2}(0) = y. We can view σ as a section of L pulled back to this torus. Now, evaluated on a typical tangent vector to the torus a\frac{∂}{∂s} + b\frac{∂}{∂t}, the derivative of σ at the intersection point is given by

\begin{aligned}D_{(y,y)}σ ∘ (C_{1} × C_{2})(a\frac{∂}{∂s} + b\frac{∂}{∂t}) = & aD_{(y,y)}σ(\frac{dC_{1}}{ds} (0), 0) \\ & + bD_{(y,y)}σ(0, \frac{dC_{2}}{dt} (0)) \\ = & − aD_{(y,y)}σ(0, \frac{dC_{1}}{ds} (0)) \\ & + bD_{(y,y)}σ(0, \frac{dC_{2}}{dt} (0)). \end{aligned} (20)

(We have used the chain rule and the derivative of the relation that σ(C_{1}(s), C_{1}(s)) ≡ 0.) Transversality of the intersection of C_{1} and C_{2} at 0 ensures that the image of this differential is

D_{(y,y)}σ(0 ⊕ T Σ_{y});

so transversality of the section corresponding to q(y × p_{2} × ⋯ × p_{k}) at its zero y ensures that the image of the differential surjective onto the fiber of E over y; i.e., the canonical section is transverse. The sign is correct, as one can see by inspecting Equation (20). q.e.d.

Remark 6.11. With the help of the above results, we can describe explicitly the invariant of the tube:

ŜW_{(C,J)} : H^{∗}(C) ⟶ H^{∗}(J) ⊂ 𝔸(X_{0}),

which we do now for completeness. Let Λ = Λ^{∗}H_{1}(Σ) ⊂ 𝔸(Y). According to the proof of Lemma 6.7, ŜW_{(C,J)} is a homomorphism of Λ-modules; so, since 𝔸(Y) = Λ[U] surjects onto H^{∗}(C), the invariant is determined by ŜW_{(C,J)}(U^{i}), as i ranges over the non-negative integers. Since the Poincaré dual of Sym^{k}(Σ) ⊂ T^{2g} (which is the image of M̂(C, J) under ρ_{J}, according to Theorem 4.1), is

\frac{(∑_{i=1}^{g} μ(A_{i})μ(B_{i}))^{k}}{k!} ,

it follows that

ŜW_{(C,J)}(U^{ℓ}) = \frac{(∑_{i=1}^{g} A_{i} · B_{i})^{k+ℓ}}{(k + ℓ)!} .

We will not use this formula, however. The results we prove in this paper require only the general properties of ŜW_{(C,J)} which follow from its definition, together with Proposition 6.9.

According to Lemma 6.7, if γ ⊂ Y is a curve which is null-homolo-gous in X_{0}, then it annihilates the relative invariants, in the sense that

SW_{(X_{0},C)}(a ⊗ μ(γ) · ω) = 0.

If sufficiently many curves in Y become null-homologous in X_{0}, then any class of sufficiently high degree in 𝔸(Y) annihilates the invariant, as follows.

Proposition 6.12. Fix natural numbers k, ℓ with ℓ ≥ k. Let I denote the ideal generated by μ(A_{1}), μ(A_{2}), …, μ(A_{ℓ}) in H^{∗}(Sym^{k}(Σ)). Then every element of H^{∗}(Sym^{k}(Σ)) of degree greater than k lies in I.

Proof. The vector space H^{∗}(Sym^{k}(Σ)) is generated by homogeneous elements of the form

U^{a} · ∏_{q=1}^{b}(A_{i_{q}} · B_{i_{q}}) · ∏_{r=1}^{c} A_{i_{b+r}} · ∏_{s=1}^{d} B_{i_{b+c+s}},

where {i_{1}, …, i_{b+c+d}} is a subset of {1, …, g}, and a, b, c, d are integers with a + b + c + d ≤ k.

Clearly, it suffices to prove the proposition for homogeneous generators of degree k + 1. Modulo I, such an element is equivalent to the element

∏_{p=1}^{a}(U − A_{p} · B_{p}) · ∏_{q=1}^{b}(A_{i_{q}} · B_{i_{q}} − A_{a+q} · B_{a+q}) · ∏_{r=1}^{c} A_{i_{b+r}} · ∏_{s=1}^{d} B_{i_{b+c+s}}.

Indeed, in light of the fact that a + b ≤ k − d ≤ ℓ − d, we can arrange (after possibly simultaneously permuting the indices of the {A_{i}}_{i=1}^{g} and {B_{i}}_{i=1}^{g}) that for each s = 1, … d, a+b < i_{b+c+s}. Moreover, the original homogeneous element would automatically lie in I unless we had that a + b < k ≤ ℓ < i_{b+r} for all r = 1, …, c. Put together, must consider elements of the above form which satisfy the constraint that a + b < i_{j} for all j > b. If the degree of such an element is k + 1, it must vanish in H^{∗}(Sym^{k}(Σ)).

This vanishing can be seen geometrically: U is Poincaré dual to the subset (identified with Sym^{k−1}(Σ)) of Sym^{k}(Σ) where one point is constrained to lie in a specified point on Σ: A_{i} (resp. B_{i}) is Poincaré dual to the cycle where one point is constrained to lie on A_{i} (resp. B_{i}). Thus, (if one chooses the point representing U to be A_{i} ∩ B_{i}), then U − A_{i} · B_{i} is Poincaré dual to the locus where two distinct points are constrained; one is to lie on A_{i}, the other on B_{i}. Similarly, the manifold Poincaré dual to A_{i}·B_{i}−A_{j} ·B_{j} gives a constraint on two distinct points in the symmetric power. Finally, the remaining A_{i_{b+r}} and B_{i_{b+c+s}} give additional, disjoint constraints (these are disjoint, if one chooses that representing curves to be disjoint from the A_{i} and B_{i} for i = 1, …, a+b, which can be arranged since a + b < i_{b+r} for all r ≥ 1). Thus, since the total degree of the expression considered is k+1, we have put constraints on k + 1 distinct points, forcing the intersection to be empty. q.e.d.

Corollary 6.13. Suppose that C = Sym^{k}(Σ), and let ℓ ≥ k be an integer so that there is a symplectic basis {A_{i}, B_{i}}_{i=1}^{g} for H_{1}(Σ) so that i_{∗}(A_{i}) = 0 in H_{1}(X_{0}; ℝ) for i = 1, …, ℓ. Then, for each b ∈ 𝔸(Σ) of degree d(b) > k, and each a ∈ 𝔸(X_{0}), ω ∈ H^{∗}(C), we have

SW_{(X_{0},C)}(a ⊗ b · ω) ≡ 0.

Proof. By Proposition 6.12, b lies in the ideal generated by μ(A_{1}), …, μ(A_{ℓ}). Now the proposition follows from Lemma 6.7. q.e.d.

We now have the promised proof of Proposition 3.8.

Proof of Proposition 3.8. Recall that we have constructed SW_{(X_{0},C)} so that

SW_{𝔰}^{irr}(a · i_{∗}(b)) = SW_{(X_{0},C)}(a · i_{∗}(b) ⊗ 1).

By Lemma 6.7 and Proposition 6.9, we can write

SW_{(X_{0},C)}(a · i_{∗}(b) ⊗ 1) = SW_{(X_{0},C)}(a ⊗ b) + SW_{(X_{0},J)}(a · c).

for some c ∈ 𝔸(Σ). Note that c lies in the ideal generated by H_{1}(Σ), as it can be expressed in terms of Seiberg-Witten invariants of the tube, which take values in H^{∗}(J) ≅ Λ^{∗}(H_{1}(Σ)) ⊂ H_{1}(Σ) · 𝔸(Σ). By Corollary 6.13, the first term vanishes (using the homological hypothesis of the inclusion of Σ in X). The remaining term is identified with an absolute invariant, according to Proposition 5.9. q.e.d.

The proof of Proposition 3.10, follows from the same argument as Proposition 3.8; only in that case, one must use the following (much simpler) analogue of Corollary 6.13.

Lemma 6.14. Suppose that C = Sym^{k}(Σ). Then, for each b ∈ 𝔸(Σ) of degree d(b) > 2k, and each a ∈ 𝔸(X_{0}), ω ∈ H^{∗}(C), we have

SW_{(X_{0},C)}(a ⊗ b · ω) ≡ 0.

Proof. This follows immediately from the fact that dim C = 2k. q.e.d.

7. The moduli spaces over N

The purpose of this section is to give the results about the neighborhood of Σ which were used in Section 5. Most of these results are applications of [22] and [24]. We assume for the duration of this section that the Spin_{ℂ} structure over N satisfies ⟨c_{1}(𝔰), [Σ]⟩ ≢ n (mod 2n). We return to the excluded cases in Section 8.

Over N, endowed with a cylindrical-end metric and a certain torsion connection on T N , the Seiberg-Witten equations admit a complex interpretation analogous to the complex interpretation of the equations over a Kähler manifold (see Section 5 of [24] for an explicit description of this connection, and especially Proposition 5.6 where the complex interpretation is proved). The Seiberg-Witten equations over N can be written as equations for a connection A over E, α ⊕β ∈ (Ω^{0,0} ⊕Ω^{0,1})(N, E):

(21)  2ΛF_{A} − ΛF_{K_{N}} = \frac{i}{2}(|α|^{2} − |β|^{2})
(22)  TrF_{A}^{0,2} = \overline{α} ⊗ β
(23)  \overline{∂}_{A}α + \overline{∂}_{A}^{∗}β = 0,

where Λ denotes projection onto the (1, 1) form of the metric. As noted in [24], for finite energy solutions, decay estimates justify the usual integration-by-parts which shows that one of α or β must vanish identically; i.e., the solutions over N correspond to vortices over N.

When β ≡ 0, then A induces an integrable \overline{∂}-operator on E, \overline{∂}_{A}, with respect to which α is holomorphic. Moreover, by the usual exponential decay results, together with the understanding of the solutions over Y (Theorem 4.1), (A, α) exponentially approaches the pull-back of a vortex solution over Σ. According to [22], the underlying holomorphic data extends to the ruled surface R obtained by attaching a copy of Σ (denoted Σ_{+}) to N “at infinity.” We state the results here for convenience.

Definition 7.1. Let Φ ∈ Γ(N, W^{+}), Ψ ∈ Γ(Y, W) be a pair of spinors, and δ > 0 be some real number. Then, Ψ is said to δ-decay to Ψ if for each k ≥ 0,

lim_{t↦∞} sup_{{t}×Y} e^{δt}|∇^{(k)}Ψ − ∇^{(k)}π^{∗}(Ψ)| = 0,

where ∇^{(k)} denotes the k-fold covariant derivative. More generally, Φ is said to decay to Ψ if there is some δ > 0 so that Ψ δ-decays to Ψ. A similar notion can be defined for objects other than spinors, such as connections, differential forms, etc.

Definition 7.2. Given a line bundle E over Z, a holomorphic pair (A, α) in E is a pair consisting of a \overline{∂}-operator \overline{∂}_{A} over E, and a section α of E, so that F_{A}^{0,2} = 0, and \overline{∂}_{A}α = 0.

Theorem 7.3. Let (A, α) be a holomorphic pair on N which decays to a the pull-back of a holomorphic pair (A_{0}, α_{0}) over Σ. Then, there is a naturally associated line bundle Ê over R and holomorphic pair (Â, α̂) in Ê, so that (\overline{∂}Â, α̂)|_{R−Σ_{+}} ≅ (\overline{∂}_{A}, α̂) and (\overline{∂}Â, α̂)|_{Σ_{+}} ≅ (\overline{∂}_{A_{0}}, α_{0}).

The above theorem is essentially a restatement of Theorem 7.7 of [22], where it is stated for the cylinder, thought of as R minus two copies of Σ, rather than the neighborhood of Σ, thought of as R minus one copy of Σ (though the proof is no different). Analogous results for the antiself-dual equations were obtained by Guo [9].

In a similar vein we have the following result, which allows us to deal with solutions with reducible boundary values. We state the result slightly differently from the above, since we will apply it in other contexts later.

Theorem 7.4. Let A be a connection on a line bundle E over N, with F_{A}^{0,2} = 0 and E|_{(0,∞)×Y} ≅ π^{∗}(E_{0}), so curvature form F_{A} decays to the pull-back of a closed two-form F_{0} over Σ with

(\frac{i}{2π} ∫_{Σ} F_{0}) − ⟨c_{1}(E_{0}), [Σ]⟩ ∉ nℤ.

Then, there is an associated line bundle Ê over R and complex structure \overline{∂}Â with

(Ê, \overline{∂}Â)|_{R−Σ_{+}} ≅ (E, \overline{∂}_{A}),

and ⟨c_{1}(Ê), [Σ_{+}]⟩ is the greatest integer congruent to ⟨c_{1}(E), [Σ]⟩ moduli n smaller than \frac{i}{2π} ∫ F_{0}. Furthermore, there is a natural identification

KerD̸_{A} ∩ L^{2} ≅ H^{0}(R, Ê) ⊕ H^{2}(R, Ê),

and

CokerD̸_{A} ∩ L^{2} ≅ H^{1}(R, Ê).

The above is proved in Proposition 9.2 (see Corollary 9.11 and Theorem 10.6) of [22].

These results allow us to rule out the existence of certain solutions. Recall first the following standard fact about the cohomology of R (see for example [10]):

Proposition 7.5. Let R denote the ruled surface over Σ, which is given as the projectivization of ℂ ⊕ L, ℙ(ℂ ⊕ L) (here, L is some line bundle over Σ). Let Ê be a line bundle over the ruled surface R and let E_{0} denote the restriction of of Ê to ℙ(ℂ ⊕ 0) ≅ Σ and let ℓ be the evaluation of c_{1}(Ê) on a fiber in the ruling. Then, if ℓ ≥ 0,

H^{0}(R, Ê) ≅ ∑_{j=0}^{ℓ} H^{0}(Σ, E_{0} ⊗ L^{⊗j}); H^{1}(R, Ê) ≅ ∑_{j=0}^{ℓ} H^{1}(Σ, E_{0} ⊗ L^{⊗j});

H^{2}(R, Ê) = 0;

and if ℓ < 0,

H^{0}(R, Ê) = 0; H^{1}(R, Ê) ≅ ∑_{j=1}^{−ℓ−1} H^{0}(Σ, E_{0} ⊗ L^{⊗−j});

H^{2}(R, Ê) ≅ ∑_{j=1}^{−ℓ−1} H^{1}(Σ, E_{0} ⊗ L^{⊗−j}).

In particular, if ℓ = −1, then H^{∗}(R, Ê) = 0.

We can apply these results to the case where N is a neighborhood of a surface of self-intersection number with −Σ · Σ > 2g − 2. Recall that according to Theorem 4.1, for each Spin_{ℂ} structure 𝔰 on N, there are at most two components to the moduli space of the boundary, the reducible component J and the irreducible component C.

Proposition 7.6. If

|⟨c_{1}(𝔰), [Σ]⟩| < n,

the moduli space M_{N}(J) contains only reducibles. Moreover, the space of reducibles is smoothly identified with the Jacobian torus J (i.e., the kernel and the cokernel of the Dirac operator coupled to any reducible vanishes). Furthermore, M_{N}(C) is empty.

Proof. We prove that both moduli spaces M_{N}^{∗}(J) and M_{N}(C) are empty. Suppose there were some finite energy solution to the Seiberg-Witten equations in a Spin_{ℂ} structure with |⟨c_{1}(𝔰), [Σ]⟩| < n. We know that the spinor lies entirely in one of the two summands in the splitting of the spinor bundle W^{+} ≅ E⊕(K_{N}^{−1}⊗E) (i.e., it is an α- or a β-spinor, in the notation of Equations (21)-(23)). By conjugating if necessary (which switches the two summands and sends the Spin_{ℂ} structure 𝔰 to another one J𝔰 with c_{1}(J𝔰) = −c_{1}(𝔰)), we can assume without loss of generality that the solution is an α-solution.

According to Theorem 7.3 (and Theorem 7.4, when the boundary value is reducible), we can extend the data (E, \overline{∂}_{A}, α) over the associated ruled surface R, obtained by attaching the curve Σ_{+} at infinity. The fact that Ê is an extension of E says that

⟨c_{1}(Ê), [Σ_{−}]⟩ = ⟨c_{1}(E), [Σ]⟩
= \frac{1}{2}⟨c_{1}(𝔰) + c_{1}(K_{N}), [Σ]⟩
= g − 1 + \frac{n + ⟨c_{1}(𝔰), [Σ]⟩}{2}

where Σ_{−} is the curve in Ê with self-intersection number −n (which is identified with Σ ⊂ N). By our hypothesis, then,

g − 1 < ⟨c_{1}(Ê), [Σ_{−}]⟩ < n + g − 1.

On the other hand, Equation (21) says that \frac{i}{2π} F_{A} converges to the pullback of a form over Σ whose integral is g −1, so Theorem 7.4 guarantees that the Chern number of restriction to the other section of the ruling satisfies the bound

−n + g − 1 < ⟨c_{1}(Ê), [Σ_{+}]⟩ < g − 1.

Now, since the Poincaré dual of a fiber is (PD[Σ_{+}] − PD[Σ_{−}])/n, we see that the evaluation of c_{1}(Ê) on a fiber is given by

ℓ = \frac{⟨c_{1}(Ê), [Σ_{+}]⟩ − ⟨c_{1}(Ê), [Σ_{−}]⟩}{n} = −1.

According to Proposition 7.5, it follows that α̂ (and hence also α) must vanish identically, contradicting the irreducibility hypothesis on (A, α).

The fact that the reducibles are smoothly cut out in this range follows in an analogous manner, using Theorem 7.4 and Proposition 7.5.

q.e.d.

Remark 7.7. Most of this result can be found in Proposition 2.5 of [24].

The above vanishing result is special to the particular Spin_{ℂ} structures considered, as it used the fact that the Dolbeault cohomology of certain line bundles over the ruled surface vanish. In general, the moduli spaces over N typically do contain irreducibles. To study the deformation theory around these irreducibles, we use an infinitesimal version of Theorem 7.3; but first, we pause for a brief discussion of deformation theory for the Seiberg-Witten equations in general.

In general, on a four-manifold X_{0} with a cylindrical end, the deformation complex around a solution (A, Φ) whose boundary value is smooth and irreducible, is given by

Ω^{0}(X_{0}, iℝ) ⟶ Ω^{1}(X_{0}, iℝ) ⊕ Γ(X_{0}, W^{+}) ⟶ Ω^{+}(X_{0}, iℝ) ⊕ Γ(X_{0}, W^{−}).

Here, terms in Ω^{0}(X_{0}, iℝ) are required to lie in L_{δ,k}^{2}, the δ-decaying Sobolev space with k derivatives (here we can choose any k ≥ 3); i.e., functions for which

(∥f∥_{δ,k})^{2} = ∫_{X_{0}}(|f|^{2} + |∇f|^{2} + ⋯ + |∇^{(k)}f|^{2})e^{δτ} < ∞,

where τ is a smooth function on X_{0} which agrees with the t coordinate over the cylindrical end. Terms in Ω^{1}(X_{0}, iℝ)⊕Γ(X_{0}, W^{+}) are required to lie in L_{δ,k−1}^{2} extended by the tangent space to the moduli space at infinity at ρ[A, Φ]. Finally, terms in Ω^{+}(X_{0}, iℝ) ⊕ Γ(X_{0}, W^{−}) are required to lie in L_{k−2}^{2}. The first map in the deformation complex is the linearization of the gauge group action on [A, Φ] around the identity, while the second is the linearization of the Seiberg Witten equations around [A, Φ]. When the boundary value of [A, Φ] is a smooth reducible, then the above specifies the deformation theory for the moduli space based at infinity. In either case, the moduli space of solutions about [A, Φ] is transversally cut out by the Seiberg-Witten equations on X_{0} if the H^{2} of the above complex vanishes. (This discussion is modeled on the theory developed in [20].)

The space of divisors in a compact, complex surface X admits a deformation theory, defined as follows. Consider the pair (\overline{∂}Â, α̂) where \overline{∂}Â is an integrable \overline{∂}-operator, and α̂ is \overline{∂}Â-holomorphic; i.e.,

F 0,2 ; Â = 0
\overline{∂}Âα = 0.

This has a deformation complex

Ω^{0,0} ⟶ Ω^{0,1} ⊕ Ω^{0,0}(E) ⟶ Ω^{0,2} ⊕ Ω^{0,1}(E) ⟶ Ω^{0,2},

whose cohomology groups are identified with the cohomology groups of the quotient sheaf E/α̂, obtained from the short exact sequence of sheaves:

0 ⟶ O_{X} \underrightarrow{α̂} E ⟶ E/α̂ ⟶ 0.

Theorem 7.8. Let (A, α) correspond to a solution to the Seiberg-Witten equations over N, with irreducible boundary values. Then, the cohomology groups of the deformation complex of the Seiberg-Witten deformation complex are naturally isomorphic to the cohomology groups deformation complex of the divisor [\overline{∂}Â, α̂] in the line bundle Ê over R, provided by Theorem 7.3. When (A, α) has a reducible boundary value, then H^{2} of the Seiberg-Witten deformation complex is identified with H^{1}(R, E/α̂), while the tangent space of the based moduli space is identified with ℂ ⊕ H^{0}(R, E/α̂).

Proof. This follows exactly as in Theorem 9.14 (for irreducible boundary values) and Theorem 10.12 (for reducible boundary values) of [22]. The key observation at this point is to note that

Λ∂\overline{∂} + |α|^{2} : L_{δ,k}^{2} ⟶ L_{δ,k−2}^{2}

is an isomorphism, which allows one to “unroll” parts of the Seiberg-Witten deformation complex to identify it with the deformation theory of divisors in N. As in [22] (see also [13]), we can identify Λ∂\overline{∂} with the operator over the cylindrical end with

−e^{−2λt} \frac{∂}{∂t}e^{2λt} \frac{∂}{∂t} + Δ_{Y} ,

where λ = \frac{πn}{Vol(Σ)}. According to the theory of [16], the operator

Λ∂\overline{∂}: L_{k,δ}^{2} ⟶ L_{k−2,δ}^{2}

is Fredholm for all weights 0 < δ < 4λ. In particular, it has the same index for all small 0 < δ as it has on the weight δ = 2λ, where it can be connected via Fredholm operators to the manifestly self-adjoint operator

d^{∗_{λ}}d: L_{k,λ}^{2} ⟶ L_{k−2,λ}^{2},

where d^{∗_{λ}} denotes the formal λ-weighted adjoint of d. It follows from the homotopy invariance of the index that Λ∂\overline{∂} + |α|^{2} has index zero on L_{k,δ}^{2}. From the maximum principle, it has no kernel, so it induces an isomorphism as claimed, identifying the deformation theory of the Seiberg-Witten equations with the deformation theory of divisors in N. Passing to the ruled surface then follows from Corollary 9.4 of [22].

q.e.d.

Proposition 7.9. Let N be a disk bundle over a surface Σ with

Σ · Σ = −n < 2 − 2g,

endowed with a Spin_{ℂ} structure 𝔰 with

n < |⟨c_{1}(𝔰), [Σ]⟩| ≤ n + 2g − 2.

Let

e = \frac{n + 2g − 2 − |⟨c_{1}(𝔰), [Σ]⟩|}{2}.

Then, the expected dimensions of the moduli spaces over N and X_{0} are given by:

(24)  e-dimM_{N}(J) = 2e + 1
(25)  e-dimM_{N}(C) = 2e.

Moreover, M_{N}^{∗}(J), M_{N}(C), are transversally cut out by the Seiberg-Witten equations (in particular, they are manifolds of the expected dimension). Furthermore, the boundary map

ρ: M_{N}(C) ⟶ C

is an orientation-preserving diffeomorphism onto its image.

Proof. The deformation theory around a solution [A, α] ∈ M_{N}(C) is identified the deformation theory around a corresponding divisor in the line bundle Ê with ⟨c_{1}(Ê), [Σ_{±}]⟩ = e; i.e., with divisors in a line bundle which (topologically) pulls back from Σ. According to Proposition 7.5, all such divisors actually pull back from the base Σ; and indeed, the deformation theory corresponds to deformation theory of degree e divisors in the base Σ, which is unobstructed. Thus, M_{N}(C) is a manifold of real dimension 2e, transversally cut out by the Seiberg-Witten equations.

The above transversality applies to M_{N}(J) as well, except that the expected dimension is greater by one, as we saw in Theorem 7.8.

This identification of deformation theories of M_{N}(C) proves that ρ is an orientation-preserving local diffeomorphism onto its image in Sym^{e}(Σ) ≅ C. In fact, it is injective, as follows. As we saw, any two solutions with the same boundary values actually vanish over the same disks (with the same multiplicities). By the usual analysis of the vortex equations, any two such solutions must differ by a complex gauge transformation; i.e., a function u which satisfies

Λ\overline{∂}∂u + |α|^{2}(e^{2u} − 1) = 0,

where u is a function which decays on the cylinder. By the maximum principle, such a function must vanish identically. q.e.d.

Having analyzed the moduli spaces over neighborhoods of Σ, we close with a some general results concerning the rest of the moduli spaces of the complement of Σ.

Proposition 7.10. Let X_{0} be as in Proposition 5.1. Then, letting e-dimM(X) = d, we have

(26)  e-dimM_{X_{0}}(J) = d + 2g − 2e − 2
(27)  e-dimM_{X_{0}}(C) = d.

Moreover, M_{X_{0}}(J), and M_{X_{0}}(C) are transversally cut out by the Seiberg-Witten equations (in particular, they are manifolds of the expected dimension).

Proof. By a standard excision argument, we have

e-dimM_{X_{0}}(J) + e-dimM_{N}(J) − 2g + 1 = e-dimM_{X}(𝔰) = d,

which calculates e-dimM_{X_{0}}(J), given Proposition 7.9. Similarly, we have

e-dimM_{X_{0}}(C) + e-dimM_{N}(C) − 2e = d,

which gives us e-dimM_{X_{0}}(C).

Smoothness of M_{X_{0}}(J) and M_{X_{0}}(C) follows from adapting methods of [20]. q.e.d.

8. Perturbations when e = g − 1

In our earlier discussion, we had to exclude one Spin_{ℂ} structure over Y. In this section, we introduce a perturbation of the equations which allows us to handle this case. We begin by adapting results of Section 4 to this perturbed equation, and then, we give a discussion which is parallel to that of Section 7. The perturbations used here are analogues of Taubes’ perturbations in the symplectic category [29], [30]; see also [15] for a related discussion.

Recall that the Seiberg-Witten equations over Y are obtained as the critical points of the Chern-Simons-Dirac functional CSD defined over the configuration space

B(Y, W) = A(W) ⊕ Γ(Y, W)/Map(Y, S^{1}),

where A(Y, W) denotes the space of connection in the spinor bundle W which are compatible with some fixed connection ∇ on T Y. The functional is defined by

(28) CSD(B, Ψ) = ∫_{Y} (B − B_{0}) ∧ Tr(F_{B} + F_{B_{0}}) − ∫_{Y} ⟨Ψ,D̸_{B}Ψ⟩,

where B_{0} ∈ A(Y, W) is some reference connection, B − B_{0} ∈ Ω^{1}(Y; iℝ) denotes the difference 1-form, and Tr denotes the trace of the corresponding connection on W. Its Euler-Lagrange equations (the threedimensional Seiberg-Witten equations) are

∗Tr(F_{B}) − iτ(Ψ) = 0 (29) D̸_{B}Ψ = 0, (30) where

τ: Γ(Y, W) → Ω^{1}(Y; ℝ)

is adjoint to Clifford multiplication, in the sense that for all γ ∈ Ω^{1}(Y; ℝ), Ψ ∈ Γ(Y, W), we have

\frac{1}{2}⟨iγ · Ψ, Ψ⟩_{W} = −⟨γ, τ(Ψ)⟩_{Λ^{1}}. (31)

Moreover, its upward gradient flow equations are the usual Seiberg-Witten equations on the four-manifold ℝ × Y.

When Y is a circle-bundle over a Riemann surface with Euler number −n satisfying

n > 2g − 2,

recall that these equations are inconvenient in the Spin_{ℂ} structure when e = g−1 (in the notation of Section 4). We will find it useful to consider a perturbed functional

CSD_{u}: B(Y, 𝔱) ⟶ ℝ,

where u ∈ ℝ, given by

(32) CSD_{u}(B, Ψ) = CSD(B, Ψ) + u ∫_{Y} iη ∧ (TrF_{B} − TrF_{B_{0}}),

where η is the connection form for Y over Σ, and the reference connection B_{0} satisfies Tr(F_{B_{0}}) ≡ 0 (i.e., B_{0} ∈ J). These give rise to perturbed Seiberg-Witten equations of the form

∗Tr(F_{B}) − iτ(Ψ) + iu(∗dη) = 0 (33) D̸_{B}Ψ = 0, (34)

whose moduli space of solutions is denoted N_{u}(Y). The gradient flow equations of the perturbed functional are solutions to the Seiberg-Witten equations on ℝ × Y, perturbed by the self-dual component of iu(dη), which can be collected into moduli spaces, denoted M_{u}(C_{1}, C_{2}), or their unparameterized versions

M̂_{u}(C_{1}, C_{2}) = M_{u}(C_{1}, C_{2})/ℝ.

We have the following analogue of Theorem 4.1.

Theorem 8.1. Let Y be a circle-bundle over a Riemann surface with genus g > 0 and Euler number −n < 2 − 2g. Let 𝔱 be the Spin_{ℂ} structure corresponding to g − 1 ∈ ℤ/nℤ ⊂ H^{2}(Y; ℤ). For all u with 0 < u < 2, the moduli space contains two components, a reducible one J, identified with the Jacobian torus H^{1}(Σ; ℝ/ℤ), and a smooth irreducible component C diffeomorphic to Sym^{g−1}(Σ). Both of these components are non-degenerate in the sense of Morse-Bott. There is an inequality CSD_{u}(J) > CSD_{u}(C), so the space M_{u}(J , C) is empty. The space M̂_{u}(C, J) is smooth of expected dimension 2g − 2; indeed it is diffeomorphic to Sym^{g−1}(Σ).

Proof. Most of this is a straightforward adaptation of [22].

We begin with the identification of the moduli spaces over Y. As in [22], the equations over Y reduce to vortex equations over Σ. More specifically, the components of the moduli spaces N_{Y} (𝔱) correspond to line bundles E_{0} over Σ with the property that

π^{∗}(E_{0} ⊕ K^{−1} ⊗ E_{0}) ≅ W,

the spinor bundle of 𝔱 (here K denotes the canonical line bundle over Σ). The vortex equations are are equations for B ∈ A(E_{0}), α ⊕ β ∈ Γ(Σ, E_{0} ⊕ K^{−1} ⊗ E_{0}), which, in the case at hand, take the form

(35)  2F_{B} − F_{K} + iu(dη) = i(|α|^{2} − |β|^{2})(∗1)
(36)  \overline{∂}_{B}α + \overline{∂}_{B}^{∗}β = 0
(37)  α ⊗ β = 0.

Thus, one of α or β must vanish. In fact, in our case,

deg E_{0} ≡ g − 1 (mod n).

In fact, if

deg E_{0} ≠ g − 1,

then the solution space to these equations (0 < u < 2) is empty. More specifically, letting deg E_{0} = g − 1 + nℓ, we see that when β ≢ 0, then by integrating Equation (35) over Σ against i/2π, we get

2(g − 1 + nℓ) − (2g − 2) + u deg Y = 2nℓ − un ≥ 0,

which forces ℓ ≥ 1 (since u > 0). Since in this case deg(E_{0}) > g − 1, H^{1}(Σ, E_{0}) = 0, so β must vanish. If, on the other other hand, it is α ≢ 0, then we obtain in the same manner that

2nℓ − un ≤ 0,

which forces ℓ ≤ 0 (since u < 2). Since α represents a class in H^{0}(Σ, E_{0}), it follows that ℓ = 0.

So, all irreducibles correspond to α-vortices in the line bundle E_{0} with deg E_{0} = g − 1. The identification of this space of vortices with the symmetric product follows from [2] (see also [11]).

Non-degeneracy of the irreducible manifold C follows exactly as in [22]. To see non-degeneracy of J, we appeal to results of Section 5.8 of [22]. Consider the Dirac operator on the Spin_{ℂ} structure with spinors W = E ⊗ (ℂ ⊕ K^{−1}) with connection induced from a connection B ∈ A(E) whose curvature pulls up from Σ. It is shown in Proposition 5.8.4 of [22] that this Dirac operator admits no harmonic spinors unless the holonomy around a fiber circle in Y is trivial. In fact this holonomy is trivial when the following integral is congruent to g − 1 modulo nℤ:

\frac{i}{4π^{2}} ∫_{Y} F_{B} ∧ η = g − 1 − \frac{u deg(Y)}{2},

(we have used here Equation (33)). Since 0 < u < 2, this holonomy is non-trivial, so the reducibles admit no harmonic spinors, i.e., J is smoothly cut out by the equations.

We now perform the Chern-Simons calculations (see the proof of Proposition 5.23 of [22]). Suppose [(B_{1}, Ψ_{1})] ∈ C, and [(B_{0}, 0)] ∈ J. Then, we have

2 deg B_{0} − deg K + u deg(Y) = 0;
2 deg B_{1} − deg K = 0,

where by deg B, we mean the integral \frac{i}{4π^{2}} ∫_{Y} F_{B} ∧ η, which when B is induced from a line bundle over Σ, agrees with the degree of that line bundle. So,

CSD_{u}(B_{1}) = ∫_{Y} (B_{1} − B_{0}) ∧ (2F_{B_{1}} + 2F_{B_{0}} − 2F_{K})
+u ∫ iη ∧ (2F_{B_{1}} − 2F_{B_{0}})
= \frac{8π^{2}}{deg Y} (deg B_{1} − deg B_{0})(deg B_{1} + deg B_{0} − deg K)
+u ∫ iη ∧ (2F_{B_{1}} − 2F_{B_{0}})
= 2π^{2}u^{2} deg Y,

which is negative; while CSD_{u}(B_{0}) = 0.

The smoothness of the space of flows, and its identification with the symmetric product, follows exactly as in the unperturbed case (see Section 4). q.e.d.

We now turn to the neighborhood of Σ. We use a perturbation over N which is compatible with the above perturbation over Y. Specifically, let

f: N ⟶ ℝ

be a smooth function which is identically zero on the complement of the cylinder [0, ∞) × Y ⊂ N, and identically one on the subcylinder [1, ∞) × Y. We consider the Seiberg-Witten equations perturbed by the self-dual part of iuf(dη). Note that this perturbing two-form is iuλf times the (1, 1) form of the standard cylindrical-end metric on N (see [24]), where

λ = −\frac{2π deg Y}{Vol(Σ)}

(here, Vol(Σ) denotes the volume of Σ). Similarly, we can extend the perturbation over Y to a self-dual two-form perturbation of the equations over X_{0} (and, consequently, X(T) to all T > 2). Denote the corresponding moduli spaces by M_{N,u}(J), M_{N,u}(C), M_{X_{0},u}(J), M_{X_{0},u}(C), and M_{X(T),u}. Strictly speaking, we still have to show that these perturbed moduli spaces M_{X(T),u}(𝔰) can be used to calculate the Seiberg-Witten invariant in either chamber. This is clear because we can always choose a compactly-supported perturbing two-form η_{0} whose integral against ω_{g} dominates the integral of ω_{g} against iuf(dη)^{+}. The key point is that the latter integral is finite, since ω_{g} decays exponentially (see [1]).

We now have the following analogue of Proposition 5.5:

Proposition 8.2. Suppose ⟨c_{1}(𝔰), [Σ]⟩ = n, and let u be a real number with 0 < u < 2. Then the perturbed moduli space M_{N,u}(J) contains only reducibles. Moreover, the space of reducibles is smoothly identified with the Jacobian torus J (i.e., the kernel and the cokernel of the Dirac operator coupled to any reducible vanishes). Furthermore, M_{N,u}(C) is empty.

Proof. We begin by proving M_{N,u}(C) is empty. Note that C consists entirely of α-solutions, hence so must any section in M_{N,u}(C). Thus, a solution (A, α) ∈ M_{N,u}(C) induces a non-zero element in H^{0}(Ê) with

⟨c_{1}(Ê), [Σ_{−}]⟩ = n + g − 1 and ⟨c_{1}(Ê), [Σ_{+}]⟩ = g − 1.

But H^{∗}(R, Ê) ≡ 0, according to Proposition 7.5. The same argument, now appealing to Theorem 7.4, shows that M_{N,u}^{∗}(J) is empty, and that J is smooth. q.e.d.

Proposition 8.3. Suppose that

⟨c_{1}(𝔰), [Σ]⟩ = −n,

and let u be any real number with 0 < u < 2. Then according to Theorem 8.1, N_{u}(𝔰|_{Y}) has two components, J and C, where C is diffeomorphic to Sym^{g−1}(Σ). Furthermore, the expected dimensions of the moduli spaces over N and X_{0} are given by:

(38)  e-dimM_{N,u}(J) = 2g − 1
(39)  e-dimM_{N,u}(C) = 2g − 2
(40)  e-dimM_{X_{0},u}(J) = 2d
(41)  e-dimM_{X_{0},u}(C) = 2d.

Moreover, M_{N,u}^{∗}(J), M_{N,u}(C), M_{X_{0},u}(J), and M_{X_{0},u}(C) are transversally cut out by the Seiberg-Witten equations (in particular, they are manifolds of the expected dimension). Furthermore, the boundary map

ρ: M_{N,u}(C) ⟶ C

is an orientation-preserving diffeomorphism onto its image.

Proof. The proofs of Propositions 7.9 and 7.10 apply directly in this perturbed context. q.e.d.

9. Cohomology

The Seiberg-Witten invariant is obtained from pairings of certain canonical cohomology classes on the Seiberg-Witten moduli space. These cohomology classes are inherited from the configuration spaces in which the moduli spaces live. In this section, we recall the definitions of these classes and discuss natural geometric representatives for them. (See Chapter 5 of [4] for the corresponding discussion of cohomology relevant to Donaldson invariants.)

Let X be a Riemannian four-manifold with a Spin_{ℂ} structure 𝔰 specified by the pair of Hermitian ℂ^{2} bundles W^{+} and W^{−}, and the Clifford action

ρ: T X ⊗ W^{+} ⟶ W^{−}.

The Seiberg-Witten pre-configuration space is the space

C(W^{+}) = A(W^{+}) × Γ(X; W^{+}) ≅ Ω^{1}(W; ℝ) × Γ(X; W^{+}),

where A(W^{+}) denotes the space of connections compatible with some fixed connection ∇ on T X, and the isomorphism above is induced by comparing any connection A against some fixed connection A_{0}. The irreducible pre-configuration space C^{∗}(W^{+}) is the subset of C(W^{+}) consisting of pairs (A, Φ), where Φ ≢ 0. Now, C^{∗}(W^{+}) is weakly contractible, and the space Map(X; S^{1}) acts freely on it, so the irreducible configuration space, which is

B^{∗}(W^{+}) = C^{∗}(W^{+})/Map(X; S^{1})

is weakly homotopy equivalent to the classifying space of Map(X; S^{1}). Now,

Map(X; S^{1}) ∼ Map(X; S^{1})_{e} × π_{0}(Map(X; S^{1})) ∼ S^{1} × H^{1}(X; ℤ);

so

B^{∗}(W^{+}) ∼ ℂℙ^{∞} × \frac{H^{1}(X; ℝ)}{H^{1}(X; ℤ)},

and

H^{∗}(B^{∗}(W^{+}); ℤ) ≅ ℤ[U] ⊗ Λ^{∗}H^{1}(X; ℤ),

where U is a generator with grading two. More invariantly, we define

𝔸(X) = ℤ[H_{0}(X; ℤ)] ⊗ Λ^{∗}H_{1}(X; ℤ),

graded by declaring H_{0}(X; ℤ) to have grading two and H_{1}(X; ℤ) to have grading one. Then, we have seen that

H^{∗}(B^{∗}(W^{+}); ℤ) ≅ 𝔸(X).

We describe two functorial mechanisms for constructing generators in H^{∗}(B^{∗}(W^{+}); ℤ). Over the space X × B^{∗}(W^{+}), there is a universal line bundle L = X × S^{1} × C^{∗}(W^{+})/Map(X, S^{1}), where the action is defined by

u(x, ζ, A, Φ) = (x, u(x)ζ, A + u^{−1}du, uΦ).

Using this class we can define a “μ-map”

μ: (H_{0} ⊕ H_{1})(X; ℤ) ⟶ H^{∗}(B^{∗}(W^{+})),

which sends a homology class of degree i to a cohomology class of degree 2 − i, by

μ(x) = c_{1}(L)/x;

i.e., μ(x) is the cohomology class on B^{∗}(W^{+}) with the property that for any homology class c ∈ H_{∗}(B^{∗}(W^{+})),

⟨μ(x), c⟩ = ⟨c_{1}(L), x × c⟩.

We describe another convenient mechanism for constructing onedimensional cohomology in C(W^{+}) as follows. A closed curve x: S^{1} ⟶ X induces a map

Hol_{x}: B^{∗}(W^{+}) ⟶ S^{1}

which is defined to be the holonomy of the connection A around the curve x. The pull-back of the volume form dθ of S^{1} by this map gives rise to a one-dimensional cohomology class Hol_{x}^{∗}(dθ) associated to x, which we call the holonomy class around x.

Proposition 9.1. The cohomology groups of the configuration space B^{∗}(W^{+}) are generated by the image of the μ-map. Moreover, given x ∈ H_{1}(X; ℤ), μ(x) is the holonomy class around x, Hol_{x}^{∗}(dθ)|_{B^{∗}(W^{+})}.

Remark 9.2. Note that Hol_{x}^{∗}(dθ) is naturally defined over the entire configuration space

B(W^{+}) = C(W^{+})/Map(X, S^{1}) ∼ \frac{H^{1}(X; ℝ)}{H^{1}(X; ℤ)}.

Proof. We begin by proving the second claim. Note that L comes with a tautological connection along the X factor, with the property that for any path β: S^{1} ⟶ X and connection A ∈ C(W^{+}),

Hol_{β×A}(L) = Hol_{β}(A). (42)

C(W^{+}) Now, fix a path in X,

β: S^{1} ⟶ X.

We need to show that for all paths in the configuration space

α: S^{1} ⟶ B(W^{+})(𝔰),

we have that

⟨c_{1}(L), α × β⟩ = deg(Hol_{β} ∘ α: S^{1} ⟶ S^{1}).

This follows from the fact that for a line bundle L over the the torus S^{1} × S^{1}, the first Chern number is the degree of the map from S^{1} × S^{1} defined by

x ↦ Hol_{x×S^{1}}L

(a map which makes sense only after one puts a connection on L, but the degree is independent of this connection, so we left it out of the notation), together with the universal property of Equation (42). Thus, we have identified μ on any one-dimensional homology class.

The rest of the proposition is established, once we see that for a point x ∈ X, μ[x] generates H^{2} of the configuration space. But this follows easily from the fact that Map(X, S^{1})_{e} acts freely on the space of irreducible configurations. q.e.d.

With this concrete understanding of the μ-classes, we turn to a discussion of submanifold representatives for them.

Given a point x ∈ X, let L_{x} denote the line bundle associated to the base fibration of X; i.e., it is the restriction of the universal line bundle L to the slice B^{∗}(W^{+}) ≅ {x} × B^{∗}(W^{+}) ⊂ X × B^{∗}(W^{+}). Given a point in the fiber Ψ(x) ∈ W_{x}^{+}, we can construct a canonical section over B(X, W^{+}) by

[A, Φ] ↦ [A, Φ, ⟨Φ(x), Ψ(x)⟩].

The zero set of this section in B(X, W^{+}) is a codimension-two submanifold representing μ[x]. The restriction of this section to a moduli space M_{X}(𝔰) ⊂ B(X, 𝔰) is not, in general, transverse. However, by mollifying the construction appropriately, we can find a section which is generic over the moduli space, and hence obtain a divisor V (x) representing μ[x], as follows.

Definition 9.3. Fix a ball B ⊂ X around x and a non-vanishing section Ψ of W^{+}|_{B}. Given a self-dual two-form λ which is compactly supported over B, the λ-mollified section is the section of L_{x} defined by

[A, Φ] ↦ [A, Φ, ∫_{B}⟨λ · Φ, Ψ⟩].

Lemma 9.4. There are L^{2} sections λ compactly supported in B so that the λ-mollified section of L_{x}, restricted to the moduli space M_{X}, vanishes transversally.

Proof. Fix a compactly-supported cut-off function β in B, and consider the section

[A, Φ] × λ ↦ [A, Φ, ∫_{B}⟨λ · Φ, Ψ⟩β]

of π_{2}^{∗}(L_{x}), thought of as a line bundle over Ω^{+}(B) × M(X, 𝔰), giving Ω^{+}(B) the L^{2} topology. This transversality follows from the fact that, for any [A, Φ] ∈ M_{X}, as we vary λ, the integral ∫_{B}⟨λ·Φ, Ψ⟩β can take on any complex value. This, in turn, follows from the unique continuation theorem for elliptic differential operators, which guarantees that the section Φ cannot vanish identically over B. q.e.d.

Remark 9.5. In effect, the above lemma tells us how to construct a divisor representative V (x) for μ[x] when [x] ∈ H_{0}(X); this divisor is represented by the zero-set of the λ-mollified section of L_{x}. Finding codimension-one representatives for μ[γ], where γ ∈ H_{1}(X) is even easier: one need only find a regular value θ for the map

Hol_{γ}: M_{X}(𝔰) → S^{1}.

Then, Hol_{γ}^{−1}(θ) is the submanifold V (γ) representing μ[γ] over M_{X}(𝔰).

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Institute for Advanced Study, New Jersey Princeton University