Gauge theory for embedded surfaces, II

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$Math 126 Solution Set 2$

$Math S?101. Worksheet 1. Axioms of Integer Arithmetic$

42 P. B. Kronheimer and T. S. Mrowka will change as we cross from on chamber to the next. The walls between the chambers move as α varies, so qk,l will depend on α. Our argument relies on the fact that, when α becomes small and the metric on X acquires a long neck, we are placed firmly in one particular chamber. It is for the value of qk,l in this chamber that we shall prove a counterpart to Theorem 5.10. We shall pursue the invariant qk,l only in the case k =0andl= 1 4 (2g − 2), and in this case we christen the invariant Γ so as to recall the fact that it is chamber-dependent [D2]. Throughout this section, (X,Σ) will be the ruled surface as above, with genus g and degree n. WewriteΣ ′ for the section at infinity in the ruled surface (the section we add to form the compactification), and F for the typical 2-sphere fibre. We note that H2(X; Z) is generated by [Σ] and [F ] with the intersection form Σ·Σ = n Σ·F =1 F·F =0. (7.1) The homology class of Σ ′ is [Σ ′ ]=[Σ]−n[F]. Before beginning, it is worthwhile mentioning that there is another possible route to understanding p0,l. We have chosen to pass to the compact ruled surface X because this has the structure of an algebraic surface, and on such a manifold, one can compare anti-self-dual connections with holomorphic bundles via Donaldson’s solution of the hermitian Yang-Mills equations [D1]. The alternative line is to work directly with the non-compact manifold W o , equipped with a complete metric which is conformally Kähler, and to solve the hermitian Yang-Mills equations in this non-compact setting to obtain an interpretation of the moduli space M α 0,l (W o ,Σ) in terms of holomorphic bundles. Thus one should obtain, on the level of the moduli spaces, a correspondence which we shall arrive at only on the level of the invariants. It seems that the necessary ingredients of such an approach are contained in the recent work of Guo [Gu]. (ii) Invariants for the ruled surface. We shall consider the moduli space M α 0,l (X,Σ) for the ruled surface, and the lower moduli spaces M α 0,l ′(X,Σ) forl′ ≤l. Eventually we shall be concerned only with small values of α. Lemma 7.2. The moduli space M α 0,l ′(X,Σ) contains no flat connections, and is empty unless l ′ is strictly positive. Proof. The manifold X\Σ contracts onto Σ ′ , so the holonomy of any flat connection is trivial on the loop linking Σ. There are therefore no flat connections with α ∈ (0, 1 2 ). The other clause follows from the Chern-Weil formula, as in (4.12). ⊓⊔ Since b + (X) = 1 there is, up to scale, a unique self-dual harmonic 2-form for each metric on X. We can therefore write the space of self-dual forms as H + =span(Σ+tF ) (7.3)
Gauge theory for embedded surfaces, II 43 for some unique t. Since the self-dual forms have positive square, a necessary constraint on t is that t>−n/2. Lemma 7.4. For l ′ ≤ l, the moduli space M α 0,l ′ contains reducible solutions if and only if t = l′ − αn α . Proof. By [KrM] (section 5(iii)), a solution exists only if there is a line bundle L with c1(L) 2 =0 c1(L)[Σ]=−l ′ c1(L)+α[Σ]∈H − If we write c1(L) asλ[Σ]+µ[F] and solve the first two constraints above, we find that either λ =0andµ=−l ′ ,orλ=−2l ′ /n and µ = l ′ . The latter option is ruled out for l ′ ≤ l since λ must be an integer. So c1(L)=−l ′ [F]. The third condition now says that α[Σ] − l ′ [F ] must be orthogonal to H + . From (7.1) and (7.3) we obtain αn+αt−l ′ = 0, which gives the condition in the lemma. ⊓⊔ Lemma 7.5. Let α0 be given as in the proof of (4.13). Then for α
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Gauge theory for embedded surfaces, II

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