Gauge theory for embedded surfaces, II

More documents

Recommendations

Info

$Math 126 Solution Set 2$

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

62 P. B. Kronheimer and T. S. Mrowka Proposition 9.13. For all [A] ∈ M α 0,l ′, the obstruction space H2 A in the antiself-dual deformation complex vanishes; so the moduli space is regular. Proof. According to Proposition 8.23, the obstruction space is isomorphic to H 2 (X,T ). The short exact sequence defining T gives a long exact sequence ···→H 1 (X,End0 E) → H 1 (Σ,L −2 ) → H 2 (W, T ) → H 2 (X,End0 E) →···. Because E is pulled back from Σ (9.6) with spheres as fibres, its cohomology groups on X are just the pull-backs of its cohomology groups on Σ (see the proof of (9.6)). In particular, H2 (X,End0 E) vanishes and the map H1 (X,End0 E) → H1 (Σ,L−2 )isonto.SoH2 (X,T ) vanishes. ⊓⊔ We can now consider two maps from the compactified moduli space ¯ M α 0,l , with l = 1 4 (2g −2), to Bo Σ ′, (the space of connections on Σ′ which are irreducible or reducible of degree 0). On the one hand there is the usual restriction map ¯ρ : ¯ M α 0,l −→ B o Σ ′ , considered previously at (7.7) and used to define the invariant Γ at (7.8). On the other hand, we can identify the moduli space as a space of parabolic bundles and so restrict the holomorphic bundles E to Σ ′ ; by (9.12) this forgetful map maps the moduli space ¯ M α 0,l to the moduli space of semi-stable bundles, Mss (Σ ′ ). This in turn can be identified with the moduli space of flat SU(2) connections over Σ ′ ,sogivingamap σ: ¯ M α 0,l −→ M ss = Mflat ↩→B o Σ ′. The nature of the correspondence between M ss and Mflat means that these two maps are homotopic. Indeed, because the restriction of each bundle to Σ ′ is semistable, the image of the restriction map ¯ρ is contained in B ss , the semi-stable set in B o , and the heat-flow defines a retraction of B ss onto Mflat: this heat-flow realizes the isomorphism between Mflat and M ss (see [DK] for example), and defines the homotopy from ¯ρ to σ. Recall now that Γ is the degree of ¯ρ[ ¯ M α 0,l ], measured using the fundamental 2-dimensional class µ(Σ ′ ), while ∆ was defined to be the degree of Mflat. The latter is non-zero because, for example, it is related to the degree of a projective embedding of Mflat. The homotopy between ¯ρ and σ means that Γ = p∆, where pis the degree of the map σ ′ : ¯ M α 0,l −→ M ss (Σ ′ ) (9.14) given by restricting the complex bundles. By (9.13), the lower strata in the compactification of M α 0,l have dimension strictly less than 6g − 6, so we can choose a point of Mss which is not in their image. The inverse image of such a , the smooth open stratum, and we can point is therefore a set of points in M α 0,l assume these are regular points of σ ′ . This set of points is non-empty because, by Proposition 9.1, the map σ ′ is surjective. Also, all points contribute +1 to the degree because, by Proposition 8.23, the map σ ′ is holomorphic on the complex
Gauge theory for embedded surfaces, II 63 manifold M α 0,l . The degree p is therefore strictly positive, and so Γ is non-zero. This completes the proof of Proposition 9.2, and with it Corollary 9.3 (also known as Theorem 6.1), and finally Theorem 1.1 also. ⊓⊔ Remark. As it stands, the argument here rests on the fact that the ‘standard’ defined in Appendix 2 coincides with its standard orienta- orientation of M α 0,l tion as a complex manifold. Because these orientations are rather elusive, it is reassuring to observe that we can avoid this route. The point is that M α 0,l is connected (this is easy to demonstrate from its description in terms of stable bundles), so even if the two orientations were not the same, they would simply differ by an overall change of sign: Proposition 9.2 would not be affected. (iii) Calculating p0,l. For completeness, we now refine Corollary 9.3 to show that p0,l =2 g . The proof of Proposition 9.2 has shown that p0,l is the degree of the map σ ′ in (9.14). For a stable bundle E on Σ ′ ,letS(E) denote its inverse image under σ ′ . From our identification of ¯ M α 0,l ,weseethat Our aim is therefore to prove S(E)={L|L⊂E and 0 < − deg L ≤ l }. Proposition 9.15. For generic stable bundles E, thesetS(E)has exactly 2 g elements. The origin of the number 2 g is in the proof of Proposition 9.1, but in order to carry out the argument we need some transversality results first. Lemma 9.16. For generic stable bundles E on Σ, the following conditions hold for all L in S(E): (a) degL=−l ; (b) dim H0 (L−1 ⊗E)=1; (c) the natural map H1 (O) → Hom H0 (L−1 ⊗E),H1 (L−1⊗E) is onto. Proof. The content of (a) isthatEdoes not lie in the image of the lower strata in the compactified moduli space ¯ M α 0,l under σ′ ; we have already used the fact that the lower strata have dimension less than 6g − 6, from which this part of the lemma follows. Assuming now that E is chosen so that (a) holds, consider (b). If the dimension here is 2 or more, then there are two independent copies of L as subbundles of E. By projecting the first copy L1 onto the quotient E/L2 ∼ = L−1 , one obtains a non-zero section of L−2 .SoL−2lies in the theta divisor Θ in the Jacobian Jg−1 (since L has degree −l = − 1(2g − 2)). We now count parameters 4 to see the dimension of the family of bundles E which can be obtained as extensions L→E→L−1 ,withL−2∈Θ. Counting complex parameters, we have
Page 1 and 2:
Gauge theory for embedded surfaces,
Page 3 and 4:
Page 5 and 6:
Page 7 and 8:
Page 9 and 10:
Page 11 and 12: Gauge theory for embedded surfaces,
Page 61: Gauge theory for embedded surfaces,
show all

Gauge theory for embedded surfaces, II

Create successful ePaper yourself

Delete template?

Save as template?