Gauge theory for embedded surfaces, II

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$Math 126 Solution Set 2$

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

56 P. B. Kronheimer and T. S. Mrowka Theorem 8.21 [SW]. Let ¯ X be,asaboveacomplexsurfacewithaKähler form ¯ω, having a cone-like singularity along Σ with cone-angle 2π/ν. Then there is a one-to-one correspondence between: (i) the set of irreducible connections in the moduli space M α k,l ( ¯ X,Σ) of α-twisted connections, anti-self-dual with respect to the cone-like metric determined by ¯ω, with holonomy parameter α = a/ν; and (ii) the set of stable parabolic SL(2, C) bundles ( Ē, ¯ L,α) on ¯ X,withthesame weight α, satisfying c2( Ē)=k and c1( ¯ L)=−l. Proof. For weight α = a/ν, the moduli space M α k,l coincides with the moduli space of anti-self-dual orbifold connections on X, a fact we have exploited before. Next, because we can treat the hermitian Yang-Mills equations in an orbifold setting, the irreducible orbifold anti-self-dual connections on X are in one-to-one correspondence with the orbifold stable bundles. For example, one can follow the proof for Kähler surfaces given in [DK]; this is carried out in [W]. Finally, by (8.20), orbifold stable bundles on X correspond to parabolic stable bundles on ¯ X. It remains to check that we have the correct definition of k and l, which means showing that β and ¯ L have the same topological type as the bundles Ē and ¯ L which went into the definition of the space of connections Aα k,l . This slightly awkward exercise is left to the reader. ⊓⊔ To complete the picture of the correspondence between twisted connections and parabolic bundles, we need to discuss their deformation theory briefly. Given a parabolic bundle (Ē, ¯L,α), let End oĒ denote the sheaf of trace-free endomorphisms of the bundle Ē on ¯ X, and let T denote the subsheaf consisting of those endomorphisms which preserve the line bundle ¯ L on Σ; thusTappears as the kernel in the short exact sequence T−→End o Ē−→ ¯ L −2 , (8.22) in which the third term arises as Hom( ¯ L, Ē/ ¯ L). This is the sheaf appropriate to the description of the deformation theory of parabolic bundles, as the next proposition makes clear. Proposition 8.23. In the situation of Theorem 8.21, the irreducible part of the moduli space M α k,l has the structure of a complex analytic variety. If [A] ∈ M α k,l is an irreducible anti-self-dual connection, then the cohomology groups H1 A and H2 A in its deformation complex are isomorphic to the sheaf cohomology groups H1 ( ¯ X,T ) and H2 ( ¯ X,T ) for the corresponding parabolic object, and as a complex-analytic space, M α k,l is modelled on the zero-set of an analytic map ψ : H1 ( ¯ X,T ) → H2 ( ¯ X,T ) in the neighbourhood of zero. Under the correspondence described in (8.21), the family of parabolic bundles parametrized by ψ−1 (0) is a complex analytic family. Proof. Again, the orbifold is a stepping stone. If E is the orbifold holomorphic bundle corresponding to A, we can follow Kuranishi’s construction in the orb-
Gauge theory for embedded surfaces, II 57 ifold setting and construct a universal deformation of E parametrized by a space K which is the zero-set of a complex analytic map ψ from a neighbourhood of zero in H0,1 (X,End oE) toH0,2 (X,End oE). These orbifold Dolbeault groups are isomorphic to H 1 A and H2 A by the orbifold version of the Hodge theory, and under this isomorphism the Kuranishi families for the anti-self-dual and holomorphic objects coincide (see [DK] for a discussion). Since the ‘bar’ construction which takes orbifold holomorphic bundles to parabolic bundles can be applied to families, their is a universal complex-analytic deformation of the quasi-parabolic bundle ( Ē, ¯ L), parametrized by the same space K. Our main concern is to identify the orbifold Dolbeault groups H 0,i (X,End oE) with the sheaf cohomology groups H i ( ¯ X,T ). By the Dolbeault theorem on the orbifold, the groups H 0,i (X,End oE) are isomorphic to the Čech groups on the orbifold, Hi (X,End oE). Since the Čech groups are made from local invariant sections, they can be identified with the Čech groups of µ(End oE), a fact we have mentioned before. So it remains to show that µ(End oE)=T. Certainly µ(End oE) is a subsheaf of End o( Ē)⊗M,asisT; so to verify that they coincide it is sufficient to take a local case, where E = K⊕K−1and K ∼ = O (a) . Here µ(End oE)=µ(K 2 ⊕O⊕K −2 ) =µ O (2a) ⊕O⊕O (−2a) =O⊕O⊕O[−Σ] = ¯ K 2 ⊕O⊕( ¯ K) −2 [−Σ]. On the other hand End o Ē is ¯ K 2 ⊕O⊕ ¯ K −2 , and the subsheaf preserving ¯ L = ¯ K|Σ is the same as that above. ⊓⊔ 9. Parabolic bundles on the ruled surface (i) An index of a family. We begin with an important proposition concerning line subbundles of vector bundle on a curve. It is the origin of the non-vanishing of the invariants qk,l. Proposition 9.1. Let E→Σbe a stable SL(2, C) bundle over a curve Σ of odd genus. Then E has a line-subbundle L of degree −l ′ for some l ′ with 0
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