Gauge theory for embedded surfaces, II

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$Math 126 Solution Set 2$

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

60 P. B. Kronheimer and T. S. Mrowka comes down to classifying the stable parabolic bundles (E, L) ofweightαon (X,Σ), for the Kähler class [ω]. From now on only the cohomology class of ω is material. Since b + (X) =1, the space H + is spanned by ω, and as in (7.3) we therefore right [ω]=λ([Σ]+t[F]), (9.4) for some t ∈ R. Since the integrals of ω on the complex curves F and Σ ′ must both be positive, λ and t are greater than zero. Since α is fixed, we shall drop the distinction between quasi-parabolic and parabolic structures: as explained in section 8(i), given a map j : F→Efrom an ordinary line bundle F, we define the α-degree of F by the formula (8.2), and define E to be α-stable if this is always negative. The hypothesis that α is less than α1 is used only in the form of its corollary: (n + t)α
Gauge theory for embedded surfaces, II 61 Now a cannot be negative, for if it were then (9.10) would give −na − 2b ≥ 0, and using this to eliminate b from (9.7) would then give ( 1 2n + t)a ≥ 0, which is impossible if a
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Gauge theory for embedded surfaces, II

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