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By Gunning R.
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Example text
A cochain s ∈ C p (U, O(λ)) consists of sections sα0 ···αp ∈ Γ(Uα0 ∩ · · · ∩ Uαp , O(λ)) for each ordered set of p + 1 open subsets Uα0 , . . , Uαp of the covering U, where these sections are skew-symmetric in the indices α0 , . . , αp ; and the section sα0 ···αp can be identified with a holomorphic function fα0 ···αp in the intersection Uα0 ∩ · · · ∩ Uαp in terms of the fibre coordinate over Uαp when the intersection is viewed as a subset of the last coordinate neighborhood Uαp . This identification will be used consistently in the subsequent discussion; thus cochains in C p (U, O(λ)) will be identified without further comment as collections of holomorphic functions in the intersections Uα0 ∩ · · · ∩ Uαp ⊂ M in terms of the fibre coordinates in λ over Uαp .
The alternating sum of the dimensions of the spaces in an exact sequence of vector spaces such as this is zero, as can be seen most simply by decomposing the exact sequence into a collection of short exact sequences; consequently 0 = dim H 0 (M, O(λ)) − dim H 0 (M, O(λζp )) + 1 − dim H 1 (M, O(λ)) + dim H 1 (M, O(λζp )) = χ(λ) − χ(λζp ) + 1. Since c(λζp ) = c(λ) + 1 this yields the desired result, thereby concluding the proof. An immediate consequence of this lemma is the fundamental existence theorem for compact Riemann surfaces.
Furthermore if the distance from z to U exceeds δ/2 then g(z) = 0 since f (z + ζ) = 0 whenever |ζ| ≤ δ/2 while r(ζ) = 0 whenever ζ ≥ δ/2; thus the support of the function g is contained in V . For any fixed point z ∈ U let D be a disc centered at the origin in the plane of the complex variable ζ with radius sufficiently large that f (z + ζ) = 0 whenever ζ ∈ D, and let D be another disc centered at the origin in the plane of the complex variable ζ with radius . Then ∂f (z + ζ) dζ ∧ dζ i ∂f (z + ζ) dζ ∧ dζ r(ζ) = r(ζ) ∂z ζ 2π C ζ ∂ζ C i ∂f (z + ζ) r(ζ) = lim dζ ∧ dζ →0 2π D∼D ζ ∂ζ i ∂g(z) = ∂z 2π i →0 2π = lim = lim →0 −i 2π d f (z + ζ) D∼D ∂D r(ζ) dζ − ζ i f (z + ζ) dζ − ζ 2 f (z + ζ) D∼D ∂ r(ζ) dζ ∧ dζ ζ ∂ζ f (z + ζ)s(ζ)dζ ∧ dζ, C where Stokes’s Theorem is used to replace the integral over D ∼ D with the integral over the boundary of this region and the integrand vanishes on the boundary of D while r(ζ) = 1 for ζ ∈ ∂D for sufficiently small .