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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 .

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. 14 (Existence Theorem) A holomorphic line bundle on a compact Riemann surface has nontrivial meromorphic cross-sections. 13) χ(λζpn ) − c(λζpn ) = χ(λ) − c(λ) for any integer n, or explicitly dim H 0 (M, O(λζpn )) − dim H 1 (M, O(λζpn )) − c(λ) − n = dim H 0 (M, O(λ)) − dim H 1 (M, O(λ)) − c(λ) since c(λζpn ) = c(λ) + n; and hence dim H 0 (M, O(λζpn )) = n + dim H 0 (M, O(λ)) − dim H 1 (M, O(λ)) + dim H 1 (M, O(λζpn )) ≥ n + dim H 0 (M, O(λ)) − dim H 1 (M, O(λ)).

Fn ∈ Γ(M, O(λζp )) where n = γ(λζp ). If fi (p) = 0 for all of these cross-sections then the mapping ×h is surjective and γ(λ) = γ(λζp ). If for instance f1 (p) = 0 then the mapping ×h is not surjective, so γ(λ) ≤ γ(λζp ) − 1; the differences gi (z) = fi (z) − fi (p)/f1 (p) f1 (z) for 2 ≤ i ≤ n are n − 1 linearly independent holomorphic cross-sections of the bundle λζp that vanish at the point p so are the images under the injective homomorphism ×h of n − 1 linearly independent holomorphic cross-sections of λ, and consequently γ(λ) ≥ n − 1 = γ(λζp ) − 1.

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