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By Bobenko A.

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When does the integral γP Q ω depend on the points P, Q and not on the integration path? 2 A differential ω is closed, dω = 0, if and only if for any two homological paths γ and γ˜ ω= ω γ γ ˜ holds. Proof The difference of two homological curves γ − γ˜ is a boundary for some D. Applying (53) we have γ ω= ω= ω− γ ˜ dω = 0. D ∂D The differential ω is closed since D is arbitrary. 3 Let ω be a closed differential, Fg be a simply connected model of Riemann surface of genus g (see Section 3) and P0 be some point in Fg .

D˜z g−1 hg d˜ z g−1 1  ˜ h1 ... d ˜ g(g−1)/2  h . dz  dz 1 det  = .. d˜ z  . dg−1 ˜ ... g−1 h1 dz = dz d˜ z g(g−1)/2 ˜g  h d ˜  dz hg   ..  . g−1 d ˜ g−1 hg dz dz dz . ∆ h1 , . . , hg d˜ z d˜ z (98) On the other hand algebraic properties of determinant imply also ∆[f h1 , . . , f hg ] = f g ∆[h1 , . . , hg ], where f is an arbitrary holomorphic function. Combined with (98) for f = ˜ = ∆ dz d˜ z (99) dz d˜ z this yields g(g+1)/2 ∆. Since the differentials ωi are linearly independent ∆ ≡ 0.

The map L(D2 ) → L(D1 ) defined by the multiplication L(D2 ) f −→ hf ∈ L(D1 ) is an isomorphism, which proves l(D2 ) = l(D1 ). Let Ω0 be a non-zero Abelian differential and C = (Ω0 ) be its divisor. The map H(D) → L(D − C) defined by Ω ∈ L(D − C) H(D) Ω −→ Ω0 is an isomorphism of linear spaces, which proves i(D) = l(D − C). 4 (Riemann-Roch). Let R be a compact Riemann surface of genus g and D a divisor on R. Then l(−D) = deg D − g + 1 + i(D). (88) We prove the Riemann-Roch theorem in several steps.

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Compact Riemann surfaces by Bobenko A.


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