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G ) of holomorphic differentials. In this case the second condition of the theorem reads N Pi ω ˜≡0 (mod periods of ω ˜ ). 2 The Riemann-Roch theorem Let D∞ be a positive divisor on R. A natural problem is to describe the vector space of meromorphic functions with poles at D∞ only. More generally, let D be a divisor on R. Let us consider the vector space L(D) = {f meromorphic functions on R | (f ) ≥ D or f ≡ 0}. Let us split D = D0 − D∞ into negative and positive parts D0 = ni Pi , D∞ = mk Qk , where both D0 and D∞ are positive.

5 MEROMORPHIC FUNCTIONS ON COMPACT RIEMANN SURFACES 58 Let us suppose that deg D ≥ g. Then l(−D1 ) ≥ deg D2 + 1 and there exists a function in L(−D1 ) with the zero divisor ≥ D2 . This yields l(−D) > 0, which contradicts our assumption. We have proven that deg D ≤ g − 1. In the same way using i(D) = l(D − C) = 0 one gets deg (C − D) ≤ g − 1. 9 this implies deg D ≥ g − 1, and finally deg D = g − 1, which completes the proof of the Riemann-Roch theorem. 5 A positive divisor D of degree deg D = g is called special if i(D) > 0, i.

G ) is the canonical basis of holomorphic differentials and P0 ∈ R, is called the Abel map. 4 Harmonic differentials and proof of existence theorems As we mentioned in Section 1 angles between tangent vectors are well defined on Riemann surfaces. In particular one can introduce rotation of tangent spaces on angle π/2. The induced transformation of the differentials13 is called the conjugation operator ω = f dz + g d¯ z → ∗ω = −if dz + ig d¯ z. It is a map onto, since clearly ∗∗ = −1. In terms of the conjugation operator the differentials of type (1, 0) (resp.