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Example text

2 Abelian differentials of the first, second and third kind Let now R be a Riemann surface. The transition functions (49) are holomorphic and one can define more special differentials on R. 2 A differential ω on a Riemann surface R is called holomorphic (or an Abelian differential of the first kind) if in any local chart it is represented as ω = h(z)dz where h(z) is holomorphic. The differential ω ¯ is called anti-holomorphic. Holomorphic and anti-holomorphic differentials are closed. Holomorphic differentials form a complex vector space, which is denoted by H 1 (R, C).

It can be decomposed into its exact dfγ and harmonic hγ components αγ = dfγ + hγ . Note that both parts are automatically smooth. The harmonic differential hγ has the same periods as the original differential αγ . Chosing different cylces from a1 , b1 , . . , ag , bg as γ one constructs 2g linearly independent harmonic differentials. For the dimension we obtain dim H ≥ 2g. (82) Consider again holomorphic and antiholomorphic differentials and denote their spaces ¯ respectively. These spaces are obviously orthogonal H ⊥ H.

Let Pk be one of the Weierstrass points with Λ(Pk ) = ∞, Λ(Pk ) = ∞ (one can always find such a point from 2g + 2 Weierstrass points). The existence of the functions 1 1 , ˜ ˜ k) Λ(P ) − Λ(Pk ) Λ(P ) − Λ(P ˜1 + Q ˜ 2 are equivalent shows that the divisors Q1 + Q2 and Q ˜1 + Q ˜ 2. Q1 + Q2 ∼ 2Pk ∼ Q ˜1 − Q ˜ 2, There exists a meromorphic function ξ with the divisor (ξ) = Q1 + Q2 − Q ˜1 − Q ˜ 2) establishing the isomorphism of L(−Q1 − Q2 ) and L(−Q ˜1 − Q ˜ 2 ). ξL(−Q1 − Q2 ) = L(−Q ˜ form the basises of L(−Q1 − Q2 ) and L(−Q ˜1 − Q ˜ 2 ) respectively Since {1, Λ} and {1, Λ} we get ˜ = αξΛ + βξ1 Λ 1 = γξΛ + δξ1, and finally eliminating ξ ˜ = αΛ + β .

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